# Faculty Activities

*Abstract:*

In 1949, Fermi proposed a mechanism for the heating of particles in cosmic rays. He suggested that on average, charged particles gain energy from collisions with moving magnetic mirrors since they hit the mirrors more frequently with heads on collisions. Fermi, Ulam and their followers modeled this problem by studying the energy gain of particles moving in billiards with slowly moving boundaries. Until 2010 several examples of such oscillating billiards leading to power-law growth of the particles averaged energy were studied. In 2010 we constructed an oscillating billiard which produces exponential in time growth of the particles energy [1]. The novel mechanism which leads to such an exponential growth is robust and may be extended to arbitrary dimension. Moreover, the exponential rate of the energy gain may be predicted by utilizing adiabatic theory and probabilistic models [2,3]. The extension of these results to billiards with mixed phase space leads to the development of adiabatic theory for non-ergodic systems [4]. Finally, such accelerators lead to a faster energy gain in open systems, when particles are allowed to enter and exit them through a small hole [5]. The implications of this mechanism on transport in extended systems [6] and on equilibration of energy in closed systems like "springy billiards" will be discussed [7]. The latter application provides a key principle: to achieve ergodicity in slow-fast systems (in the adiabatic limit), the fast subsystems should NOT be ergodic [7].These are joint works, mainly with with K. Shah, V. Gelfreich and D. Turaev [1-5],[7] and [6] is with M. Pinkovezky and T. Gilbert ;

[1] K. Shah, D. Turaev and V. Rom-Kedar, Exponential energy growth in a Fermi accelerator, Phys. Rev. E 81, 056205, 2010.

[2] V. Gelfreich, V. Rom-Kedar, K. Shah, D. Turaev, Robust exponential accelerators, PRL 106, 074101, 2011.

[3] V.Gelfreich, V. Rom-Kedar and D. Turaev, "Fermi acceleration and adiabatic invariants for non-autonomous billiards", Chaos **22**, 033116 (2012); (*21 pages*).

[4] V. Gelfreich, V. Rom-Kedar, D. Turaev Oscillating mushrooms: adiabatic theory for a non-ergodic system , 2014, Journal of Physics A : Mathematical and Theoretical, Volume 47 (Number 39). Article number 395101 . ISSN 1751-8113

[5] Leaky Fermi Accelerators, K. Shah, V. Gelfreich, V. Rom-Kedar, D. Turaev, Phys. Rev. E **91**, 062920 (2015).

[6] Fermi acceleration in a dispersive medium yields exponential diffusion, M. Pinkovezky, T. Gilbert and V. Rom-Kedar, 2017, draft.

[7] Equilibration of energy in slow-fast systems, K. Shah, D. Turaev, V. Gelfreich, V. Rom-Kedar, PNAS Vol 114, no. 49, E10514, 2017

*Abstract:*

In this talk I will discuss a general method for obtaining sharp lower bounds for the constants associated with certain functional inequalities on weighted Riemannian manifolds, whose (generalized) Ricci curvature is bounded from below. Using this method we prove new sharp lower bounds for the Poincar\'{e} and log-Sobolev constants.The first major ingredient of the method is the localization theorem which has recently been developed by B. Klartag. The proof of this theorem is based on optimal transport techniques; it leads to a characterization of the constants associated with the pertinent functional inequalities as solutions to a mixed optimization problem over a set of functions and a set of measures, both of which are supported in R.The second major ingredient of the method is a reduction of the optimization problem to a subclass of measures, which are referred to as 'model-space' measures. This reduction is based on functional analytical arguments, in particular an important characterization of the extreme points of a subset of measures, and corollaries of the Krein-Milman theorem.The third ingredient of the method is an explicit solution to the reduced optimization problem over the subset of 'model-space' measures; this solution is approached by ad-hoc methods.In my talk I will discuss each of the ingredients of the method, with an emphasis on solving the problem of finding the sharp lower bound for the Poincar\'{e} constant.Our results show that within a certain range of the pertinent parameters (specifically 'the effective dimension N'), the characterization of the sharp lower bound for the Poincar\'{e} constant is of an utterly different nature from what has been known to this date.

*Abstract:*

We will discuss the field of definition of a rational function and in what ways it can change under iteration, in particular when the degree over the base field drops. We present two families of rational functions with the property above, and prove that in the special case of polynomials, only one of these families is possible. We also explain how this relates to Ritt's decomposition theorem on polynomials. Joint work with F. Veneziano.

*Abstract:*

A theorem by Gelander shows that the number of generators of a lattice in a semi-simple Lie group is bounded by its co-volume. We prove a generalization of this result to an arbitrary connected Lie group with the appropriate modifications. This is one aspect of the phenomenon where the volume of locally symmetric spaces determines their topological complexity. Joint work with Tsachik Gelander.

*Abstract:*

An observation by Marklof implies that the primitive rational points of denominator n along the stable horocycle orbits of large volume determined by n equidistribute within a proper submanifold of the unit tangent bundle to the modular surface. We examine the general behavior of primitive rational points along expanding horospheres and prove joint equidistribution in products of the torus and the unit tangent bundle to the modular surface using effective mixing for congruence quotients.

*Announcement:*

**ד"ר חן מאירי**

הפקולטה למתמטיקה

טכניון

**Dr. Chen Meiri**

The Faculty of Mathematics

Technion

**MATH CLUB 16.1.19**

**המשפט של הסה-מינקובסקי**

מהפתרון של מאטיאשביץ׳ לבעיה העשירית של הילברט נובע כי אין אלגורתים שבהינתן פולינום f עם מקדמים שלמים ו-n משתנים, מחליט האם למשוואה f=0 יש פתרון עם מספרים שלמים. לעומת זאת, עבור תבניות ריבועיות שלמות (כלומר פולינומים הומוגנים ממעלה 2 עם מקדמים שלמים) קיים אלגוריתם כזה: המשפט של הסה-מינקובסקי מראה שפתרון כזה קיים אם ורק אם קיים פתרון עם מספרים ממשיים ובנוסף לכל ראשוני p ולכל טבעי m קיים פתרון מודולו p^m. בהרצאה נוכיח את המשפט במקרה הפרטי בו התבנית הריבועית היא בעלת שלושה משתנים.

**Hasse-Minkowski Theorem**

It follows from Matiyasevich's solution to Hilbert's 10th problem that there does not exist an algorithm for deciding whether for an arbitrary integral polynomial f in n variables there exists an integral solution to the equation f=0. However, for integral quadratic forms, i.e. integral homogeneous polynomials of degree 2, such an algorithm does exist: the Hasse-Minkowski theorem states that such a solution exists if and only if there exists a real solution and for every prime p and natural number m there exists a solution modulo p^m. In this talk we will prove this theorem in the special case where the quadratic form has 3 variables.

**ההרצאה תהיה בעברית**

**The lecture will be in Hebrew**

*Abstract:*

We say that a family F of k-element sets is a j-junta if there is a set J of size j such that, for any set, its presence in F depends on its intersection with J only. Approximating arbitrary families by j-juntas with small j is a recent powerful technique in extremal set theory. The weak point of all known approximation by juntas results is that they work in the range n>Ck, where C is an extremely fast growing function of the input parameters. In this talk, we present a simple and essentially best possible junta approximation result for an important class of families, called shifted. As an application, we present some progress in the question of Aharoni and Howard on families with no cross-matching. Joint work with Peter Frankl.

*Abstract:*

The problem of computational super-resolution asks to recover fine features of a signal from inaccurate and bandlimited data, using an a-priori model as a regularization. I will describe several situations for which sharp bounds for stable reconstruction are known, depending on signal complexity, noise/uncertainty level, and available data bandwidth. I will also discuss optimal recovery algorithms, and some open questions.

*Abstract:*

Aldous' Spectral gap conjecture, proved in 2009 by Caputo, Liggett and Richthammer, states the following a priori very surprising fact: the spectral gap of a random walk on a finite graph is equal to the spectral gap of the interchange process on the same graph. This seminal result has a very natural interpretation in terms of Cayley graphs, which leads to natural conjectural generalizations. In joint works with Gadi Kozma and Ori Parzanchevski we study some of these possible generalizations, clarify the picture in the case of normal generating sets and reach a more refined, albeit bold, conjecture.

*Abstract:*

In the talk I will present the recent advances in the complexity of Nash equilibrium for the query complexity and the communication complexity model. In particular, I will discuss lower bounds for computing an approximate Nash equilibrium in these models.

*Abstract:*

One of the classical enumerative problems in algebraic geometry is that of counting of complex or real rational curves through a collection of points in a toric variety. We explain this counting procedure as a construction of certain cycles on moduli of rigid tropical curves. Cycles on these moduli turn out to be closely related to Lie algebras. In particular, counting of both complex and real curves is related to the quantum torus Lie algebra. More complicated counting invariants (the so-called Gromov-Whitten descendants) are similarly related to the super-Lie structure on the quantum torus. No preliminary knowledge of tropical geometry or the quantum torus algebra is expected.

*Abstract:*

We prove that a two-dimensional laminar flow between two plates (x_1,x_2)\in{\mathbb R}_+\times [-1,1] given by {\mathbf v}=(U,0) is linearly stable in the large Reynolds number limit, when |U^{\prime\prime}| \ll |U^\prime| (nearly Couette flow). We assume no-slip conditions on the plates and an arbitrary large (but fixed) period in the x_1 direction. Stronger results are obtained when the no-slip conditions on the plates are replaced by a fixed traction force condition. This is joint work with Bernard Helffer.

*Abstract:*

A celebrated result of C.L. Siegel from 1929 shows that the multiplicity of eigenvalues for the Laplace eigenfunctions on the unit disk is at most two. To show this, Siegel shows that positive zeros of Bessel functions are transcendental. We study the fourth order clamped plate problem, showing that the multiplicity of eigenvalues is at most by six. In particular, the multiplicity is uniformly bounded. Our method is based on Siegel-Shidlovskii theory and new recursion formulas.This is joint work with Yuri Lvovski.

*Abstract:*

In this talk I will introduce a procedure to produce interesting examples of non-positively curved cube complexes. The construction we suggest takes as input two finite simplicial complexes and gives as output a finite cube complex whose local geometry can be easily described. This local information can then be used to obtain global information, e.g. about cohomological dimension and hyperbolicity of the fundamental group of the cube complex. This is joint work with Robert Kropholler.

*Abstract:*

For the integer ring Z, it is known that every map f:Z\to Z such that the set of values of f(x+y)-f(x)-f(y) is finite is, up to a finite perturbation, a multiplication by a real scalar. What happens if we take the field of rationals, Q, instead of Z? and what are the analogue statements regarding Z[1/p] or a number field? We will discuss questions of classical algebra using the frame work of coarse spaces.

*Abstract:*

Representations of Toeplitz-Cuntz algebras were studied by Davdison, Katsoulis and Pitts via non-self-adjoint techniques, originating from work of Popescu on his non-commutative disk algebra. This is accomplished by working with the WOT closed algebra generated by operators corresponding to vertices and edges in the representation. These algebras are called free semigroup algebras, and provide non-self-adjoint invariants for representations of Toeplitz-Cuntz algebras.

The classication of Cuntz-Krieger representations of directed graphs up to unitary equivalence was used in producing wavelets on Cantor sets by Marcolli and Paolucci and in the study of semi-branching function systems by Bezuglyi and Jorgensen. With Davidson and B. Li we extended the theory of free semigroup algebras to arbitrary directed graphs, where free semigroupoid algebras provide new connections with graph theory.

In this talk I will present a characterization of those finite directed graphs that admit self-adjoint free semigroupoid algebras. We will make full circle with the theory of automata, as we will use a periodic version of the Road Coloring theorem due to Beal and Perrin, originally proved by Trahtman in the aperiodic case. This is based on joint work with Christopher Linden.

*Abstract:*

Abstract: By a theorem of Stanley, the distribution of descent number over all the shuffles of two permutations depends only on the descent numbers of these permutations. For a quantitative version of this result and its cyclic analogue, we use a new cyclic counterpart of Gessel's ring of quasi-symmetric functions, together with an unusual homomorphism and a mysterious binomial identity. No previous acquaintance assumed. Based on the recent preprint arXiv:1811.05440; joint work with Ira Gessel, Vic Reiner and Yuval Roichman.

*Abstract:*

A matrix is called totally nonnegative (TN) if all its minors are nonnegative, and totally positive (TP) if all its minors are positive. Multiplying a vector by a TN matrix does not increase the number of sign variations in the vector. In a largely forgotten paper, Schwarz (1970) considered matrices whose exponentials are TN or TP. He also analyzed the evolution of the number of sign changes in the vector solutions of the corresponding linear system.

In a seemingly different line of research, Smillie (1984), Smith (1991), and others analyzed the stability of nonlinear tridiagonal cooperative systems by using the number of sign variations in the derivative vector as an integer-valued Lyapunov function.

We provide a tutorial on these fascinating research topics and show that they are intimately related. This allows to derive generalizations of the results by Smillie (1984) and Smith (1991) while simplifying the proofs. This also opens the door to many new and interesting research directions.

Joint work with and Eduardo D. Sontag, Northeastern University.

The paper on which this talk is based can be accessed via the link: https://www.sciencedirect.com/science/article/pii/S000510981830548X

*Abstract:*

We propose an index for pairs of a unitary map and a clustering state on many-body quantum systems. We require the map to conserve an integer-valued charge and to leave the state invariant. This index is integer-valued and stable under perturbations. In general, the index measures the charge transport across a fiducial line. We show that it reduces to (i) an index of projections in the case of non-interacting fermions, (ii) the charge density for translational invariant systems, and (iii) the quantum Hall conductance in the two-dimensional setting without any additional symmetry. Example (ii) recovers the Lieb-Schultz-Mattis theorem, and (iii) provides a new and short proof of quantization of Hall conductivity in interacting many body systems.

*Abstract:*

On a closed Riemannian manifold, the Courant nodal domain theorem gives an upper boundon the number of nodal domains of n-th eigenfunction of the Laplacian. In contrast to that, there does not exist such bound on the number of isolated critical points of an eigenfunction. I will try to sketch a proof of the existence of a Riemannian metric on the 2-dimensional torus, whose Laplacian has infinitely many eigenfunctions, each of which hasinfinitely many isolated critical points. Based on a joint work with A. Logunov and M. Sodin.

*Abstract:*

Abstract: We introduce a novel approach addressing the global analysis of a difficult class of nonlinearly composite nonconvex optimization problems. This genuine nonlinear class captures many problems in modern disparate fields of applications. We develop an original general Lagrangian methodology relying on the idea of turning an arbitrary descent method into a multiplier method. We derive a generic Adaptive Lagrangian Based mUltiplier Method (ALBUM) for tackling the general nonconvex nonlinear composite model which encompasses fundamental Lagrangian methods. This paves the way for proving global convergence results to a critical point of the problem in the broad semialgebraic setting. The potential of our results is demonstrated through the study of two major Lagrangian schemes whose convergence was never analyzed in the proposed general setting: the proximal multiplier method and the proximal alternating direction of multipliers scheme. This is joint work with Jerome Bolte (Toulouse 1 Capitole University) and Marc Teboulle (Tel Aviv University).

*Abstract:*

In this talk I will survey several methods used in order to solve number theoretical questions over function fields. These methods involve Galois Theory, Characters, and some Random Matrix theory.

*Abstract:*

The following question is well-studied: Are almost commuting matrices necessarily close to commuting matrices? Both positive and negative answers were given, depending on the types of matrices considered and the metrics used to measure proximity. Variants of the question replace the matrices by different objects and/or replace the commutativity relation by another one. This suggests the following framework:

Fix a family \calG of pairs (G,d), where G is a group and d is a bi-invariant metric on G. For example, one may take \calG to be the family of finite symmetric groups endowed with the normalized Hamming metrics, or take it to be the family of unitary groups endowed with your favorite bi-invariant metrics on the groups U(n). Fix a word w over S±, where S is a finite set of formal variables. We say that w is \calG-stable if for every \epsilon>0 there is \delta>0 such that for every (G,d) in \calG and f:S-->G, if d(f(w),1_G) <= \delta, then there is f':S-->G such that f' is \epsilon-close to f and f'(w)=1_G. The stability of a set of words, representing simultaneous equations, is defined similarly.

It turns out that the \calG-stability of a set E of words depends only on the group \Gamma generated by S subject to the relations E. In other words, stability is a group property. A finitely generated group \Gamma is \calG-stable if one (hence all) of its presentations corresponds to a \calG-stable set of words.

We will give a survey of this topic, and then focus on recent results on stability of a finitely generated group \Gamma w.r.t. symmetric groups and unitary groups. These results relate stability to notions such as amenability, Invariant Random Subgroups, Property (T) and mapping class groups.

Based on joint works with Alex Lubotzky, Andreas Thom and Jonathan Mosheiff.

*Abstract:*

In this talk I will present a generalization of the Euclidean lattice point counting problem in the context of a certain type of homogeneous groups, the so-called Heisenberg groups. This problem was first considered in a paper by Garg, Nevo & Taylor, in which various upper bounds for the lattice point discrepancy were obtained with respect to a certain family of homogeneous norms. In the case of the first Heisenberg group, we will show that the upper bounds obtained by Garg, Nevo & Taylor are sharp when the norm under consideration is the Cygan-Koranyi norm, and I will present the main ideas needed for the proof. If time permits, I will present some recently obtained results regarding the higher dimensional case.

*Abstract:*

We say a graph $G$ has a Hamiltonian path if it has a path containing all vertices of $G$. For a graph $G$, let $\sigma_2(G)$ denote the minimum degree sum of two nonadjacent vertices of $G$; restrictions on $\sigma_2(G)$ are known as Ore-type conditions. It was shown by Mon\'ege that if a connected graph $G$ on $n$ vertices satisfies $\sigma_2(G) \geq {3 \over 2}n$, then $G$ has a Hamiltonian path or an induced subgraph isomorphic to $K_{1,4}$. In this talk, I will present the following analogue of the result by Mom\`ege. Given an integer $t\geq 5$, if a connected graph $G$ on $n$ vertices satisfies $\sigma_2(G)>{t-3 \over t-2}n$, then $G$ has either a Hamiltonian path or an induced subgraph isomorphic to $K_{1, t}$. This is joint work with Ilkyoo Choi.

*Abstract:*

I will discuss some models for the shape of liquid droplets on rough solid surfaces. These are elliptic free boundary problems with oscillatory coefficients. The framework of homogenization theory allows to study the large scale effects of small scale surface roughness, including interesting physical phenomena such as contact line pinning, hysteresis, and formation of facets. The talk is partly based on joint work with Charles Smart.

*Abstract:*

The probability that there is no point in a given region for a given point process is known as the hole probability. The infinite Ginibre ensemble is a determinant point process in the complex plane with kernel $e^{z\bar{w}}$ with respect the standard complex Gaussian measure. Alternatively, it can be thought as the limiting point process of the finite Ginibre ensembles, which is the eigenvalues of nxn Ginibre matrices. We compute the exact decay rate of the hole probabilities for finite and infinite Ginibre ensembles as size of the regions increase. We show that the precise decay rate of the hole probabilities is determined by a solution to a variational problem from potential theory.

*Abstract:*

T.B.A.

*Abstract:*

**Advisors**: Leshansky Alexander, Morozov Konstantin

**Abstract**: Controlled steering of chiral magnetic micro-/nano propellers by rotating

magnetic field is a promising technology for targeted delivery in various biomedical applications. In most of these applications magnetized chiral (helical) propellers are actuated by rotating magnetic field as they propel unidirectionally through a fluidic environment similar to a rotating corkscrew. It was shown recently that achiral magnetized objects, such as planar V-shaped motors, can propel as well. However, unidirectionality of the propulsion is not guaranteed due to the high symmetry of the problem. We study dynamics of planar V-shaped propellers actuated by a precessing magnetic fi eld provided by superimposing the dc magnetic fi eld onto the uniform rotating fi eld along the the field rotation axis. We demonstrate that the dc fi eld reduces the above symmetry and can, in fact, lead to unidirectional propulsion of planar and in-plane magnetized objects, that otherwise exhibit no net propulsion in a plane rotating field.

*Abstract:*

In this talk I will discuss applications of geometric invariant theory to the study of Hopf algebras. The question which will be considered is the classification of Hopf 2-cocycles on a given finite dimensional Hopf algebra. I will explain why this is in fact a geometric problem, and how geometric invariant theory can helpus here. I will give some examples arising from Bosonizations of nonabelian group algebras and dual group algebras, and present some new family of Hopf algebras arising from such cocycle deformations. If time permits, I will also explain the connection with the universal coefficients theorem, and how some of these invariants relate to surfaces.

*Abstract:*

Zeta-functions and L-functions play in important role in various areas of mathematics. In this talk, I will start by describing the Riemann zeta function, its role in the analytic number theory and how this function can be interpreted in representation theoretic terms. I will then introduce the Langlands L-function, which is a generalization of the Riemann Zeta Function for irreducible representation of (adelic) reductive groups and talk about their some interesting problems involving these L-functions.

*Abstract:*

Every k entries in a permutation can have one of k! different relative orders, called patterns. How many times does each pattern occur in a large random permutation of size n? The distribution of this k!-dimensional vector of pattern densities was studied by Janson, Nakamura, and Zeilberger (2015). Their analysis showed that some component of this vector is asymptotically multinormal of order 1/sqrt(n), while the orthogonal component is smaller. Using representations of the symmetric group, and the theory of U-statistics, we refine the analysis of this distribution. We show that it decomposes into k asymptotically uncorrelated components of different orders in n, that correspond to representations of Sk. Some combinations of pattern densities that arise in this decomposition have interpretations as practical nonparametric statistical tests.

*Abstract:*

A point scatterer, or the Laplacian perturbed with a delta potential, is a model for studying the transition between chaos and integrability in quantum systems. The eigenfunctions of this operator consist of the Laplace eigenfunctions which vanish at the scatterer, and a set of new, perturbed eigenfunctions. We discuss the mass distribution of the new eigenfunctions of a point scatterer on a flat torus, and present some of our recent results.

*Abstract:*

$$ \mathrm dX_t = - A X_t \mathrm dt + f(t,X_t)\mathrm dt + \mathrm dW_t , $$ where $A$ is a positive, linear operator, $f$ is a bounded Borel measurable function and $W$ a cylindrical Wiener process. If the components of $f$ decay to 0 in a faster than exponential way we establish path-by-path uniqueness for mild solutions of this SDE. This extends A. M. Davie’s result from $\mathbb R^d$ to Hilbert space-valued stochastic differential equations. In this talk we consider the so-called path-by-path approach where the above SDE is considered as a random integral equation with parameter $\omega\in\Omega$. We show that there is a set $\Omega'$ of measure 1 such that for every $\omega\in\Omega'$ the corresponding integral equation for this $\omega$ has atmost one solution. This notion of uniqueness (called path-by-path uniqueness) is much stronger than the usual pathwise uniqueness considered in the theory of SDEs.

*Abstract:*

The hydrogen atom system is one of the most thoroughly studied examples of a quantum mechanical system. It can be fully solved, and the main reason why is its (hidden) symmetry. In this talk I shall explain how the symmetries of the Schrödinger equation for the hydrogen atom, both visible and hidden, give rise to an example in the recently developed theory of algebraic families of Harish-Chandra modules. I will show how the algebraic structure of these symmetries completely determines the spectrum of the Schrödinger operator and sheds new light on the quantum nature of the system.

No prior knowledge on quantum mechanics or representation theory will be assumed.

*Abstract:*

In his influential disjointness paper, H. Furstenberg proved that weakly-mixing systems are disjoint from irrational rotations (and in general, Kronecker systems), a result that inspired much of the modern research in dynamics. Recently, A. Venkatesh managed to prove a quantitative version of this disjointness theorem for the case of the horocyclic flow on a compact Riemann surface. I will discuss Venkatesh's disjointness result and present a generalization of this result to more general actions of nilpotent groups, utilizing structural results about nilflows proven by Green-Tao-Ziegler. If time permits, I will discuss certain applications of such theorems in sparse equidistribution problems and number theory.

-----------------------

We will have two seminars this week. This is the first one and please note the unusual time. The second seminar will be at the usual time.

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*Abstract:*

Abstract: An important and challenging class of two-stage linear optimization problems are those without relative-complete recourse, wherein there exist first-stage decisions and realizations of the uncertainty for which there are no feasible second-stage decisions. Previous data-driven methods for these problems, such as the sample average approximation (SAA), are asymptotically optimal but have prohibitively poor performance with respect to out-of-sample feasibility. In this talk, we present a data-driven method for two-stage linear optimization problems without relative-complete recourse which combines (i) strong out-of-sample feasibility guarantees and (ii) general asymptotic optimality. Our method employs a simple robustification of the data combined with a scenario-wise approximation. A key contribution of this work is the development of novel geometric insights, which we use to show that the proposed approximation is asymptotically optimal. We demonstrate the benefit of using this method in practice through numerical experiments.

*Abstract:*

We will start by discussing a special class of automorphisms of a Poisson point process on an infinite measure space called Poisson suspensions and explain that the space of Poisson suspensions is a Polish group. After this we will explain an if and only if criteria for existence of an absolutely continuous invariant measure and show that a group has Kazhdan's property T if and only if all of its actions as Poisson suspensions are not properly nonsingular. If time permits we will show how one can use the previous construction to obtain a simple proof of a result of Bowen, Hartman and Tamuz that a group does not have Kazhdan property T if and only if it does not have a Furstenberg entropy gap in the sense of Nevo.

*Announcement:*

**דר' יובל פילמוס**

הפקולטה למדעי המחשב

טכניון

**Dr. Yuval Filmus**

The Faculty of Computer Science

Technion

**MATH CLUB 19.12.18**

**לאחר ההרצאה יתקיים טקס הענקת הפרסים של התחרות ע"ש גרוסמן**

**כמה מסובך לצבוע גרפים?**

בהינתן גרף, כמה קל לבדוק האם ניתן לצבוע את קדקודיו ב-* c *צבעים כך שכל קשת מחברת בין שני קדקודים בעלי צבעים שונים?

איך אפשר להשתכנע שצביעה כזו קיימת? ואיך אפשר להשתכנע שצביעה כזו אינה קיימת?

נדון בשאלות אלה כצוהר לעולם של סיבוכיות חישובית.

**Complexity of coloring graphs**

Given an undirected graph, how easy is it to determine whether we can color its vertices with *c* colors, so that no edge is monochromatic?

How easy is it to demonstrate that such a coloring exists? How can we prove that no such coloring exists?

We use these questions as a window to the world of computational complexity.

**ההרצאה תהיה בעברית**

**The lecture will be in Hebrew**

*Abstract:*

Abstract: Where extremal combinatorialists wish to optimise a discrete parameter over a family of large objects, probabilistic combinatorialists study the statistical behaviour of a randomly chosen object in such a family. In the context of representable matroids (i.e. the columns of a matrix) over $\mathbb{F}_2$, one well-studied distribution is to fix a small $k$ and large $m$ and randomly generate $m$ columns with $k$ 1’s. Indeed, when $k = 2$, this is the graphic matroid of the Erdos-Renyi random graph $G_{n,m}$. We turn back to the simplest corresponding extremal question in this setting. What is the maximum number of weight-$k$ columns a matrix of rank $\leq n$ can have? We show that, once $n \geq N_k$, one cannot do much better than taking only $n$ rows and all available weight-$k$ columns. This partially confirms a conjecture of Ahlswede, Aydinian and Khachatrian, who believe one can take $N_k=2k$. This is joint work with Wesley Pegden.

*Abstract:*

We shall discuss strengthening of the ballistic RWRE annealed functional CLT from the standard uniform topology to the rough path topology. An interesting phenomenon appears: the scaling limit of the area process is not only the Stratonovich Levy area but there is an addition of a linear term called the area anomaly. Moreover, the latter is identified in terms of the walk on a regeneration interval and naturally provides an extra information on the limiting process in case the correction is non-zero. Our result holds more generally, namely for any discrete process with bounded jumps which has a regular enough regenerative structure. An application to simulations is known for such limits in the rough path topology, which is generally not true in the uniform topology. Consider a difference equation driven by the walk, then a scaling limit to the corresponding SDE holds, with a correction expressed in terms of the area anomaly. This is a joint work with Olga Lopusanschi (Sorbonne).

*Abstract:*

Nonholonomic mechanics concerns with mechanical systems whose velocity is constrained. If these velocity constraints are linear, they define k-planes at every point of the configuration space of the system. In more complex situations further constraints appear: the movement of the system not only has to be tangent to these k-planes, but must obey conditions in which tangent vectors to the trajectories have constant length, or satisfy other, in general nonlinear, relations. This equips kinematics of nonholonomic mechanical systems with various geometric structures. These are: vector distributions on manifolds, their symmetry groups, differential invariants, associated exterior differential systems, Cartan connections, etc.

In the lectures we will discuss these geometric structures in simple examples of existing (or possible to exist) mechanical systems. We will concentrate on systems whose kinematics is described by a low dimensional parabolic geometry i.e. a geometry modeled on a homogeneous space G/P, with G being a simple Lie group, and P being its parabolic subgroup. The considered systems will include a movement of ice skaters on an ice rink, rolling without slipping or twisting of rigid bodies, movements of snakes and ants, and even movements of flying saucers. Geometric relations between these exemplary nonholonomic systems will be revealed. An appearance of the simple exceptional Lie group G2 will be a repetitive geometric phenomenon in these examples.

*Abstract:*

In this talk, I will discuss a question which originates in complex analysis but is really a problem in non-linear elliptic PDE. A finite Blaschke product is a proper holomorphic self-map of the unit disk, just like a polynomial is a proper holomorphic self-map of the complex plane. A celebrated theorem of Heins says that up to post-composition with a M\"obius transformation, a finite Blaschke product is uniquely determined by the set of its critical points. Konstantin Dyakonov suggested that it may be interesting to extend this result to infinite degree. However, one must be a little careful since infinite Blaschke products may have identical critical sets. I will show that an infinite Blaschke product is uniquely determined by its "critical structure” and describe all possible critical structures which can occur. By Liouville’s correspondence, this question is equivalent to studying nearly-maximal solutions of the Gauss curvature equation $\Delta u = e^{2u}$. This problem can then be solved using PDE techniques, using the method of sub- and super-solutions.

*Abstract:*

The algebra $H^{\infty}(\mathbb{D})$ of bounded analytic functions on the unit disc in the complex plane is a well-studied object. This algebra arises frequently in various areas of mathematics, in particular, function theory, hyperbolic geometry, and operator algebras. The classical Schwarz-Pick lemma tells us that analytic functions bounded by $1$ on the disc are necessarily contractions with respect to the Poincare metric. Furthermore, preserving metric between two points is equivalent to being an isometry and thus a Moebius map. In its other incarnation $H^{\infty}(\mathbb{D})$ is an operator algebra. The connection between the operator algebraic structure and the hyperbolic geometric of the disc was exploited to obtain interpolation and classification results.

However, operator algebras are generally noncommutative, hence it is common to think of them as quantized function algebras. The goal of my talk is to present a noncommutative generalization of this interplay between bounded functions on the disc and its geometry. To this end, I will introduce functions of noncommutative variables and explain how they arise naturally in many (even classical commutative) contexts. The focus of my talk is on bounded nc functions, that turns out to be automatically analytic. We will discuss the generalization of a classical fixed point theorem of Rudin and Herve and give an operator algebraic application.

Only basic familiarity with operators on Hilbert spaces and complex analysis is assumed.

*Abstract:*

**Advisors: **Dan Garber and Sabach Shoham** **

**Abstract**: Composite convex optimization problems that include a low-rank promoting term have important applications in data and imaging sciences. However, such problems are highly challenging to solve in large-scale: the low-rank promoting term prohibits efficient implementations of proximal based methods and even simple subgradient methods are very limited. On the other hand, methods which are tailored for low-rank optimization, such as conditional gradient-type methods, are usually slow. Motivated by these drawbacks, we present new algorithms and complexity results for some optimization problems in this class. At the heart of our results is the idea of using a low-rank SVD computations in every iteration. This talk is based on joint works with Dan Garber and Shoham Sabach.

*Abstract:*

Composite convex optimization problems that include a low-rank promoting term have important applications in data and imaging sciences. However, such problems are highly challenging to solve in large-scale: the low-rank promoting term prohibits efficient implementations of proximal based methods and even simple subgradient methods are very limited. On the other hand, methods which are tailored for low-rank optimization, such as conditional gradient-type methods, are usually slow. Motivated by these drawbacks, we present new algorithms and complexity results for some optimization problems in this class. At the heart of our results is the idea of using low-rank SVD computations in every iteration. This talk is based on joint works with Dan Garber and Shoham Sabach.

*Abstract:*

The Kobayashi pseudometric on a complex manifold M is the maximal pseudometric such that any holomorphic map from the Poincare disk to M is distance-decreasing. Kobayashi has conjectured that this pseudometric vanishes on Calabi-Yau manifolds. It is still out of reach for general Calabi-Yau manifolds. The proof of Kobayashi conjecture for hyperkahler manifolds is based on ergodic theory. I would explain its proof in application to K3 surfaces.

*Abstract:*

In Diophantine approximation we are often interested in the Lebesgue and Hausdorff measures of certain lim sup sets. In 2006, Beresnevich and Velani proved a remarkable result — the Mass Transference Principle — which allows for the transference of Lebesgue measure theoretic statements to Hausdorff measure theoretic statements for lim sup sets arising from sequences of balls in R k . Subsequently, they extended this Mass Transference Principle to the more general situation in which the lim sup sets arise from sequences of neighbourhoods of “approximating” planes. In this talk I will discuss a recent strengthening (joint with Victor Beresnevich, York, UK) of this latter result in which some potentially restrictive conditions have been removed from the original statement. This improvement gives rise to some very general statements which allow for the immediate transference of Lebesgue measure Khintchine–Groshev type statements to their Hausdorff measure analogues and, consequently, has some interesting applications in Diophantine approximation

*Abstract:*

The Bonnet-Myers theorem states that a complete manifold with Ricci curvature bounded below by a positive threshold is compact with an explicit diameter bound and that its fundamental group is finite. The talk will consist of a review of several extensions of this result. In particular, we will explain how assumptions on the Schrödinger operator with Ricci curvature as potential imply finiteness of the fundamental group of a compact manifold. Those are implied by the so-called Kato condition on the negative part of Ricci curvature. We will also give a purely geometric condition that suffices for the Ricci curvature to be Kato.

*Abstract:*

The nodal distribution of a given standard quantum graph have been shown to hold information about the topology of the graph, and it was explicitly calculated for specific families of graphs. In all of those cases, and in any numerical simulation, the nodal statistics appears to obey a central limit type convergence to a normal distribution as the number of edges (more specifically, the first Betti number) goes to infinity. We conjecture that this central limit type convergence of the nodal statistics is a universal property of quantum graphs. In the talk I will define the nodal statistics, state the conjecture and describe the proof of convergence for specific families of graphs.

*Abstract:*

The Teichmuller space of symplectic structuresis the quotient of the space of all symplectic forms by the action of the connected component of the diffeomorphism group. Teichmuller space of symplectic structures was first considered by Moser, who proved that it is a smooth manifold. The mapping class group acts on the Teichmuller space by diffeomorphism.

I would describe the Teichmuller space of symplectic structures in the few examples when it is understood (torus, K3 surface, hyperkahler manifold) and explain how the ergodic properties of the mapping group action can be used to obtain information about symplectic geometry.

*Abstract:*

A construction of Stallings encodes the information of a Heegaard splitting as a continuous map between 2-complexes. We investigate this construction from a more geometric perspective and find that irreducible Heegaard splittings can be encoded as square complexes with certain properties.

*Abstract:*

The talk will address the following problem; Can one start with two arbitrary three-dimensional manifolds, each with an Anosov flow, and glue them along their boundary to form a closed three manifold M with a new Anosov flow?I'll review a classical example due to Franks and Williams, recent general results due to Beguin, Bonatti and Yu, and work in progress extending their results, together with Adam Clay.

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Coffee and cookies at 14:20 on the 8th floor!

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*Abstract:*

There are two natural ways to measure how far is a pair of permutations A and B over {1,...,n}, from commuting with each other. The pair's local defect L(A,B) is the distance between the permutation compositions AB and BA (in the normalized Hamming metric). The global defect G(A,B) ts the distance of (A,B) to the nearest commuting pair of permutations. Generalizing this setting, we consider properties defined by a simultaneous system of equations. For example, the requirement that two given permutations commute corresponds to the system consisting of the single equation XY=YX. For an equation system E we define the functions L_E and G_E, which take an assignment of permutations to the variables of E, and map it to that assignment's local and global defect, respectively. We seek an upper bound on G_E, in terms of L_E (but not of n). In particular, if G_E is bounded by O(L_E^(1/d)), we say that E is polynomially stable (of degree d). Following a link to the theory of group stability, which concerns similar questions in a non-quantitative framework, we associate an equation set with a group, and show that polynomial stability, and the corresponding degree, are group invariants.. Our main result is that any equation system associated with an abelian group is polynomially stable. In particular, this includes the equation systems indicating pairwise commutativity of k permutations, for any k. Specifically, k=2 yields our initial example of commuting permutation pairs. We also note a connection between our result and efficient property testing algorithms. This is a joint work with Oren Becker.

*Abstract:*

--- THE CORRECT HOUR IS 13:30 --- There are two natural ways to measure how far is a pair of permutations A and B over {1,...,n}, from commuting with each other. The pair's local defect L(A,B) is the distance between the permutation compositions AB and BA (in the normalized Hamming metric). The global defect G(A,B) ts the distance of (A,B) to the nearest commuting pair of permutations. Generalizing this setting, we consider properties defined by a simultaneous system of equations. For example, the requirement that two given permutations commute corresponds to the system consisting of the single equation XY=YX. For an equation system E we define the functions L_E and G_E, which take an assignment of permutations to the variables of E, and map it to that assignment's local and global defect, respectively. We seek an upper bound on G_E, in terms of L_E (but not of n). In particular, if G_E is bounded by O(L_E^(1/d)), we say that E is polynomially stable (of degree d). Following a link to the theory of group stability, which concerns similar questions in a non-quantitative framework, we associate an equation set with a group, and show that polynomial stability, and the corresponding degree, are group invariants.. Our main result is that any equation system associated with an abelian group is polynomially stable. In particular, this includes the equation systems indicating pairwise commutativity of k permutations, for any k. Specifically, k=2 yields our initial example of commuting permutation pairs. We also note a connection between our result and efficient property testing algorithms. This is a joint work with Oren Becker.

*Abstract:*

This will be the FIFTH and FINAL LECTURE out of a series of FIVE lectures that Satish Pandey will give in the OA/OT learning seminar.

*Abstract:*

(Joint work with Alexey Kulik & Michael Scheutzow) I will present new techniques for analyzing ergodicity in nonlinear stochastic PDEs with an additive forcing. These techniques complement the Hairer-Mattingly approach. The first part of the talk is devoted to SPDEs that satisfy comparison principle (e.g., stochastic heat equation with a drift). Using a new version of the coupling method, we show how the corresponding Hairer-Mattingly results can be refined and we establish exponential ergodicity of such SPDEs in the hypoelliptic setting. In the second part of the talk, we show how a generalized coupling approach can be used to study ergodicity for a broad class of nonlinear SPDEs, including 2D stochastic NavierStokes equations. This extends the results of [N. Glatt-Holtz, J. Mattingly, G. Richards, 2017]. [1] O. Butkovsky, A.Kulik, M.Scheutzow (2018). Generalized couplings and ergodic rates for SPDEs and other Markov models. arXiv:1806.00395.

*Abstract:*

We consider a simultaneous small noise limit for a singularly perturbed coupled diffusion described by \begin{eqnarray*} dX^{\varepsilon}_t &=& b(X^{\varepsilon}_t, Y^{\varepsilon}_t)dt + \varepsilon^{\alpha}dB_t, \\\n dY^{\varepsilon}_t &=& - \frac{1}{\varepsilon} \nabla_yU(X^{\varepsilon}_t, Y^{\varepsilon}_t)dt + \frac{s(\varepsilon)}{\sqrt{\varepsilon}} dW_t, \end{eqnarray*} where $B_t, W_t$ are independent Brownian motions on ${\mathbb R}^d$ and ${\mathbb R}^m$ respectively, $b : \mathbb{R}^d \times \mathbb{R}^m \rightarrow \mathbb{R}^d$, $U : \mathbb{R}^d \times \mathbb{R}^m \rightarrow \mathbb{R}$ and $s :(0,\infty) \rightarrow (0,\infty)$. We impose regularity assumptions on $b$, $U$ and let $0 < \alpha < 1.$ When $s(\varepsilon)$ goes to zero slower than a prescribed rate as $\varepsilon \rightarrow 0$, we characterize all weak limit points of $X^{\varepsilon}$, as $\varepsilon \rightarrow 0$, as solutions to a differential equation driven by a measurable vector field. Under an additional assumption on the behaviour of $U(x, \cdot)$ at its global minima we characterize all limit points as Filippov solutions to the differential equation. This is joint work with V. Borkar, S. Kumar and R. Sundaresan.

*Abstract:*

A standard approach to statistically analyze a set S of polynomials is by grouping it into a family, that is, a polynomial whose coefficients are parameters, and each polynomial in S is obtained by specializing those parameters. The smaller the number of parameter is, the more accurate the statistics are. We shall discuss the problem of determining the minimal number of parameters (the essential dimension), a local approach to it (reducing mod p), and its connection to a recent conjecture of Colliot-Thelene.

*Abstract:*

We consider the trust region subproblem which is given by a minimization of a quadratic, not necessarily convex, function over the Euclidean ball. Based on the well-known second-order necessary and sufficient optimality conditions for this problem, we present two sufficient optimality conditions defined solely in terms of the primal variables. Each of these conditions corresponds to one of two possible scenarios that occur in this problem, commonly referred to in the literature as the presence or absence of the ``hard case". We consider a family of first-order methods, which includes the projected and conditional gradient methods. We show that any method belonging to this family produces a sequence which is guaranteed to converge to a stationary point of the trust region subproblem. Based on this result and the established sufficient optimality conditions, we show that convergence to an optimal solution can also be guaranteed as long as the method is properly initialized. In particular, if the method is initialized with the zero vector and reinitialized with a randomly generated feasible point, then the best of the two obtained vectors is an optimal solution of the problem with probability 1.This is joint work with Amir Beck.

*Abstract:*

Let M be a Riemannian manifold with a volume form. We will explain how to construct coclasses in the cohomology of thegroup of volume preserving diffeomorphisms (or homeomorphisms) of M. As an application, we show that 3-rd bounded cohomology of those groupsis highly non-trivial.

*Abstract:*

We shall explain how to use Eisenstein series to give asymptotics for for discrete orbits of lattices of SL2R when acting on the plane. Selberg's bounds on their polynomial growth properties come in and will be used as black box. Our point of view will be of 'how' to use them. Based on joint work with Claire Burrin, Amos Nevo and Barak Weiss.

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Coffee and cookies at 14:20 on the 8th floor!

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*Announcement:*

**דר' רון רוזנטל**

הפקולטה למתמטיקה

טכניון

**Dr. Ron Rosenthal**

The Faculty of Mathematics

Technion

**Math Club 21.11.18**

**חוק חצי המעגל של ויגנר**

מטריצות מהוות את אחד האובייקטים הבסיסיים ביותר במתמטיקה ויש להן שימושים כמעט בכל ענף של המדע.

מטריצות מקריות החלו להופיע בשנות ה- 50 כמודלים לתיאור של מערכות פיזיקליות מסובכות ומאז מצאו את דרכן באופן טבעי אל תורת ההסתברות.

בהרצאה זו נעסוק בהתנהגות הטיפוסית של ערכים עצמיים ווקטורים עצמיים של מטריצות מקריות וכן בקשר שלהן לתחומים שונים במתמטיקה ובמדע.

ההרצאה תניח ידע בסיסי בהסתברות.

**Wigner’s semi-circle law**

Matrices are one of the most fundamental objects in mathematics and applications of matrices are found in most scientific fields.

Random matrices appeared in the early 50’s as models for complex physical systems and naturally found their way to probability theory.

In the lecture we will discuss the typical behavior of eigenvalues and eigenvectors of random matrices and their relation to different areas in mathematics and science in general.

Basic knowledge in probability will be assumed.

**ההרצאה תהיה בעברית**

**The lecture will be in Hebrew**

*Abstract:*

A family of sets F is said to satisfy the (p,q)-property if among any p sets in F, some q have a non-empty intersection. Hadwiger and Debrunner (1957) conjectured that for any p > q > d there exists a constant c = c_d(p,q), such that any family of compact convex sets in R^d that satisfies the (p,q)-property, can be pierced by at most c points. Helly's Theorem is equivalent to the fact that c_d(p,p)=1 (p > d). In a celebrated result from 1992, Alon and Kleitman proved the conjecture. However, obtaining sharp bounds on the minimal such c_d(p,q), called `the Hadwiger-Debrunner numbers', is still a major open problem in combinatorial geometry. In this talk we present improved upper and lower bounds on the Hadwiger-Debrunner numbers, the latter using the hypergraph container method. Based on joint works with Shakhar Smorodinsky and Gabor Tardos.

*Abstract:*

This will be the FOURTH LECTURE out of a series of FIVE lectures that Satish Pandey will give in the OA/OT learning seminar. See previous announcements for the summary of the series.

Remaining topics to be covered:

6. Zauner's conjecture and its connection to SIC POVM

7. Separability and entanglement

8. Entanglement breaking channels

9. Entanglement breaking rank

10. Equivalences of Zauner's conjecture

*Abstract:*

Green’s functions are essential tools for solving linear partial differential equations with source terms, being the fundamental solution of a unit impulse. Closed form expressions are known only for a limited number of simple geometries, and in general, numerical simulation is the only available option where the computational burden can be prohibitive. A highly efficient yet simple solution is modal expansion, which involves expanding the inhomogeneous partial differential equation via the “modes” of the homogeneous partial differential equation. Modes are calculated once and for all, and are applicable to any arbitrary configuration of sources.Modal expansion techniques have long been used for closed systems, which the formulation is exceedingly simple because the partial differential operator is Hermitian. Recent research in nanophotonics, for example, has generated an explosion of interest in generalizing modal expansion to non-Hermitian open systems. We present a simple expansion method that bypasses all complexities usually associated with open systems, and recovers the simplicity of modal expansion in closed systems. We furthermore present a highly-efficient exponentially-convergent method of generating the modes themselves. We apply our methods to generate the electromagnetic Green’s tensor, which is fundamental to the photonic density of states, and thus the rate of quantum light-matter interaction under a semi-classical treatment.

*Abstract:*

A Boolean function f:{0,1}^n -> {0,1} is called 'noise sensitive' if flipping each of its input bits with a small probability affects its output ‘significantly’. Otherwise, it is called 'noise resistant'. Noise sensitivity is a fundamental property of Boolean functions that has been studied extensively over the last two decades. Its applications span several areas, including percolation theory and machine learning. A main result of the seminal work of Benjamini, Kalai and Schramm [BKS, 1999] which initiated the study of noise sensitivity, is that an unbiased Boolean function is noise resistant if and only if it has a 'strong' correlation with a halfspace. The original definition of noise sensitivity in [BKS] is meaningless for biased functions (i.e., functions whose expectation is close to 0 or 1). In this talk we propose a definition of noise sensitivity for biased functions, and prove an analogue of the main result of [BKS] for biased functions, with respect to the new definition. We then use our results to prove a conjecture of Kalai, Keller and Mossel in analysis of Boolean functions. A main tool we use is a local type of Chernoff's inequality, proved by Devroye and Lugosi (2008), which compares the rates of decay of the function Pr[ \sum a_i x_i > t] (where {x_i} are {-1,1} random variables and {a_i} are constants), for different ranges of t.

*Announcement:*

The talk will be held in the University of Haifa. Please email dneftin@technion.ac.il if you need a ride or an entry for your car into the campus.

Abstract: Chebotarev's theorem is one of the central theorems in algebraic number theory, and in its quantitative form it counts primes up to x with certain Frobenius in a number field. Some applications of the theorem necessitate a short-interval version of the theorem; that is to say, sample the primes in the interval [x, x+x^{a}) for large x and for a fixed 0<a<1. While the General Riemann Hypothesis implies short-interval theorem with any a>1/2, getting below 1/2 is beyond reach even with the GRH.

In this talk, we will discuss a short-interval Chebotarev theorem in the function field setting, where we replace the ring of integers by the ring of polynomials over a finite field. The key tools are a multi-dimensional explicit Chebotarev theorem and a novel computation of a Galois group.

Based on a joint work with T. Karidi, O. Gorodetsky, and W. Sawin.

*Abstract:*

In this talk we discuss the fine scale $L^2$-mass distribution of toral Laplace eigenfunctions with respect to random position. For the 2-dimensional torus, under certain flatness assumptions on the Fourier coefficients of the eigenfunctions and generic restrictions on energy levels, both the asymptotic shape of the variance and the limiting Gaussian law are established, in the optimal Planck-scale regime. We also discuss the 3-dimensional case, where the asymptotic behaviour of the variance is analysed in a more restrictive scenario. This is joint work with Igor Wigman.

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Coffee and cookies at 14:20 on the 8th floor!

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*Abstract:*

Abstract: "In a deductive game for two players, SF and PGOM, SF conceals an n-digit number x = x_1, ... , x_n, and PGOM, who knows n, tries to identify x by asking a number of questions, which are answered by SF. Each question is an n-digit number y = y_1,..., y_n; each answer is the number of subscripts i such that x_i = y_i. Moreover, we require PGOM send all the questions at once. In this talk, I will show that the minimum number of questions required to determine x is (2+o(1))n / lg n. A more general problem is to determine the asymptotic formula of the metric dimension of Cartesian powers of a graph. I will state the class of graphs for which the formula can be determined, and the smallest graphs for which we did not manage to settle. Joint work with Zilin Jiang.

*Abstract:*

(This will be the THIRD LECTURE out of a series 3 or 4 lectures that Satish Pandey will give in the OA/OT learning seminar)

Title: A brief introduction to quantum information theory aimed at studying entanglement breaking rank

================================================================================

Abstract of the series: We will begin with a quick review of the basics of Quantum Mechanics. This primarily entails the four postulates of Quantum Mechanics which provide connections between the physical world and the mathematical formalism of Quantum Mechanics. The notion of von Neumann's density matrices will subsequently be defined and we will use it to reformulate the above postulates in the language of density matrices. We next define and study quantum channels and present the celebrated Choi-Kraus representation theorem. These constitute the key concepts and results that are required to move forward and define a special class of quantum channels called "entanglement breaking channels" --- the quantum channels that admit a Choi-Kraus representation consisting of rank-one Choi-Kraus operators.

We then introduce the entanglement breaking rank of an entanglement breaking channel and define it to be the least number of rank-one Choi-Kraus operators required in its Choi-Kraus representation. We shall show how this rank parameter for a certain map links to an open problem in linear algebra: Zauner's conjecture. In particular, we show that the problem of computing the entanglement breaking rank of the channel $$X \mapsto \frac{1}{d+1}(X+\text{Tr}(X)I_d),$$ is equivalent to the existence problem of SIC POVM in dimension $d$.

Here is an outline of the series of talks:

1. Postulates of Quantum Mechanics

2. Mixed States

3. von Neumann's density matrices

4. Quantum channels

5. Choi-Krauss representation theorem

6. Zauner's conjecture and its connection to SIC POVM

7. Separability and entanglement

8. Entanglement breaking channels

9. Entanglement breaking rank

10. Equivalences of Zauner's conjecture

*Abstract:*

We study special classes of the stationary solutions of the Vlasov-Maxwell-Fokker-Planck system and their connection with nonlinear elliptic equations of the double Liouville type. We consider the coupling between the double Liouville system and a new integrable two component evolutionary dispersive Schwartz - KdV system of third order.

*Abstract:*

This talk provides upper and lower bounds on the optimal guessing moments of a random variable taking values on a finite set when side information may be available. These moments quantify the number of guesses required for correctly identifying the unknown object and, similarly to Arikan's bounds, they are expressed in terms of the Arimoto-R\'{e}nyi conditional entropy. Although Arikan's bounds are asymptotically tight, the improvement of the bounds which are considered in this talk is significant in the non-asymptotic regime. Relationships between moments of the optimal guessing function and the MAP error probability are provided, characterizing the exact locus of their attainable values. * This is a joint work with Sergio Verdu.

*Abstract:*

Nonholonomic mechanics concerns with mechanical systems whose velocity is constrained. If these velocity constraints are linear, they define k-planes at every point of the configuration space of the system. In more complex situations further constraints appear: the movement of the system not only has to be tangent to these k-planes, but must obey conditions in which tangent vectors to the trajectories have constant length, or satisfy other, in general nonlinear, relations. This equips kinematics of nonholonomic mechanical systems with various geometric structures. These are: vector distributions on manifolds, their symmetry groups, differential invariants, associated exterior differential systems, Cartan connections, etc.

In the lectures we will discuss these geometric structures in simple examples of existing (or possible to exist) mechanical systems. We will concentrate on systems whose kinematics is described by a low dimensional parabolic geometry i.e. a geometry modeled on a homogeneous space G/P, with G being a simple Lie group, and P being its parabolic subgroup. The considered systems will include a movement of ice skaters on an ice rink, rolling without slipping or twisting of rigid bodies, movements of snakes and ants, and even movements of flying saucers. Geometric relations between these exemplary nonholonomic systems will be revealed. An appearance of the simple exceptional Lie group G2 will be a repetitive geometric phenomenon in these examples.

*Abstract:*

We define a natural topology on the collection of (equivalence classes up to scaling of) locally finite measures on a homogeneous space and prove that in this topology, pushforwards of certain infinite volume orbits equidistribute in the ambient space. As an application of our results we prove an asymptotic formula for the number of integral points in a ball on some varieties as the radius goes to infinity. This is a joint work with Uri Shapira.

*Abstract:*

We say that I^{n}(F) is m-linked if every m bilinear n-fold Pfsiter forms have a common (n-1)-fold factor. In a recent publication, Karim Becher pointed out that when F is a global field, I^{n}(F) is m-linked for every positive integer m, and raised the question of whether I^{n}(F) being 3-linked implies that it is m-linked for every positive integer m. In the special case of characteristic 2, this question can be phrased in two versions - one for bilinear forms and another for quadratic forms. We will provide negative answers to both versions of the question in characteristic 2, and discuss some open problems.

*Abstract:*

We prove that if the edges of a graph G can be colored blue or red in such a way that every vertex belongs to a monochromatic k-clique of each color, then G has at least 4(k-1) vertices. This confirms a conjecture of Bucic, Lidicky, Long, and Wagner, and thereby solves the 2-dimensional case of their problem about partitions of discrete boxes with the k-piercing property. We also characterize the case of equality in our result.

*Abstract:*

(This will be the SECOND LECTURE out of a series 3 or 4 lectures that Satish Pandey will give in the OA/OT learning seminar)

Title: A brief introduction to quantum information theory aimed at studying entanglement breaking rank

================================================================================

Abstract of the series: We will begin with a quick review of the basics of Quantum Mechanics. This primarily entails the four postulates of Quantum Mechanics which provide connections between the physical world and the mathematical formalism of Quantum Mechanics. The notion of von Neumann's density matrices will subsequently be defined and we will use it to reformulate the above postulates in the language of density matrices. We next define and study quantum channels and present the celebrated Choi-Kraus representation theorem. These constitute the key concepts and results that are required to move forward and define a special class of quantum channels called "entanglement breaking channels" --- the quantum channels that admit a Choi-Kraus representation consisting of rank-one Choi-Kraus operators.

We then introduce the entanglement breaking rank of an entanglement breaking channel and define it to be the least number of rank-one Choi-Kraus operators required in its Choi-Kraus representation. We shall show how this rank parameter for a certain map links to an open problem in linear algebra: Zauner's conjecture. In particular, we show that the problem of computing the entanglement breaking rank of the channel $$X \mapsto \frac{1}{d+1}(X+\text{Tr}(X)I_d),$$ is equivalent to the existence problem of SIC POVM in dimension $d$.

Here is an outline of the series of talks:

1. Postulates of Quantum Mechanics

2. Mixed States

3. von Neumann's density matrices

4. Quantum channels

5. Choi-Krauss representation theorem

6. Zauner's conjecture and its connection to SIC POVM

7. Separability and entanglement

8. Entanglement breaking channels

9. Entanglement breaking rank

10. Equivalences of Zauner's conjecture

*Abstract:*

The Boltzmann equation without angular cutoff is considered when the initial data is a perturbation of a global Maxwellian with algebraic decay in the velocity variable. Global solution is proved by combining the analysis in moment propagation, spectrum of the linearized operator and the smoothing effect of the linearized operator when initial data in Sobolev space with negative index.

This is a joint work with Ricardo Alonso, Yoshinori Morimoto and Weiran Sun.

*Abstract:*

Consider a Gaussian stationary function on the real line (that is, a random function whose distribution is shift-invariant and all its finite marginals have centered multi-normal distribution). What is the probability that it has no zeroes at all in a long interval? What is the probability that it has a significant deficiency or abundance in the number of zeroes? These questions were raised more than 70 years ago, but even modern tools of large deviation theory do not directly apply to answer them. In this talk we will see how real, harmonic and complex analysis shed light on these questions, yielding new results and many open questions. Based on joint works with O. Feldheim and S. Nitzan (arXiv:1709.00204) and R. Basu, A. Dembo and O. Zeitouni (arXiv:1709.06760).

*Abstract:*

In 2008 Agol showed that a 3-manifold with a certain condition on its fundamental group is virtually fibered, i.e. it has a finite covering that is a surface bundle over the circle. A few years later it was shown by Agol and Wise that the fundamental groups of most 3-manifold satisfy Agol's condition, i.e. most 3-manifodls are virtually fibered. We will outline a proof of Agol's theorem following an approach taken by myself and Kitayama.

*Abstract:*

In the 1940’s Graham Higman initiated the study of finite subgroups of the unit group of an integral group ring. He proved for example that if the normalized unit group of the integral group ring of a finite group G contains an element of prime order p then so does G itself. Here a unit is called normalized if its coefficients sum up to 1.

Define the prime graph of a group X to be the graph whose vertices are primes appearing as orders of elements in X and two vertices p and q are connected by an edge if and only if X contains an element of order pq. Then the Prime Graph Question asks if the the prime graph of a finite group G coincides with the prime graph of the normalized unit group of the integral group ring of G. Note that the vertices of the two graphs are equal by the result of Higman mentioned above.

Kimmerle proved the question for solvable groups and Frobenius groups. Contrary to most other statements on units in group rings the Prime Graph Question allows a reduction - to almost simple groups. I will report on the proof of the question for alternating and symmetric groups which is the first answer for a whole class of almost simple groups. The proof is a consequence of a theorem which states that the two prime graphs are equal “around a prime p” if the principal p-block of G is particularly well-behaved, namely it is a Brauer tree algebra whose Brauer tree is a line with no exceptional vertex.

This is joint work with A. Bächle.

*Abstract:*

Classical knot theory is the study of smooth 1-manifolds in R^3. We start with a beautiful connection between notions of positivity for knots with the study of algebraic curves in C^2 due to Rudolph and Boileau-Orevkov. We discuss applications of knot theory to complex curve questions and vice versa. Based on joint works with Lewark-Lobb and Borodozik.

*Abstract:*

In 1955, Rogers proved a series of higher moment formulas for the Siegel transform on the space of unimodular lattices extending Siegel's classical mean value theorem. Among them, the second moment formula is of most interest due to its many applications to counting problems. For example, using the second moment formula, Schmidt proved a very good bound for the discrepancy for generic lattices with respect to an increasing family of sets with unbounded volumes. In this talk, I will describe a second moment formula for the Siegel transform restricted to the subspace of symplectic lattices. As an application, we prove a similar bound for the discrepancy for a generic symplectic lattice.

*Abstract:*

**Advisor: **Emanuel Milman

**Abstract: **We establish new sharp inequalities of Poincare or log-Sobolev type, on weighted Riemannian manifolds whose (generalized) Ricci curvature is bounded from below. To this end we establish a general method which complements the 'localization' theorem which has recently been established by B. Klartag. Klartag's theorem is based on optimal transport techniques, leading to a disintegration of the manifold measure into marginal measures supported on geodesics of the manifold. This leads to a reduction of the problem of proving a n-dimensional inequality into an optimization problem over a class of measures with 1-dimensional supports. Our method is based on functional analytic techniques, and leads to a further reduction of the optimization problem into a simpler problem over a sub-class of model-space measures. By employing ad-hoc analytical techniques we solve the optimization problems associated with the Poincare and the log-Sobolev inequalities. Quiet unexpectedly the solution to the problem of characterizing the sharp Poincare constant reveals anomalous behavior within a certain domain of the generalized-dimension parameter, hinting on a new phenomena.

*Abstract:*

(This will be the FIRST LECTURE out of a series 3 or 4 lectures that Satish Pandey will give in the OA/OT learning seminar)

Title: A brief introduction to quantum information theory aimed at studying entanglement breaking rank.

================================================================================

Abstract: We will begin with a quick review of the basics of Quantum Mechanics. This primarily entails the four postulates of Quantum Mechanics which provide connections between the physical world and the mathematical formalism of Quantum Mechanics. The notion of von Neumann's density matrices will subsequently be defined and we will use it to reformulate the above postulates in the language of density matrices. We next define and study quantum channels and present the celebrated Choi-Kraus representation theorem. These constitute the key concepts and results that are required to move forward and define a special class of quantum channels called "entanglement breaking channels" --- the quantum channels that admit a Choi-Kraus representation consisting of rank-one Choi-Kraus operators.

We then introduce the entanglement breaking rank of an entanglement breaking channel and define it to be the least number of rank-one Choi-Kraus operators required in its Choi-Kraus representation. We shall show how this rank parameter for a certain map links to an open problem in linear algebra: Zauner's conjecture. In particular, we show that the problem of computing the entanglement breaking rank of the channel $$X \mapsto \frac{1}{d+1}(X+\text{Tr}(X)I_d),$$ is equivalent to the existence problem of SIC POVM in dimension $d$.

Here is an outline of the series of talks:

1. Postulates of Quantum Mechanics

2. Mixed States

3. von Neumann's density matrices

4. Quantum channels

5. Choi-Krauss representation theorem

6. Zauner's conjecture and its connection to SIC POVM

7. Separability and entanglement

8. Entanglement breaking channels

9. Entanglement breaking rank

10. Equivalences of Zauner's conjecture

*Abstract:*

The Cantor uniqueness theorem states that any trigonometric series converging to zero everywhere must be trivial. We investigate the question: is it still true when taking a limit along subsequences? Joint work with A. Olevskii.

*Abstract:*

A main goal of geometric group theory is to understand finitely generated groups up to a coarse equivalence (quasi-isometry) of their Cayley graphs. Right-angled Coxeter groups, in particular, are important classical objects that have been unexpectedly linked to the theory of hyperbolic 3-manifolds through recent results, including those of Agol and Wise. I will give a brief background of what is currently known regarding the quasi-isometric classification of right-angled Coxeter groups. I will then describe a new computable quasi-isometry invariant, the hypergraph index, and its relation to other invariants such as divergence and thickness.

*Abstract:*

Mathematical epidemiology uses modelling to study the spread of contagious diseases in a population, in order to understand the underlying mechanisms and aid public health planning. In recent years there is growing interest in applying similar models to the study of `social contagion': the spread of ideas and behaviors. It is of great interest is to consider the ways in which social contagion differs from biological contagion at the individual level, and to use mathematical modelling to understand the population-level consequences of these differences. In this talk I will present simple `two-stage' contagion models motivated by social-science literature, and study their dynamics. It turns out that these models give rise to some interesting and non-intuitive nonlinear phenomena which do not arise in the `classical' models of mathematical epidemiology, and which might have relevance to understanding some real-world observations.

*Abstract:*

The octopus lemma states that certain operators on the symmetric group are positive semi-definite. Its original application was to resolve a long-standing conjecture of Aldous related to the spectral gap of interacting particle systems. Since then it has found other applications. We will survey this new topic, perhaps some proofs will be involved.

*Abstract:*

The octopus lemma states that certain operators on the symmetric group are positive semi-definite. Its original application was to resolve a long-standing conjecture of Aldous related to the spectral gap of interacting particle systems. Since then it has found other applications. We will survey this new topic, perhaps some proofs will be involved.

*Abstract:*

I will discuss some recent results on minimal actions of general countable groups. In particular I will describe a new property of such minimal actions called the DJ property which is defined in terms of the notion of disjointness of actions and explain how it is related to an old question of Furstenberg on the algebra spanned by the minimal functions on a group. All concepts above will be explained.

*Abstract:*

The Whitehead conjecture asks whether a subcomplex of an aspherical 2-complex is always aspherical. This question is open since 1941. Howie has shown that the existence of a finite counterexample implies (up to the Andrews-Curtis conjecture) the existence of a counterexample within the class of labelled oriented trees. Labelled oriented trees are algebraic generalisations of Wirtinger presentations of knot groups. In this talk we start with an introduction into the field. Then we present several possibilities to show asphericity in the class of labelled oriented trees. There are many known classes of aspherical LOTs given by the weight test of Gersten, the I-test of Barmak/Minian, LOTs of Diameter 3 (Howie), LOTs of complexity two (Rosebrock) and several more.

*Abstract:*

It is in general a very difficult (algebraic) problem to ascertain whether a group $H$ is a subgroup of a group $G$. The geometric version of this question (for finitely generated groups) is to ask whether there exists a coarse embedding of a Cayley graph of $H$ into a Cayley graph of $G$. We can control the growth and asymptotic dimension of $H$ in terms of $G$, but these are not sufficient to dismiss a (relatively naive) statement like For every $k$ there exists some $l$ such that any hyperbolic group of asymptotic dimension $k$ coarsely embeds into real hyperbolic space of dimension $l$. In this talk I will introduce a new class of geometric invariants and use them to disprove the statement above, and also prove that the only Baumslag-Solitar group which can be coarsely embedded into a hyperbolic group is $\mathbb Z^2=BS(1,1)$.

*Abstract:*

**Advisor: **Prof**. **Amos Nevo

**Abstract: **Counting lattice points has a long history in number theory, that can be traced back to Gauss. I will introduce this subject, and its relation to questions regarding asymptotic properties of integral points in the plane. The main focus will be an arithmetic result regarding equidistribution of parameters that characterize primitive integral points. I will also present a similar result for rings of integers in C (e.g., Gaussian integers).

Both these results are achieved via counting lattice points w.r.t. the Iwasawa decomposition in certain simple rank-one Lie groups, a topic which I will discuss more generally, if time permits.

*Abstract:*

**Advisor:** Savir Yonatan

**Abstract: ** One of the main determinants of the the fitness of biological systems is their ability to accurately sense their environment and respond accordingly. In particular, cells have to sense the nutrients in their environment and coordinate their metabolic gene program appropriately. One of the well-studied examples of such regulation is catabolite repression - a phenomenon where a preferred carbon source represses the transcription of genes which encode enzymes required for the utilization of alternative carbon sources. The decision of when to switch from a preferred to a less-preferred carbon source is akin to a general optimal switching problem which has long been of interest in management science and operations research. Using dynamical models we determine the optimal switching strategy - the optimal time in which cells will induce genes required for metabolizing the less-preferred carbon source. We derive the conditions in which the optimal switching strategy includes preparation for impending depletion of the preferred nutrient by inducing early, and the conditions for when the switching mechanism depends on either threshold-sensing or ratio-sensing of the external nutrient concentrations. We formalize the problem in terms of evolutionary game theory and show that an optimal switching strategy which involves preparation is not always evolutionarily stable and determine the particular environment in which preparing is evolutionarily stable.

*Abstract:*

**Advisor: **Uri Shapira

**Abstract: **

The highlight of my presentation is the wedding of p-arithmetic extension and the thickening trick. The result is a beautiful offspring: an equidistribution result regarding periodic geodesics along certain paths in the p-Hecke graph. This result calls on an ingenious use of the decay of matrix coefficients in tandem with the astonishing notion of tubes around compact orbits, which is the diva of this talk so we handle it with care.

*Abstract:*

Let K=F_q(C) be the global function field of rational functions over a smooth and projective curve C defined over a finite field F_q. The ring of regular functions on C - S where S is any non-empty finite set of closed points on C, is a Dedekind domain O_S of K. For a semisimple O_S-group G with a smooth fundamental group F, we describe both the set of genera of G and its principal genus in terms of abelian groups depending on O_S and F only. This leads to a necessary and sufficient condition to the Hasse local-global principle to hold for G, and facilitates the computation of the Tamagawa number of some twisted K-groups. We finally describe the all set of O_S-classes of twisted-forms of G in terms of the above O_S-invariants and the absolute type of the Dynkin diagram of G. This finite set turns out to biject in many cases to a disjoint union of abelian groups.

Joint work with Ralf Kohl and Claudia Schoemann.

*Abstract:*

Abstract: Semicocycles appear naturally in the study of the asymptotic behavior of non-autonomous differential equations. They play an important role in the theory of dynamical systems and are closely connected to semigroups of weighted composition operators. In this talk we consider semicocycles whose elements are either continuous or holomorphic mappings on a domain in a real/complex Banach space which take values in a unital Banach algebra. We study properties of semicocycles employing, in particular, their link with semigroups of nonlinear mappings. We show analogies as well as differences between the theory of semigroups and the theory of semicocycles. One of our main aims is to establish conditions for differentiability of a semicocycle and prove the existence of a "generator". The simplest semicocycles are those independent of the spatial variable. An interesting problem is to determine whether a given semicocycle is cohomologous to an independent one (in other words, is linearizable). We provide some criteria for a semicocycle to be linearizable as well as several easily verifiable sufficient conditions. This is joint work with Fiana Jacobzon and Guy Katriel.

*Abstract:*

Abstract: Poincar\'{e}'s inequality, which is probably best known for its applications in PDEs and the calculus of variations, is one of the simplest examples of an inequality that lies at the crossroads of Analysis, Probability and Semigroup/Spectral theory. It can be understood as the functional inequality that arises from attempting to understand convergence of the so-called heat flow to its equilibrium state. This approach can be generalized to the setting of Markov semigroups, with a non-positive generator that posseses a spectral gap. A natural question that one can consider is: What happens if the generator does not have a spectral gap? Can we still deduce a rate of convergence from a functional setting? In this talk we will discuss a new approach to this question and see how an understanding of the way the spectrum of the generator behaves near the origin, in the form of a density of states estimate, can lead to weak Poincar\'{e} type inequalities, from which a quantitative estimation of convergence can be obtained. This talk is based on a joint work with Jonathan Ben-Artzi.

*Abstract:*

Abstract: Consider a polygon-shaped billiard table on which a ball can roll along straight lines and reflect off of edges infinitely. In work joint with Moon Duchin, Viveka Erlandsson and Chris Leininger, we have characterized the relationship between the shape of a polygonal billiard table and the set of possible infinite edge itineraries of balls travelling on it. In this talk, we will explore this relationship and the tools used in our characterization (notably a new rigidity result for flat cone metrics).

*Abstract:*

The classical results of Atiyah (1967) and Atiyah, Singer (1969) provide the homotopy types of the space FB(H) of Fredholm bounded operators on a Hilbert space H and of its subspace FBsa(H) consisting of self-adjoint operators. Namely, FB(H) is a classifying space for the functor K^0, while FBsa(H) is a classifying space for the functor K^1.

However, in many applications (e.g. to differential operators) one deals with unbounded operators rather than bounded. The space B(H) of bounded operators is then replaced by the space R(H) of regular (that is, closed and densely defined) operators. The homotopy types of the spaces FR(H) and FRsa(H) were unknown for a long time. Finally, an analog, for regular operators, of results of Atiyah and Singer was obtained in 2003 by Joachim. His proof is based on the theory of C*-algebras and Kasparov KK-theory.

I will describe in the talk how this result of Joachim can be included into broader picture. In particular, I will show connections of the spaces FR(H) and FRsa(H) with other classical spaces. I will also give a simple definition of the family index for unbounded operators. All terminology will be explained during the talk.

*Abstract:*

Affine Sobolev inequality of G. Zhang is a refinement of the usual limiting Sobolev inequality which possesses additional invariance with respect to action of the group SL(N) of unimodular matrices. For p=2 we find a simplified form of the affine Sobolev functional and study the related affine Laplacian. For general p<N we study compactness properties of the functional and existence of minimizers. This is a joint work with Ian Schindler.

*Abstract:*

Let $(X,\|\cdot\|)$ be a uniformly convex Banach space and let $C$ be a bounded, closed and convex subset of $X$. Assume that $C$ has nonempty interior and is locally uniformly rotund. Let $T$ be a nonexpansive self-mapping of $C$. If $T$ has no fixed point in the interior of $C$, then there exists a unique point $\tilde{x}$ on the boundary of $C$ such that each sequence of iterates of $T$ converges in norm to $\tilde{x}$. We also establish an analogous result for nonexpansive semigroups. This is joint work with Aleksandra Grzesik, Wieslawa Kaczor and Tadeusz Kuczumow.

*Abstract:*

Abstract: I will describe the rich connections between homogeneous dynamics and Diophantine approximation on manifolds with an emphasis on some recent developments.

*Abstract:*

**Advisor: **Orr Shalit

**Abstract: **In this talk I will give a brief survey on my Ph.D. thesis which mainly focus on certain types of operator-algebras. The talk, correspondingly to my thesis, is divided into two parts.

The first part is about subalgebras (and also other subsets) of graph C*-algebras. I will present some results from a joint work with Adam Dor-On, in which we studied maximal representations of graph tensor algebra. I will first provide a complete description of these maximal representations and then show some dilation theoretical applications, as well as a characterization of a certain rigidity phenomenon, called hyperrigidity, that may or may not occur for a subset of a C*-algebra. I will then present an independent follow-up work in which I studied, in addition to hyperrigidity, other types of rigidity of other types of subsets of graph C*-algebras and obtained some more delicate results.

The second part is devoted to operator-algebras arising from noncommutative (nc) varieties and is based on a joint work with Orr Shalit and Eli Shamovich. The algebra of bounded nc functions over a nc subvariety of the nc ball can be identified as the multiplier algebra of a certain reproducing kernel Hilbert space consisting of nc functions on the subvariety. I will try to answer the following question: in terms of the underlying varieties, when are two such algebras isomorphic? Along the way, if time allows, I will show that while in some aspects the nc and the classical commutative settings share a similar behavior, the first enjoys – and also suffers from – some unique noncommutative phenomena.

*Abstract:*

How can $d+k$ vectors in $\mathbb{R}^d$ be arranged so that they are as close to orthogonal as possible? In particular, define $\theta(d,k):=\min_X\max_{x\neq y\in X}|\langle x,y\rangle|$ where the minimum is taken over all collections of $d+k$ unit vectors $X\subseteq\mathbb{R}^d$. In this talk, we focus on the case where $k$ is fixed and $d\to\infty$. In establishing bounds on $\theta(d,k)$, we find an intimate connection to the existence of systems of ${k+1\choose 2}$ equiangular lines in $\mathbb{R}^k$. Using this connection, we are able to pin down $\theta(d,k)$ whenever $k\in\{1,2,3,7,23\}$ and establish asymptotics for general $k$. The main tool is an upper bound on $\mathbb{E}_{x,y\sim\mu}|\langle x,y\rangle|$ whenever $\mu$ is an isotropic probability mass on $\mathbb{R}^k$, which may be of independent interest. (Joint work with Boris Bukh)

*Abstract:*

In this talk we will derive sufficient conditions for the absence of embedded eigenvalues of two-dimensional magnetic Schroedinger operators. The limiting absorption principle will be discussed as well. This is a joint work with S.Avramska-Lukarska and D.Hundertmark.

*Abstract:*

NOTE THE SPECIAL TIME.

TheElisha Netanyahu Memorial Lectures

*Abstract:*

* Abstract: *Given n uniform points on the surface of a two-dimensional sphere, how can we partition the sphere fairly among them?

It turns out that if the given points apply a two-dimensional gravity force to the rest of the sphere, then the basins of attraction for the resulting flow yield such a partition—with exactly equal areas, no matter how the points are distributed. (See http://www.ams.org/publications/journals/notices/201705/rnoti-cvr1.pdf) Our main result is that this partition minimizes, up to a bounded factor, the average distance between points in the same cell. This has an application to almost optimal matching of n uniform blue points to n uniform red points on the sphere. I will also describe open problems regarding greedy and electrostatic matching (Joint work with Nina Holden and Alex Zhai) Another topic where local and global optimization sharply differ appears starts from the classical overhang problem: Given n blocks supported on a table, how far can they be arranged to extend beyond the edge of the table without falling off? With Paterson, Thorup, Winkler and Zwick we showed ten years ago that an overhang of order cube root of n is the best possible; a crucial element in the proof involves an optimal control problem for diffusion on a line segment and I will describe generalizations of this problem to higher dimensions (with Florescu and Racz).

*Abstract:*

Abstract is available here: https://noncommutativeanalysis.files.wordpress.com/2018/06/abstract.pdf

*Announcement:*

**פרופ' רון אהרוני**

הפקולטה למתמטיקה

טכניון

**Prof. Ron Aharoni**

The Faculty of Mathematics

Technion

**Math Club 20.6.18**

**טופולוגיה וחתונות:** איך אפשר להיעזר בטופולוגיה לצורך שידוכים.

**Topology and weddings:** Surprisingly, topology can help you find a matching

ההרצאה תהיה בעברית

The lecture will be in Hebrew

*Abstract:*

In 1941, Turan proved the famous Turan theorem, i.e., If G is a graph which does not contain Kr+1 as its subgraph, then the edge number of G is no more than the Turan graph Tn,r, which started the extremal theory of graphs. In this talk, we will introduce the spectral Turan-Type results which are associated with the adjacency matrix, signless Laplacian matrix. Moreover, some open problems in this field are proposed.

*Abstract:*

Free boundary minimal surfaces in the unit 3-ball have recently attracted a lot of attention, and many new examples have been constructed. In a seminal series of papers, A. Fraser and R. Schoen (2013) have shown that the existence of such surfaces is related to a maximisation problem for the first non-zero Steklov eigenvalue, on abstract surfaces with boundary. A natural quantity that is worth investigating for critical points of a functional in general, and for free boundary minimal surfaces in particular, is the so-called Morse index. More precisely, it is interesting to relate this number to the topology of the surface. This type of questions has first been considered for complete surfaces of the Euclidean space R^3, and for closed minimal surfaces of the 3-sphere. For the latter, a celebrated result of F. Urbano (1990) characterises the closed minimal surfaces of the 3-sphere with minimal index. In this talk, I will present some partial results towards a generalisation of Urbano’s theorem to free boundary minimal surfaces in the 3-ball

*Abstract:*

In 2003, Welschinger defined invariants of real symplectic manifolds of complex dimensions 2 and 3, which are related to counts of pseudo-holomorphic disks with boundary and interior point constraints (Solomon, 2006). The problem of extending the definition to higher dimensions remained open until recently (Georgieva, 2013, and Solomon-Tukachinsky, 2016-17).

In the talk I will give some background on the problem, and describe a generalization of Welschinger's invariants to higher dimensions, with boundary and interior constraints, a.k.a. open Gromov-Witten invariants. This generalization is constructed in the language of $A_\infty$-algebras and bounding chains, where bounding chains play the role of boundary point constraints. If time permits, we will describe equations, a version of the open WDVV equations, which the resulting invariants satisfy. These equations give rise to recursive formulae that allow the computation of all invariants of $\mathbb{C}P^n$ for odd $n$.

This is joint work with Jake Solomon.

No previous knowledge of any of the objects mentioned above will be assumed.

*Abstract:*

We show how to decompose the moduli-space of shapes of polyhedra and how such a decomposition can be used to solve geometric realization problems.

*Abstract:*

Abstract: Caprace and De Medts discovered that Thompson's V can be written as a group of tree almost automorphisms, allowing to embed it densely into a totally disconnected, locally compact (t.d.l.c.) group. Matui discovered that it can be written as the topological full group of the groupoid associated to a one-sided shift. Combining these, we find countably many different t.d.l.c. groups containing a dense copy of V.

*Abstract:*

Two common approximation notions in discrete geometry are ε-nets and ε-approximants. Of the two, ε-approximants are stronger. For the family of convex sets, small ε-nets exist while small ε-approximants unfortunately do not. In this talk, we introduce a new notion "one-sided ε-approximants", which is of intermediate strength, and prove that small one-sided ε-approximants do exist. This strengthens the classic result of Alon-Bárány-Füredi-Kleitman. The proof is based on a (modification of) the local repetition lemma of Feige--Koren--Tennenholtz and of Axenovich--Person--Puzynina. Joint work with Gabriel Nivasch.

*Abstract:*

Handwriting comparison and identification, e.g. in the setting of forensics, has been widely addressed over the years. However, even in the case of modern documents, the proposed computerized solutions are quite unsatisfactory. For historical documents, such problems are worsened, due to the inscriptions’ preservation conditions. In the following lecture, we will present an attempt at addressing such a problem in the setting of First Temple Period inscriptions, stemming from the isolated military outpost of Arad (ca. 600 BCE). The solution we propose comprises: (A) Acquiring better imagery of the inscriptions using multispectral techniques; (B) Restoring characters via approximation of their composing strokes, represented as a spline-based structure, and estimated by an optimization procedure; (C) Feature extraction and distance calculation - creation of feature vectors describing various aspects of a specific character based upon its deviation from all other characters; (D) Conducting an experiment to test a null hypothesis that two given inscriptions were written by the same author. Applying this approach to the Arad corpus of inscriptions resulted in: (i) The discovery of a brand new inscription on the back side of a well known inscription (half a century after being unearthed); (ii) Empirical evidence regarding the literacy rates in the Kingdom of Judah on the eve of its destruction by Nebuchadnezzar the Babylonian king.

*Abstract:*

**Advisor**: Reichart Roi

**Abstract**: Natural Language Processing (NLP) problems are usually structured, as a natural language is. Most models for such problems are designed to predict the "highest quality" structure of the input example (sentence, document etc.), but in many cases a diverse list of structures is of fundamental importance. We propose a new method for learning high quality and diverse lists using structured prediction models. Our method is based on perturbations: learning a noise function that is particularly suitable for generating such lists. We further develop a novel method (max over marginals) that can distill a new high quality tree from the perturbation-based list. In experiments with cross-lingual dependency parsing across 16 languages, we show that our method can lead to substantial gains in parsing accuracy over existing methods

*Abstract:*

I will describe joint work with Stan Alama, Lia Bronsard, Andres Contreras and Jiri Dadok giving criteria for existence and for non-existence of certain isoperimetric planar curves minimizing length with respect to a metric having conformal factor that is degenerate at two points, such that the curve encloses a specified amount of Euclidean area. These curves, appropriately parametrized, emerge as traveling waves for a bi-stable Hamiltonian system that can be viewed as a conservative model for phase transitions.

*Abstract:*

This talk will be devoted to probabilistic constructions appearing in statistics and geometry. I will introduce the classical notion of VC dimension and discuss how it arises naturally in several problems. One of the questions will be the so-called epsilon-approximation problem. That is, how well what you see in a small random sample approximates the real structure. In the last part of the talk, I will explain how a clever deterministic choice of points may improve standard guarantees provided by the random sampling.

*Abstract:*

Lagrangian Floer cohomology is notoriously hard to compute, and is typically only possible in special cases. I describe some recent results on how one can compute Lagrangian Floer cohomology when $L$ is a product in a non-trivial symplectic fiber bundle. I will then discuss inroads for the case of a non-trivially fibered $L$. For the beginning of the talk, I will assume very little background knowledge of symplectic geometry.

*Abstract:*

Stable commutator length (scl) is a well established invariant of group elements g (write scl(g)) and has both geometric and algebraic meaning. Many classes of "non-positively curved" groups have a gap in stable commutator length: This is, for every non-trivial element g, scl(g) > C for some C > 0.

One method to obtain 1/2-gaps is by mapping the group to a free group via homomorphisms. We will show that in fact one may take a generalisation of homomorphisms (letter-quasimorphisms) to obtain this bound, in particular for some non-residually free groups.

As an application we see that the scl-gap for right-angled Artin groups and the fundamental groups of special cube complexes is exactly 1/2.

*Abstract:*

NOTE THE SPECIAL TIME AND PLACE.

This is not a mathematics or a physics talk but it is a talk about mathematicians for mathematicians and physicists. Too often, we think of historical mathematicians as only names assigned to theorems. With vignettes and anecdotes, I'll convince you they were also human beings and that, as the Chinese say, "May you live in interesting times" really is a curse. Among the mathematicians with vignettes are Riemann, Newton, Poincare, von Neumann, Kato, Loewner, Krein and Noether.

*Abstract:*

The purpose of this talk is two-fold.1) I will present an extremely simple self-contained proof of the celebrated Monge-Kantorovich theorem in the discrete setting of Optimal Transport. It can be taught in less than one hour to first-year math students!2) I will discuss an application taken from my 1986 paper with Coron and Lieb on Liquid Crystals, which provides an explicit formula for the least energy required to produce a configuration with assigned topological defects. If time permits I will mention recent developments, joint with P. Mironescu and I. Shafrir, concerning the energy required to pass from a given configuration to another one.

*Abstract:*

While originated in topological data analysis, persistence modules and their barcodesprovide an efficient way to book-keep homological information contained in Morse and Floer theories.I shall describe applications of persistence barcodes to symplectic topology and geometry of Laplace eigenfunctions.Based on joint works with Iosif Polterovich, Egor Shelukhin and Vukasin Stojisavljevic.

*Abstract:*

A K\"ahler group is a group that can be realized as fundamental group of a compact K\"ahler manifold. I shall start by explaining why the question which groups are K\"ahler groups is non-trivial. Then we will address the question which functions can be realized as Dehn functions of K\"ahler groups. After explaining why K\"ahler groups can have linear, quadratic and exponential Dehn function, we show that there is a K\"ahler group with Dehn function bounded below by $n^3$ and bounded above by $n^6$. This is joint work with Romain Tessera.

*Abstract:*

I will talk about the critical exponent associated to an invariant random subgroup of a rank one simple Lie group G. We show that this critical exponent is greater than 1/2(dim(G/K)-1), and moreover the critical exponent is precisely dim(G/K)-1 if the IRS is almost surely of divergence type. This can be viewed as a generalization of Kesten's theorem for IRS in G. Whenever G has Kazhdan's property (T) it follows that an ergodic IRS of divergence type is a lattice. Most of our results hold true more generally for IRS in the isometry group of any Gromov hyperbolic metric space.This is a joint work with Ilya Gehktman."

*Abstract:*

This will be the fourth talk in Adam's lecture series.

*Announcement:*

**דר' אריאל לייטנר**

הפקולטה למתמטיקה

טכניון

**Dr. Arielle Leitner**

The Faculty of Mathematics

Technion

**Math Club 30.5.18**

**סיור וירטואלי של גאומטריות**

נחקור איך נראה העולם בגאומטריות אחרות בעזרת משחקים והדמיות במחשב. נראה את הגאומטריות של ת׳רסטון, ונדבר על משפט הגאומטריזציה המפורסם. אם יהיה זמן, נדבר גם על גבולות בין גאומטריות.

**A Virtual Tour of Geometries**

We will explore what it is like to live in different kinds of geometric universes with the aid of computer visualizations and games. We will see some of the Thurston geometries, and discuss the famous Thurston geometrization program. Time permitting, we will discuss how some of these geometries may be deformed to others.

**ההרצאה תהיה בעברית**

**The lecture will be in Hebre**

*Abstract:*

See attached file.

*Abstract:*

**Advisor: **Roy Meshulam

**Abstract**: Attached

*Abstract:*

Let P be a second-order, symmetric, and nonnegative elliptic operator with real coefficients defined on noncompact Riemannian manifold M, and let V be a real valued function which belongs to the class of small perturbation potentials with respect to the heat kernel of P in M. We prove that under some further assumptions (satisfying by a large classes of P and M) the positive minimal heat kernels of P −V and of P on M are equivalent. If time permits we shall show that the parabolic Martin boundary is stable under such perturbations. This is a joint work with Prof. Yehuda Pinchover.

*Abstract:*

The evens and odds form a partition of the integers into arithmetic progressions. It is natural to try to describe in general how the integers can be partitioned into arithmetic progressions. For example, a classic result from the 1950's shows that if a set of arithmetic progressions partitions the integers, there must be two arithmetic progressions with the same difference. Another direction is to try to determine when such a partition is a proper refinements of another non-trivial partition.

In my talk I will give some of the more interesting results on this subject, report some (relatively) new results and present two generalizations of partitioning the integers by arithmetic progressions, namely:

1. Partitions of the integers by Beatty sequences (will be defined).

2. Coset partition of a group.

The main conjecture in thefirst topic is due to A. Fraenkel and describes all the partitionshaving distinct moduli. The main conjecture in the second topic, dueto M. Herzog and J. Schonheim, claims that in every coset partition of a group there must be two cosets of the same index.

Again, we will briefly discuss the history of these conjectures, recall some of the main results and report some new results.

Based on joint projects with Y. Ginosar, L. Margolis and J. Simpson.

*Abstract:*

In this talk we examine the regularity theory of the solutions to a few examples of (nonlinear) PDEs. Arguing through a genuinely geometrical method, we produce regularity results in Sobolev and Hölder spaces, including some borderline cases. Our techniques relate a problem of interest to another one - for which a richer theory is available - by means of a geometric structure, e.g., a path. Ideally, information is transported along such a path, giving access to finer properties of the original equation. Our examples include elliptic and parabolic fully nonlinear problems, the Isaacs equation, degenerate examples and a double divergence model. We close the talk with a discussion on open problems and further directions of work.

*Abstract:*

Bass-Serre theory is a useful tool to study groups which acts on simplicial trees by isometries. In this talk I discuss group actions on quasi-trees. A quasi-tree is a geodesic metric space which is quasi-isometric to a simplicial tree. I discuss an axiomatic method to produce group actions on quasi-trees for a given group. Quasi-trees are more flexible than trees, and it turns out that a large family of finitely generated groups have non-trivial actions on quasi-trees. I also describe applications once we obtain such actions. This is a survey talk on a joint work with Bestvina and Bromberg.

*Abstract:*

The third talk in Adam's lecture series, presenting his joint work with Davidson and Li

https://arxiv.org/abs/1709.06637

*Abstract:*

Let $\Sigma$ be a Riemann surface of genus $g \geq 2$, and p be a point on $\Sigma$. We define a space $S_g(t)$ consisting of certain irreducible representations of the fundamental group of $\Sigma \setminus p$, modulo conjugation by SU(n).This space has interpretations in algebraic geometry, gauge theory and topological quantum field theory; in particular if Σ has a Kahler structure then $S_g(t)$ is the moduli space of parabolic vector bundles of rank n over Σ. For n=2, Weitsman considered a tautological line bundle on $S_g(t)$, and proved that the (2g)^th power of its first Chern class vanishes, as conjectured by Newstead.

In this talk I will outline my extension of his work to SU(n) and to SO(2n+1).

*Abstract:*

Computing homeomorphisms between surfaces is an important task in shape analysis fields such as computer graphics, medical imaging and morphology. A fundamental tool for these tasks is solving Dirichlet’s problem on an arbitrary Jordan domain with disc topology, where the boundary of the domain is mapped homeomorphically to the boundary of a specific target domain: A convex polygon. By the Rado-Kneser-Choquet Theorem such harmonic mappings are homeomorphisms onto the convex polygon. Standard finite element approximations of harmonic mappings lead to discrete harmonic mappings, which have been proven to be homeomorphisms as well. Computing the discrete harmonic mappings is very efficient and reliable as the mappings are obtained as the solution of a sparse linear system of equations.

In this talk we show that the methodology above, can be used to compute *conformal* homeomorphisms, both for planar and sphere-type domains:

By solving Dirichlet’s problem with correct boundary conditions, we can compute conformal homeomorphisms from arbitrary Jordan domains to a specific canonical domain- a triangle. The discrete conformal mappings we compute are homeomorphisms, and approximate the conformal homeomorphism uniformly and in H^1. Furthermore we show that this methodology can also be used to conformally map a sphere type surface to a planar Jordan domain, whose edges are identified so that the planar domain has the topology of a sphere.

*Abstract:*

ALL TALKS WILL BE HELD AT AMADO 232

Speakers and schedule :

09:30-10:00 Coffee and refreshments at the 8-th floor lounge

10:00-10:50 : Tali Pinsky (Technion Mathematics Department)

10:50-11:10 : Coffee break

11:10:-12:00 : Anish Ghosh (Tata Institute of Fundamental Research)

12:00-14:00 : Lunch

14:00-14:50 : Konstantin Golubev (Bar Ilan and Weizmann Institute)

14:50-15:10 : Coffee break

15:10-16:00 : Sanghoon Kwon (Korea Institute for Advanced Study)

TITLES AND ABSTRACTS

1) Tali Pinsky :

Title: An upper bound for volumes of geodesics

Abstract: Consider a closed geodesic gamma on a hyperbolic surface S, embedded in the unit tangent bundle of S. If gamma is filling its complement is a hyperbolic three manifold, and thus has a well defined volume. I will discuss how to use Ghys' template for the geodesic flow on the modular surface to obtain an upper bound for this volume in terms of the length of gamma. This is joint work with Maxime Bergeron and Lior Silberman.

2) Anish Ghosh :

Title: The metric theory of dense lattice orbits

Abstract: The classical theory of metric Diophantine approximation is very well developed and has, in recent years, seen significant advances, partly due to connections with homogeneous dynamics. Several problems in this subject can be viewed as particular examples of a very general setup, that of lattice actions on homogeneous varieties of semisimple groups. The latter setup presents significant challenges, including but not limited to, the non-abelian nature of the objects under study. In joint work with Alexander Gorodnik and Amos Nevo, we develop the first systematic metric theory for dense lattice orbits, including analogues of Khintchine's theorems.

3) Konstantin Golubev :

Title: Density theorems and almost diameter of quotient spaces

Abstract: We examine the typical distance between points in various quotient spaces. This question has an interesting approach inspired by the work of Lubetzky and Peres. They showed that the random walk on a graph expresses under the assumption of the graph being Ramanujan. We show that this condition can be relaxed to some density condition on the eigenvalues, and apply it to various settings. Joint work with Amitay Kamber.

4) Sanghoon Kwon :

Title: A combinatorial approach to the Littlewood conjecture in positive characteristic

Abstract: The Littlewood conjecture is an open problem in simultaneous Diophantine approximation of two real numbers. Similar problem in a field K of formal series over finite fields is also still open. This positive characteristic version of problem is equivalent to whether there is a certain bounded orbit of diagonal semigroup action on Bruhat-Tits building of PGL(3,K). We describe geometric properties of buildings associated to PGL(3,K), explore the combinatorics of the diagonal action on it and discuss how it helps to investigate the conjecture.

*Abstract:*

T.B.A.

*Abstract:*

The group of Hamiltonian diffeomorphisms of a symplectic manifold is an infinite dimensional Lie group, and its homotopy type is only know in a few special cases.

In this talk I will show how the fundamental group of the group of Hamiltonian diffeomorphisms of a symplectic manifold changes when one point of the symplectic manifold is blown up.

*Abstract:*

We present an elementary argument that establishes a characterization of simple complex Lie groups (among all connected, simple Lie groups of finite center) in terms of their degree-three continuous cohomology. On our way to the result, we will give some background on continuous cohomology and discuss some links to Burger--Monod's theory of continuous bounded cohomology.

*Abstract:*

(This is the second lecture in a series of lectures)

By a result of Glimm, we know that classifying representations of non-type-I $C^*$-algebras up to unitary equivalence is a difficult problem. Instead of this, one either restricts to a tractable subclass or weakens the invariant. In the theory of free semigroup algebras, initiated by Davidson and Pitts, classification within the subclasses of atomic and finitely correlated representations of Toeplitz-Cuntz algebras can be achieved.In this talk we will sketch the proof of a classification theorem for atomic representations for Toeplitz-Cuntz-\emph{Krieger} algebras, generalizing the one by Davidson and Pitts. Furthermore, we will explain how the famous road coloring theorem, proved by Trahtman, gives us a large class of directed graphs for which the free semigroupoid algebra is in fact self-adjoint. Time permitting, we will start working our way towards classification of free semigroupoid algebras.

*Abstract:*

Superhydrophobic surfaces, formed by air entrapment within the cavities of a hydrophobic solid substrate, offer a promising potential for drag reduction in small-scale flows. It turns out that low-drag configurations are associated with singular limits, which to date have typically been addressed using numerical schemes. I will discuss the application of singular perturbations to several of the canonical problems in the field.

*Abstract:*

**Advisor**: Shai Haran

**Abstract**: We discuss the notion of a non-reduced arithmetic plane of the integers Z tensored with itself over F_1, the field with one element. It is a coproduct object in the category FR_c, which is a strictly larger category than the category of commutative rings (but there is an fully- faithfull embedding). We study its combinatorics, algebraic properties and show a connection with commutative rings.

*Abstract:*

We discuss joint work with Douglas Arnold, Guy David, Marcel Filoche and Svitlana Mayboroda. Consider the Neumann boundary value problem for the operator

L u = -div(A\nabla u) + V u

on a Lipschitz domain and, more generally, on a manifold with or without boundary. The eigenfunctions of L are often localized, as a result of disorder of the potential V , the matrix of coefficients A, irregularities of the boundary, or all of the above. In earlier work, Filocheand Mayboroda introduced the function u solving Lu = 1, and showed numerically that it strongly reflects this localization. Here, we deepen the connection between the eigenfunctions and this landscape function u by proving that its reciprocal 1/u acts as an effective potential.The effective potential governs the exponential decay of the eigenfunctions of the system and delivers information on the distribution of eigenvalues near the bottom of the spectrum.

*Announcement:*

פרופ' אמריטוס אברהם ברמן

הפקולטה למתמטיקה

טכניון

**Prof. Emeritus Abraham Berman**

The Faculty of Mathematics

Technion

**Math Club 2.5.18**

**המתמטיקה של דרוג אתרים ומספרים בקודקודים של גרף**

נתחיל את ההרצאה בחידה המצורפת כתמונה מטה. נמשיך בדיון בדרוג אתרים ונתאר את אחד המשפטים החשובים בתורת המטריצות - משפט פרון-פרובניוס. נסביר את הקשר בין המשפט לבין PageRank של Google ובין הכללה של החידה למספרים שנמצאים בקודקודים של גרף.

ההרצאה מבוססת על אלגברה לינארית בסיסית ואין צורך לפתור את החידה כדי להבין אותה.

PageRank, a game of numbers and the Perron-Frobenius Tehorem

In the talk we will describe the Perron-Frobenius Theorem on non-negative matrices and present applications of the theorem to a game of numbers on vertices of a graph and to Google’s PageRank algorithm.

**ההרצאה תהיה בעברית**

**The lecture will be in Hebre**

*Abstract:*

Experiments measuring currents through single protein channels show unstable currents, a phenomena called the gating of a single channel. Channels switch between an `open' state with a well defined single amplitude of current and ?closed? states with nearly zero current. The existing mean-field theory of ion channels focuses almost solely on the open state. The theoretical modeling of the dynamical features of ion channels is still in its infancy, and does not describe the transitions between open and closed states, nor the distribution of the duration times of open states. One hypothesis is that gating corresponds to noise-induced fast transitions between multiple steady (equilibrium) states of the underlying system. Particularly, the literature focuses on the steric Poisson-Nernst-Planck model since it has been successful in predicting permeability and selectivity of ionic channels in their open state, and since it gives rise to multiple steady states.In this work, we show that the PNP-steric equation is ill-posed in the parameter regime where multiple solutions arise. Following these findings, we introduce a novel PNP-Cahn-Hilliard model that is well-posed and admits multiple stationary solutions that are smooth and stable. We show that this model gives rise to a gating-like behavior, but that important features of this switching behavior are different from the defining features of gating in biological systems. Furthermore, we show that noise prohibits switching in the system of study. The above phenomena are expected to occur in other PNP-type models, strongly suggesting that one has to go beyond over-damped (gradient flow) Nernst-Planck type dynamics to describe spontaneous gating of single channels.Joint work with Chun Liu and Bob Eisenberg

*Abstract:*

T.B.A.

*Abstract:*

This is the second of two talks concerning invariants of families of self-adjoint elliptic boundary value problems on a compact surface. With each such family, parameterized by points of a compact topological space $X$, one can associate an invariant reflecting the analytical and the spectral properties of the family. This invariant is called the analytical index. It takes values in the Abelian group $K^1(X)$, the definition of which I will also give in the talk. I will present the index theorem, which expresses the analytical index in terms of the topological data extracted from the family of boundary value problems.

*Abstract:*

We study obstructions to symplectically embedding a cube (a polydisk with all factors equal) into another symplectic manifold of the same dimension. We find sharp obstructions in many cases, including all "convex toric domains" and "concave toric domains" in C^n. The proof uses analogues of the Ekeland-Hofer capacities, which are conjecturally equal to them, but which are defined using positive S^1-equivariant symplectic homology.

This is joint work with Michael Hutchings.

*** Please note the special day and time. ***

*Abstract:*

The Keller-Segel system in two dimensions represents the evolution of living cells under self-attraction and diffusive forces. In its simplest form, it is a conservative drift-diffusion equation for the cell density coupled to an elliptic equation for the chemo-attractant concentration. It is known that in two space dimension there is a critical mass $\beta_c$ such that for initial mass $\beta \leq \beta_c$ there is global in time existence of solutions while for $\beta>\beta_c$ finite time blow-up occurs. In the sub-critical regime $(\beta < \beta_c),$ the solutions decay as time $t$ goes to infinity, while such solution concentrate, as $t$ goes to infinity for the critical initial mass $(\beta=\beta_c).$ In the sub-critical case, this decay can be resolved by a steady, self-similar solution, while no such self-similar solution is known to exist in the critical case.Motivated by the Keller-Segel system of several interacting populations, we studied the existence/non-existence of steady states in the self-similar variables,when the system has an additional drift for each component decaying in time at the rate $O(1/\sqrt{t}).$ Such steady states satisfy a modified Liouville's system with a quadratic potential.In this presentation, we will discuss the conditions for existence/non-existence of solutions of such Liouville’s systems, which,in turn, is related to the existence/non-existence of minimizers to a corresponding Free Energy functional.This a joint work with Prof. Gershon Wolansky (arXiv:1802.08975).

*Abstract:*

This is the first lecture in a special lecture series by professor Alex Kontorovich

organized by the CMS.

*Abstract:*

In this talk we present a systematic study of regular quasi-nonexpansive operators in Hilbert space. We are interested, in particular, in weakly, boundedly and linearly regular operators. We show that the type of the regularity is preserved under relaxations, convex combinations and products of operators. Moreover, in this connection, we show that weak, bounded and linear regularity lead to weak, strong and linear convergence, respectively, of various iterative methods. This applies, in particular, to projection methods, which oftentimes are based on the above-mentioned algebraic operations applied to projections. This is joint work with Andrzej Cegielski and Simeon Reich.

*Abstract:*

Ultra-low power processors designed to work at very low voltage are the enablers of the internet of things (IoT) era. Their internal memories, which are usually implemented by a static random access memory (SRAM) technology, stop functioning properly at low voltage. Some recent commercial products have replaced SRAM with embedded memory (eDRAM), in which stored data are destroyed over time, thus requiring periodic refreshing that causes performance loss. We presents a queuing-based opportunistic refreshing algorithm that eliminates most if not all of the performance loss and is shown to be optimal. The queues used for refreshing miss refreshing opportunities not only when they are saturated but also when they are empty, hence increasing the probability of performance loss. We examine the optimal policy for handling a saturated and empty queue, and the ways in which system performance depends on queue capacity and memory size. This analysis results in a closed-form performance expression capturing read/write probabilities, memory size and queue capacity leading to CPU-internal memory architecture optimization.

*Abstract:*

(Joint with Jessica Purcell) We prove that if knots in $S^3$ are ''sufficiently'' complicated then they have unique! representations as diagrams. This suggests a new way to enumerate knots.

*Abstract:*

Non-Euclidean, or incompatible elasticity is an elastic theory for bodies that do not have a reference (stress-free) configuration. It applies to many systems, in which the elastic body undergoes inhomogeneous growth (e.g. plants, self-assembled molecules). Mathematically, it is a question of finding the "most isometric" immersion of a Riemannian manifold (M,g) into Euclidean space of the same dimension, by minimizing an appropriate energy functional.Much of the research in non-Euclidean elasticity is concerned with elastic bodies that have one or more slender dimensions (such as leaves), and finding appropriate dimensionally-reduced models for them.In this talk I will give an introduction to non-Euclidean elasticity, and then focus on thin bodies and present some recent results on the relations between their elastic behavior and their curvature.Based on a joint work with Asaf Shachar.

*Abstract:*

A ("directed") lattice path is a word (a_1, ..., a_n) over an alphabet S, a prechosen set of integer numbers. It is visualized as a polygonal line which starts at the origin and consists of the vectors (1, a_i), i=1..n, appended to each other. Well-known examples include Dyck paths, Motzkin paths, etc. In 2002, Banderier and Flajolet developed a systematic study of lattice paths by means of analytic combinatorics. In particular, they found general expressions for generating functions for several classes of lattice paths ("walks", "bridges", "meanders", and "excursions") over S. We extend and refine the study of Banderier and Flajolet by considering lattice paths that avoid a "pattern" – a fixed word p. We obtain expressions that generalize those from the work by Banderier and Flajolet. Our results unify and include numerous earlier results on lattice paths with forbidden patterns (for example, UDU-avoiding Dyck paths, UHD-avoiding Motzkin paths, etc.) Our main tool is a combination of finite automata machinery with a suitable vectorial extension of the so-called kernel method.

*Abstract:*

With every family of self-adjoint Fredholm operators on a Hilbert space one can associate an invariant reflecting the analytical and the spectral properties of the family. In the case of a one-parameter family, the corresponding invariant is integer-valued and is called the spectral flow. It can be defined as the net number of eigenvalues of the operaor passing through zero with the change of parameter. In the general case, for a family parameterized by points of a compact space $X$, the corresponding invariant takes values in the Abelian group $K^1(X)$ and is called the family index. I intend to give two talks concerning the computation of these invariants for families of self-adjoint elliptic boundary value problems on a compact surface. In the first talk I will explain how to compute the spectral flow using the topological data extracted from a given one-parameter family of boundary value problems. The talk is based on the preprint arXiv:1703.06105 (math.AP). In the second talk I will show how this result can be generalized to an arbitrary base space $X$.

*Abstract:*

A group G is called *bounded* if every biinvariant metric on G has finite diameter. If G is generated by finitely many conjugacy classes then G is bounded if every biinvariant word metric has finite diameter. In this case the diameter (of course) depends on the choice of a generating set and this is where things become subtle. I will discuss these subtleties (examples: SL(n,Z), some cocompact lattices, Ham(M,w)) and present applications to finite simple groups and Hamiltonian group actions on symplectic manifolds (example: the automorphism group of a regular tree of valence at least three does not admit a faithful Hamiltonian action on a closed symplectic manifold).

Joint work with Assaf Libman and Ben Martin.

*Abstract:*

(This is the first in a series of several talks)

By a result of Glimm, we know that classifying representations of non-type-I $C^*$-algebras up to unitary equivalence is a difficult problem. Instead of this, one either restricts to a tractable subclass or weakens the invariant. In the theory of free semigroup algebras, initiated by Davidson and Pitts, classification of atomic and finitely correlated representations of Toeplitz-Cuntz algebras can achieved.

In this first talk, we introduce free semigroupoid algebras and discuss generalizations of the above results to representations of Toeplitz-Cuntz-*Krieger* algebras associated to a directed graph $G$. We prove a classification theorem for atomic representations and explain a classification theorem for finitely correlated representations due to Fuller. Time permitting, we will explain how the famous road coloring theorem, proved by Trahtman, gives us a large class of directed graphs for which the free semigroupoid algebra is in fact self-adjoint.

*Abstract:*

The celebrated Shnol theorem [4] asserts that every polynomially bounded generalized eigenfunction

for a given energy E 2 R associated with a Schrodinger operator H implies that E is

in the L2-spectrum of H. Later Simon [5] rediscorvered this result independently and proved

additionally that the set of energies admiting a polynomially bounded generalized eigenfunction

is dense in the spectrum. A remarkable extension of these results hold also in the Dirichlet

setting [1, 2].

It was conjectured in [3] that the polynomial bound on the generalized eigenfunction can be

replaced by an object intrinsically dened by H, namely, the Agmon ground state. During

the talk, we positively answer the conjecture indicating that the Agmon ground state describes

the spectrum of the operator H. Specically, we show that if u is a generalized eigenfunction

for the eigenvalue E 2 R that is bounded by the Agmon ground state then E belongs to the

L2-spectrum of H. Furthermore, this assertion extends to the Dirichlet setting whenever a

suitable notion of Agmon ground state is available.

*Abstract:*

*Was sind und was sollen die Zahlen?* (roughly: “What are numbers and what should they be?”) is the title of a booklet first published in 1888, where Richard Dedekind introduced his definition of the system of natural numbers. This definition was based on the concept of “chains” (*Kette*), and it appeared in roughly at the same time than that, better known one, of Peano. In another booklet published for the first time in 1872 and entitled *Stetigkeit und irrationale Zahlen* (“Continuity and Irrational Numbers”), Dedekind introduced his famous concept of “cuts” as the key to understanding the issue of continuity in the system of real numbers, and through it, the question of the foundations of analysis.At roughly the same time, Cantor published his own work dealing with the same question. In his work on domains of algebraic integers, published in various versions between 1872 and 1894, Dedekind crucially introduced the concept of “ideal”, on the basis of which he approached the issue of unique factorization. At that time, Kronecker published his own work dealing, from a rather different perspective, with exactly the same issue.

From a contemporary perspective, these three concepts of Dedekind (chains, cuts, ideals) seem to belong to different mathematical realms and to address different kinds of mathematical concerns. From Dedekind’s perspective, however, they arose from a single concern about the nature of the idea of number in general. In this talk I will explain the mathematical meaning of these concepts, the historical context where they arose, the deep underlying methodological unity that characterized Dedekind’s conceptual approach, and the significant impact they had on mathematics at large at the beginning of the twentieth century.

*Abstract:*

Given two closed embedded curves on a surface we say that they are at distance one if they intersect at two points or less. This defines a metric on a family of loops by considering the shortest chain of elements at distance one. By choosing various surfaces and families of curves one can obtain metric spaces with very diverse and rich geometry.

Despite the elementary construction this metric seems to be nearly unexplored. At the same time it is related to some important metrics (e.g. Hofer's metric, fragmentation metric, etc) on groups of diffeomorphisms of the surface. I will discuss [few] examples where the geometry is understood and will describe [lots of] those where nothing is known.

No symplectic preliminaries are assumed for this talk.

*Abstract:*

Delone sets in a metric space are point sets in which there is a minimal distance between points and which at the same time admits gaps of bounded size only.

With additional analytic and geometric data, one naturally obtains bounded, linear operators modeling quantum mechanical phenomena. In the realm of locally compact, second countable groups, we study the continuity behaviour of the spectral distribution of such operators with respect to the underlying geometry. We show how convergence of dynamical systems implies convergence of the density of states measure in the weak-*-topology.

Joint work with Siegfried Beckus.

*Abstract:*

In non-commutative probability there are several well known notions of independence. In 2003, Muraki's classification, which states that there are exactly five independences coming from universal (natural) products, seemingly settled the question of what independences can be considered. But after Voiculescu's invention of bi-free independence in 2014, the question came up again. The key idea that allows to define a new notion of independence with all the features of the universal independences that appear in Muraki's classification is to consider ``two-faced'' (i.e. pairs of) random variables.In the talk, we define bi-monotone independence, a new example of an independence for two-faced random variables. We establish a corresponding central limit theorem and use it to describe the joint distribution of monotone and antimonotone Brownian motion on monotone Fock space, which yields a canonical example of a quantum stochastic process with bi-monotonely independent increments.

*Abstract:*

Can one hear the shape of a drum? In mathematical terms this famous question of M. Kac asks whether two unitarily equivalent Laplacians live on the same geometric object. It is now known that the answer to this question is negative in general.Following an idea of Wolfgang Arendt, we replace the unitary transformation intertwining the Laplacians by an order preserving one and then ask how much of the geometry is preserved. In this situation the associated semigroups, which encode diffusion, are equivalent up to an order isomorphism. Therefore, the question becomes as stated in the title and we try to provide an answer in great generality. In particular, we discuss the situation for graph Laplacians and Laplacians on metric measure spaces. (this is joint work with Matthias Keller, Daniel Lenz and Melchior Wirth)

*Abstract:*

We consider ordinary differential equations of arbitrary order up to differentiable changes of variables. It turns out that starting from 2^{nd}order ODEs there exist continuous differential invariants that are preserved under arbitrary changes of variables. This was first discovered by Sophus Lie and explored in detail by A. Tresse for 2^{nd} order ODEs. However, its was E. Cartan who first understood the geometric meaning of these invariants and related them to the projective differential geometry. We outline further advances in the equivalence theory of ODEs due to S.-S. Chern (3^{rd} order ODEs) and R. Bryant (4^{th} order ODEs) and present the general solution for arbitrary (systems of) ODEs of any order. It is based on the techniques of so-called nilpotent differential geometry and cohomology theory of finite-dimensional Lie algebras. It is surprising that a part of the invariants can be understood in purely elementary way via the theory of linear ODEs and leads to classical works of E.J.Wilczynsky back to the beginning of 20^{th} century.

*Abstract:*

The problem of minimization of a separable convex objective function has various theoretical and real-world applications. One of the popular methods for solving this problem is the proximal gradient method (proximal forward-backward algorithm). A very common assumption in the use of this method is that the gradient of the smooth term in the objective function is globally Lipschitz continuous. However, this assumption is not always satisfied in practice, thus casting a limitation on the method. We discuss, in a wide class of finite and infinite-dimensional spaces, a new variant (BISTA) of the proximal gradient method which does not impose the above-mentioned global Lipschitz continuity assumption. A key contribution of the method is the dependence of the iterative steps on a certain decomposition of the objective set into subsets. Moreover, we use a Bregman divergence in the proximal forward-backward operation. Under certain practical conditions, a non-asymptotic rate of convergence (that is, in the function values) is established, as well as the weak convergence of the whole sequence to a minimizer. We also obtain a few auxiliary results of independent interest, among them a general and usefu lstability principle which, roughly speaking, says that given a uniformly continuous function on an arbitrary metric space, if we slightly change the objective set over which the optimal (extreme) values are computed, then these values vary slightly. This principle suggests a general scheme for tackling a wide class of non-convex and non-smooth optimization problems. This is a joint work with Alvaro De Pierro and Simeon Reich.

*Abstract:*

We outline a technique to prove Central Limit Theorems for various counting functions which naturally appear in the theory of Diophantine approximation.

Joint work with A. Gorodnik (Bristol).

*Abstract:*

We introduce a notion of nodal domains for positivity preserving forms in purely analytical terms. This notion generalizes the classical ones for Laplacians on domains and on graphs. This notion allows us to prove the Courant nodal domain theorem in this generalized setting.

*Abstract:*

We describe some recent musings on various connections between problems in elementary number theory and the Fourier restriction problem in harmonic analysis.

*Abstract:*

All talks will take place in Amado 814.

Schedule:

13:30-14:20 Ami Viselter (Haifa University)

Convolution semigroups on quantum groups and non-commutative Dirichlet forms

14:30-15:20 Michael Skeide (University of Molise)

Interacting Fock Spaces and Subproduct Systems (joint with Malte Gerhold)

15:20-15:50 Coffee break

15:50-16:40 Adam Dor-On (Technion)

C*-envelopes of tensor algebras and their applications to dilations and Hao-Ng isomorphisms

*Abstract:*

**Advisor:** Prof. Simeon Reich

**Abstract: **We develop new iterative methods for solving convex feasibility and common fixed point problems, based on the notion of coherence. We also present new concepts and results in Nonlinear Analysis related to the theory of coherence and Opial's demi-closedness principle. We investigate, in particular, the properties of relaxations, convex combinations and compositions of certain kinds of operators defined on a real Hilbert space, under static and dynamic controls, as well as other properties regarding the algorithmic structure of some operators. Our iterative techniques are applied, for example, to the study of various metric and subgradient projection methods. Furthermore, all the methods are presented in both weak and strong convergence versions.

*Abstract:*

We develop new iterative methods for solving convex feasibility and common fixed point problems, based on the notion of coherence. We also present new concepts and results in Nonlinear Analysis related to the theory of coherence and Opial's demi-closedness principle. We investigate, in particular, the properties of relaxations, convex combinations and compositions of certain kinds of operators defined on a real Hilbert space, under static and dynamic controls, as well as other properties regarding the algorithmic structure of some operators. Our iterative techniques are applied, for example, to the study of various metric and subgradient projection methods. Furthermore, all the methods are presented in both weak and strong convergence versions.

*Abstract:*

Numerous optimization problems are solved using the tools of distributionally robust optimization. In this framework, the distribution of the problem's random parameter $z$ is assumed to be known only partially in the form of, for example, the values of its first moments. The aim is to minimize the expected value of a function of the decision variables $x$, assuming that Nature maximizes this expression using the worst-possible realization of the unknown probability measure of $z$. In the general moment problem approach, the worst-case distributions are atomic. We propose to model smooth uncertain density functions using sum-of-squares polynomials with known moments over a given domain. We show that in this setup, one can evaluate the worst-case expected values of the functions of the decision variables in a computationally tractable way. This is joint work with Etienne de Klerk (TU Delft) and Daniel Kuhn (EPFL Lausanne).

*Abstract:*

**Advisor**: Prof. Udi Yariv

**Abstract**: Surrounded by a spherically symmetric solute cloud, chemically active homogeneous spheres do not undergo conventional autophoresis when suspended in an unbounded liquid domain. When exposed to external flows, solute advection deforms that cloud, resulting in a generally asymmetric distribution of diffusio-osmotic slip which, in turn, modifies particle motion. We illustrate this phoretic phenomenon using two prototypic configurations, one where the particle sediments under a uniform force field and one where it is subject to a simple shear flow. In addition to the Peclet number associated with the imposed flow, the governing nonlinear problem also depends upon the intrinsic Peclet number associated with the chemical activity of the particle. As in the forced-convection problems, the small-Peclet-number limit is nonuniform, breaking down at large distances away from the particle. Calculation of the leading-order autophoretic effects thus requires use of matched asymptotic expansions. We considered two problems: sedimentation and shear problems. In the sedimentation problem we find an effective drag reduction; in the shear problem we find that the magnitude of the stresslet is decreased. For a dilute particle suspension the latter result is manifested by a reduction of the effective viscosity.

*Abstract:*

Real interpolation for the coinvariant subspaces of the shift operator on the circle will be discussed in the first part of the talk.

In the second part it will be shown that, given two closed ideals in a uniform algebra such that the complex conjugate of their intersection

is not included in some of them, the sum of these ideals is not closed.

The problem about nonclosed sums of ideals stems from a detail that emerged during the study of interpolation.

This is a joint work with I. Zlotnikov.

*Abstract:*

Spectral theory for general classes of first order systems has been less popular since 1990's. In this talk, I would like to propose a new class of first order systems which generalize both Maxwell and Dirac equations. In this new class, we can treat these two equations in a unified manner, although their physical backgrounds are very different from each other. The main point of my talk is space-time estimates for the new class of first order systems. The essential part of the idea is to derive uniform boundedness of the spectral densities. This talk is based on joint work with Matania Ben-Artzi.

*Abstract:*

Using technical language, the Navier-Stokes equations with measure initial data (such as "point vortices"), in two (spatial) dimensions, have attracted much mathematical interest in the last twenty years. There are still many basic open problems, such as well-posedness in bounded domains, Hopf bifurcation into time-periodic solutions and many more. The talk will be non-technical, the only expected analytical background is advanced calculus. It will touch on the theoretical aspects as well as the indispensable accompanying numerical simulations. The general topic has fascinated many poets: "Waves, undulaing waves, liquid, uneven, emulous waves... laughing and buoyant" (Walt Whitman). However, the talk will be much more prosaic.

*Abstract:*

About 15 years ago, Bourgain, Brezis and Mironescu proposed a new characterization of BV and W^(1,q) spaces (for q > 1) using a certain double integral functional involving radial mollifiers. We study what happens when one changes the power of |x-y| in the denominator of the integrand from q to 1. It turns out that for q > 1 the corresponding functionals "see" only the jumps of the BV-function. We further identify the function space relevant to the study of these functionals as an appropriate Besov space. We also present applications to the study of singular perturbation problems of Aviles-Giga type.

*Abstract:*

I will describe joint work with Sergei Lanzat.

Tropical geometry provides a new piece-wise linear approach to algebraic geometry. The role of algebraic curves is played by tropical curves - planar metric graphs with certain requirements of balancing, rationality of slopes and integrality. A number of classical enumerative problems can be easily solved by tropical methods. Lately is became clear that a more general approach also makes sense and seem to appear in other areas of mathematics and physics. We consider a generalization of tropical curves, removing requirements of rationality of slopes and integrality and discuss the resulting theory and its interrelations with other areas. Balancing conditions are interpreted as criticality of a certain action functional. A generalized Bezout theorem involves Minkowsky sum and mixed areas. A problem of counting curves passing through an appropriate collection of points turns out to be related to quadratic Plücker relations in Gr(2,4) and some nice Lie algebra. If time permits, we will also discuss new recursive relations for this count (in the spirit of Kontsevich and Gromov-Witten).

*Abstract:*

Commutator length is a group theoretical analogue of genus. By taking a limit, stable commutator length, scl, is obtained. This is a group invariant that can be studied topologically. As scl detects surface subgroups, it is thought to be an important invariant for the study of 3-manifolds, however, there are open questions regarding its computability and its unit norm ball. This talk will give some background on scl in low dimensional topology, and will outline some work in progress of the speaker towards resolving these questions for 3-manifold groups.

***Double feature: please note the special time***

*Abstract:*

Commutator length is a group theoretical analogue of genus. By taking a limit, stable commutator length, scl, is obtained. This is a group invariant that can be studied topologically. As scl detects surface subgroups, it is thought to be an important invariant for the study of 3-manifolds, however, there are open questions regarding its computability and its unit norm ball. This talk will give some background on scl in low dimensional topology, and will outline some work in progress of the speaker towards resolving these questions for 3-manifold groups.

*Abstract:*

The Euler--Poisson equations govern gas motion underself gravitational force. In this context the density is not strictly positive, it vanishes in the vacuum region, or falls off to zero at infinity. That causes a degeneration of the hyperbolic systems.The lecture will discuss local existence theorems under these circumstances and with a polytropic equation of state $p=K\rho^\gamma$, here $p$ is the pressure, $\rho $ the density and $\gamma>1$ is the adiabatic gas exponent. In particular, we shall discuss the question whether the initial data include the static spherical solutions for various values of the adiabatic constant $\gamma$. This is a joint work with U. Brauer, Universidad Complutense Madrid.

See also link to title/abstract.

*Abstract:*

Markoff triples are integer solutions of the equation $x^2+y^2+z^2=3xyz$ which arose in Markoff's spectacular and fundamental work (1879) on diophantine approximation and has been henceforth ubiquitous in a tremendous variety of different fields in mathematics and beyond. After reviewing some of these, we will discuss joint work with Bourgain and Sarnak on the connectedness of the set of solutions of the Markoff equation modulo primes under the action of the group generated by Vieta involutions, showing, in particular, that for almost all primes the induced graph is connected. Similar results for composite moduli enable us to establish certain new arithmetical properties of Markoff numbers, for instance the fact that almost all of them are composite.

Time permitting, we will also discuss recent joint work with Magee and Ronan on the asymptotic formula for integer points on Markoff-Hurwitz surfaces $x_1^2+x_2^2 + \dots + x_n^2 = x_1 x_2 \dots x_n$, giving an interpretation for the exponent of growth in terms of certain conformal measure on the projective space.

*Abstract:*

A zone of width $\omega$ on the unit sphere is defined as the set of points within spherical distance $\omega/2$ of a given great circle. Zones can be thought of as the spherical analogue of planks. In this talk we show that the total width of any (finite) collection of zones covering the unit sphere is at least $\pi$, answering a question of Fejes T\'oth from 1973.This is a joint work with Alexandr Polyanskii.

*Abstract:*

Convex projective manifolds are a generalization of hyperbolic manifolds. They are more flexible, and some occur as deformations of hyperbolic manifolds. Generalized cusps occur naturally as ends of properly convex projective manifolds. We classify generalized cusps, discuss their geometry, and ways they can deform.

Joint work with Sam Ballas and Daryl Cooper.

*Abstract:*

I will discuss the dynamics of light rays in the trihexagonal tiling in the plane where triangles and hexagons are transparent and have equal but opposite indices of refraction. Sometimes this is called a `tiling billiards system.' It turns out that almost every light ray is dense in the plane with a periodic family of disjoint open triangles removed. The proof involves some elementary observations about invariant subspaces, an orbit equivalence to straight-line flow on an infinite periodic translation surface, and use of relatively recent results on ergodic theoretic questions for such flows. Most of the talk will be elementary. This talk is based on joint work with Diana Davis and is available at arXiv:1609.00772.

*Abstract:*

Several forms of wireless communication involve estimating the "channel" or relationship between signal transmitted and received. In this talk we will focus on the RADAR channel. I will first introduce and develop a model for this type of channel, that happens to have a sophisticated underlying algebraic structure. In the new state of the art of wireless signal processing digital signals often have extremely high dimension N>10^6, while the channel is still sparse in a certain sense. In such cases one can significantly outperform the currently used channel estimation algorithms. The main goal of this talk is to introduce you to the underlying structure that helps achieve this.

*Abstract:*

**Advisor**:Uri Peskin

**Abstract:** Hole transport is an important transport mechanism in solid state based electronic devices. In recent years charge transport through biomolecules (such as DNA) is also attributed to hole dynamics and/or kinetics. In this work we study fundamental aspects of quantum hole dynamics in nano-scale system. Using reduced models we follow the many body dynamics of interacting electrons in the presence of a few (one or two) holes, and study the validity of the interpretation of the dynamics in terms of holes dynamics. In this seminar I will describe the models and a new computational algorithm developed in order to solve the many body Schroedinger equation for these models. I will present results which demonstrate intriguing aspects of hole dynamics in small systems, such as transition from hole repulsion to hole attraction induced by changes in the system dimensions, or in the electron-electron interaction parameter. Conclusions with respect to the common interpretation of holes in terms of effective positive charges will be given.

*Abstract:*

See link to abstract

*Abstract:*

Fermat showed that every prime p = 1 mod 4 is a sum of two squares: $p = a^2 + b^2$, and hence such a prime gives rise to an angle whose tangent is the ratio b/a. Do these angles exhibit order or randomness? I will discuss the statistics of these angles and present a conjecture, motivated by a random matrix model and by function field considerations.

*Abstract:*

In 1687 Sir Isaac Newton discovered that the area cut off from an oval in the plane by a straight line never depends algebraically on the line (the question was motivated by celestial mechanics). In 1987 V. I. Arnold proposed to generalize Newton's observation to higher dimensions and conjectured that all smooth bodies, with the exception of ellipsoids in odd-dimensional spaces, have an analogous property. The talk is devoted to the current status of this conjecture.

*Abstract:*

**advisor: **Nir Gavish

**Abstract: **Concentrated electrolytes are an integral part of many electrochemical and biological systems, including ion channels, dye sensitized solar cells, fuel cells, batteries and super-capacitors. Spatiotemporal theoretical formulation for electrolytes goes back to 1890's where Poisson-Nernst-Planck (PNP) framework was originated. Extensive research efforts during the last century attempted to extend the PNP approach to concentrated electrolyte solutions. Nevertheless, recent experimental observations show qualitative features that are beyond the scope of all existing generalized PNP models. These phenomena include bulk self-assembly, multiple-time relaxation, and underscreening, which all impact the interfacial dynamics, and the transport in these systems.

In this talk, we shall present a thermodynamically consistent, unified framework for ternary media with an evolution mechanism based on a gradient flow approach . In contrast with generalized PNP models, the starting point of this work stems from models for ionic liquids with an explicit account of the solvent density. We show that the model captures the aforementioned phenomena together, and by using tools from bifurcation theory reveal their underlying mathematical origin.

*Abstract:*

Generalized complex structures, introduced by Hitchin as a common generalization of complex and symplectic structures on manifolds, found many applications in differential geometry and in physics. They also have some peculiar features, such as the the extended diffeomorphism group (the so-called B-field action), D-branes (submanifolds with additional structure), and several competing notions of a generalized holomorphic map.

I my talk I will show that these generalized geometries and related structures can be naturally described and studied in the super-geometric context (i.e. by introducing anti-commuting coordinates) and how this description helps to elucidate the above peculiarities.

*Abstract:*

Let G be a group and let r(n,G) denote the number of its n-dimensional complex irreducible representations up to isomorphism. Representation growth is a branch of asymptotic group theory that studies the asymptotic and arithmetic properties of the sequence (r(n,G)). Whenever the sequence grows polynomially it defines a Dirichlet generating function that converges on some right half-plane and known as the representation function of G. In this talk I will give an overview on the subject, describe some recent developments and mention some open problems.

*Abstract:*

Function fields of genus 0 are of interest in the study of many questions regarding polynomials and rational functions. We use group and field theoretic results to determine the subfields of genus 0 in extensions of large degree with symmetric or alternating Galois group. As time permits we shall describe the applications towards a question of Ritt concerning decompositions of rational functions, and questions concerning reducibility of bivariate polynomials.

*Abstract:*

**Advisor**: Danny Neftin

**Abstract:** Function fields of genus 0 are of interest in the study of many questions regarding polynomials and rational functions. We use group and field theoretic results to determine the subfields of genus 0 in extensions of large degree with symmetric or alternating Galois group. As time permits we shall describe the applications towards a question of Ritt concerning decompositions of rational functions, and questions concerning reducibility of bivariate polynomials.

*Announcement:*

פרופ' משה ברוך

הפקולטה למתמטיקה

טכניון

**Prof. Moshe Baruch**

The Faculty of Mathematics

Technion

**Math Club 9.1.18**

**אנא שימו לב לשעת ההרצאה הלא שגרתית**

אחרי ההרצאה יתקיים טקס הענקת פרסים של התחרות ע"ש גרוסמן

**סכום של ריבועים**

אילו מספרים שלמים הם סכום של ארבעה ריבועים של שלמים? אילו הם סכום של שלושה ריבועים? אילו שלמים הם סכום של שני ריבועים ועוד 10 פעמים ריבוע? את השאלה האחרונה שאל רמנוג'ן בשנת 1917 והיא שאלה פתוחה גם היום.

**Sum of squares**

Which integers are sums of four squares of integers? Which are sums of three squares? Which integers are sums of two squares and 10 times a square? The last question was asked by Ramanujan in 1917 and is still open today.

**ההרצאה תהיה בעברית**

**The lecture will be in Hebrew**

*Abstract:*

(Joint with Jessica Purcell) We prove that if knots in $S^3$ are ''sufficiently'' complicated then they have unique! representations as diagrams. This suggests a new way to enumerate knots.

*Abstract:*

Six years ago, I formulated a conjecture that relates a quantum knot invariant (the degree of the colored Jones polynomial) with a classical topological invariant (a boundary slope of an incompressible surface). We will review old and recent results on this conjecture, and its relations with quadratic integer programming which appears on thequantum side, whereas a linear shadow of it appers on the classical side.

*Abstract:*

Zeta functions associated to groups and rings are natural non-commutative generalizations of the Riemann and Dedekind zeta functions. I will give an overview of the subject and describe some recent developments, with an emphasize on pro-isomorphic and representation zeta functions.

*Abstract:*

Let G be a finite group. A theorem of Deligne implies that Rep-G, considered as a symmetric monoidal category, determines G. The claim is not true when we consider Rep-G only as a monoidal category (without the symmetric structure). Etingof and Gelaki called two finite groups G_1 and G_2 isocategorical if Rep-G_1 and Rep-G_2 are equivalent as monoidal categories. They also gave a characterization of isocategorical groups. To put it in other words: for a given symmetric monoidal category C (which satisfies some properties), there is a correspondence between symmetric structures on C and isomorphism classes of finite groups G for which C is equivalent to Rep-G.

Each symmetric monoidal category gives rise to a sequence of Adams operations, which are operations on the Grothendieck group of C, determined by the symmetric structure of C. In this talk we will discuss the question of to what extent do the Adams operations determine the symmetric structure on C. We will show that the Odd Adams operations are in fact independent of the specific symmetric structure (though this is not clear a-priori from the definition). We will also show that this is not true for the second Adams operation by giving some examples. We will discuss some remain open questions and describe the group of monoidal autoequivalences of Rep-G.

*Abstract:*

In this talk, I present an analogue of the Hardy-Littlewood conjecture on the asymptotic distribution of prime constellations in the setting of short intervals in function fields of smooth projective curves over finite fields. I will discuss the definition of a "short interval" on a curve as an additive translation of the space of global sections of a sufficiently positive divisor E by a suitable rational function f, and show how this definition generalizes the definition of a short interval in the polynomial setting. I will give a sketch of the proof which includes a computation of a certain Galois group, and a counting argument, namely, Chebotarev density type theorem. This is a joint work with Tyler Foster.

**Note there are two cosecutive talks.**

*Abstract:*

The Schrödinger operator $-\Delta + V$ in $R^{N}$ has been extensively studied for potentials in $L^{\infty}$ and even $L^{p}$ with any exponent $p > N/2$.Kato's inequality published in the Israel J. Math. in the 1970s was a major breakthrough in spectral problems by allowing one to consider potentials $V$ that are merely $L^{1}$.We present new counterparts of the strong maximum principle and Hopf's boundary lemma for $-\Delta + V$ on domains when $V$ has a singular behaviour.

Abstract in PDF format attached

*Abstract:*

Given $\lambda\in (0,1)$, consider the distribution of the random series $\sum_{n=0}^\infty \pm \lambda^n$, where the signs are chosen randomly and independently, with probabilities $(\half,\half)$.This is a probability measure on the real line, which can be expressed as an infinite Bernoulli convolution product. These measures have been intensively studied since the mid-1930's, because they arise, somewhat unexpectedly, in many different areas, including harmonic analysis, number theory, and number theory. The case of $\lambda < 1/2$ is simple: we get the classical Hausdorff-Lebesgue measure on a Cantor set of constant dissection ratio and zero length, hence the measure is singular. For $\lambda=1/2$ we get a uniform measure on $[-2,2]$, but the case of $\lambda>1/2$ is very challenging. The basic question is to decide whether the resulting measure is absolutely continuous or singular, which is still open. It was believed at first that since the support of the measure is an the entire interval $[-(1-\lambda)^{-1}, (1-\lambda)^{-1}]$, it should be absolutely continuous. This turned out to be false: P. Erdos showed in 1939 that the measure is singular for $\lambda$ reciprocal of a Pisot number, e.g. for $\lambda$ equal to the golden ratio $0.618...$

Since then, many mathematicians (including the speaker) worked on this problem, and much is known by now, but it is still an open question whether all numbers in $(1/2,1)$, other than reciprocals of Pisot numbers, give rise to absolutely continuous measures. In the last five years a dramatic progress has occurred, after a breakthrough by M. Hochman, followed by important results due to P. Shmerkin and P. Varju.In the first part of the talk I will outline this recent development.

Bernoulli convolution measures can be generalized in various directions, which leads to new interesting problems. In the second part of the talk I will report on the recent work, joint with M. Hochman, on the dimension of stationary (Furstenberg) measures for random matrix products, and time permitting, on a joint work with S. Saglietti and P. Shmerkin on absolute continuity of non-homogeneous self-similar measures with ``overlap''.

*Abstract:*

We focus on nonconvex and nonsmooth minimization problems with a composite objective, where the differentiable part of the objective is freed from the usual and restrictive global Lipschitz gradient continuity assumption. This long-standing smoothness restriction is pervasive in first order methods, and was recently circumvented for convex composite optimization by Bauschke, Bolte and Teboulle, through a simple and elegant framework which captures, all at once, the geometry of the function and of the feasible set. Building on this work, we tackle genuine nonconvex problems. We first complement and extend their approach to derive a full extended descent lemma by introducing the notion of smooth adaptable functions. We then consider a Bregman-based proximal gradient method for the nonconvex composite model with smooth adaptable functions, which is proven to globally converge to a critical point under natural assumptions on the problem's data, and, in particular, for semi-algebraic problems. To illustrate the power and potential of our general framework and results, we consider a broad class of quadratic inverse problems with sparsity constraints which arises in many fundamental applications, and we apply our approach to derive new globally convergent schemes for this class. The talk is based on joint work with Jerome Bolte (Toulouse), Marc Teboulle (TAU) and Yakov Vaisbroud (TAU).

*Abstract:*

After reviewing my work with Vladimir Markovic, constructing nearly geodesic closed surfaces in a given closed hyperbolic 3-manifold, I will describe recent work with Alexander Wright, in which we construct the same kind of object in finite volume (cusped) hyperbolic 3-manifolds. If time permits we will discuss the potential application of these methods to nonuniform lattices in higher rank semisimple Lie groups, and to finding convex cocompact surface subgroups in the mapping class group.

*Abstract:*

I shall review the framework of algebraic families of Harish-Chandra modules, introduced recently, by Bernstein, Higson, and the speaker. Then, I shall describe three of their applications.The first is contraction of representations of Lie groups. Contractions are certain deformations of representations with applications in mathematical physics. The second is the Mackey bijection, this is a (partially conjectural) bijection between the admissible dual of a real reductive group and the admissible dual of its Cartan motion group.The third is the hidden symmetry of the hydrogen atom as an algebraic family of Harish-Chandra modules.

*Abstract:*

A classical problem in number theory is to evaluate the number of primes in an arithmetic progression. This problem can be formulated in terms of the von Mangoldt function. I will introduce some conjectures concerning the fluctuations of the von Mangoldt function in arithmetic progressions. I will also introduce an analogous problem in the function field setting and discuss its generalization to arithmetic functions associated with higher degree L-functions (in the limit of large field size). The main example we will discuss is an elliptic curve L-function and statistics associated with its coefficients. This is a joint work with Chris Hall and Jon Keating.

*Abstract:*

Given a Galois covering over a number field k, Hilbert’s irreducibility theorem guarantees the existence of infinitely many specialization values in k such that the Galois group of the specialization equals the Galois group of the covering. I will consider problems related to the inverse Galois problem which can be attacked using the specialization approach. In particular, the Grunwald problem is a strengthening of the inverse Galois problem, asking about the existence of Galois extensions with prescribed Galois group which approximates finitely many prescribed local extensions. I will explain some of the ideas and difficulties behind solving Grunwald problems via the specialization approach. I will also present some new observations about the structure of the set of all specializations of a Galois covering and about the problem of “specialization-equivalence” of two coverings.

*Abstract:*

This will be the third lecture in which we will study the paper "Non-commutative peaking phenomena and a local version of the hyperrigidity conjecture" by Raphael Clouatre.

*Abstract:*

I shall present two (unrelated) recent applications of caustics, one to lens design and one to visual optics.

*Abstract:*

A family of lines through the origin in Euclidean space is calledequiangular if any pair of lines defines the sameangle. The problem of estimating the maximum cardinality of such afamily in $R^n$ was extensively studied for the last 70years. Answering a question of Lemmens andSeidel from 1973, in this talk we show that for every fixed angle$\theta$ and sufficiently large $n$ there are at most $2n-2$ lines in$R^n$ with common angle $\theta$.Moreover, this is achievable only when $\theta =\arccos \frac{1}{3}$.Various extensions of this result to the more general settings oflines with $k$ fixed angles and of spherical codes will be discussedas well. Joint work with I. Balla, F. Drexler and P. Keevash.

*Abstract:*

The purpose of the talk is to describe some of the main themes, concepts, and challenges in axiomatic set theory.

We will do this by following the study of algebras in set theory which was introduced in the 1960s to investigate infinitary combinatorial problems, and has been part of many fundamental developments in the last decades.

*Abstract:*

The functoriality conjecture is a key ingredient in the theory of automorphic forms and the Langlands program. Given two reductive groups G and H, the principle of functoriality asserts that a map r:G^->H^ between their dual complex groups should naturally give rise to a map r*:Rep(G)->Rep(H) between their automorphic representations. In this talk, I will describe the idea of functoriality, its connection to L-functions and recent work on weak functorial lifts to the exceptional group of type G_2.

*Abstract:*

A model geometry for a finitely generated group is a proper geodesic metric space on which the group acts properly and cocompactly. If two groups have a common model geometry, the Milnor-Schwarz Lemma tells us that the groups are quasiisometric. In contrast, two quasi-isometric groups do not, in general, have a common model geometry.

A simple surface amalgam is obtained by taking a finite collection of compact surfaces, each with a single boundary component, and gluing them together by identifying their boundary curves. We consider the fundamental groups of such spaces and show that commensurability is determined by having a common model geometry. This gives a relatively simple family of groups that are quasi-isometric, but are neither commensurable, nor act on the same common model geometry.

This work is joint with Emily Stark.

*Abstract:*

Abstract: We show that averages on geometrically finite Fuchsian groups, when embedded via a representation into a space of matrices, have a homogeneous asymptotic limit when properly rescaled. This generalizes some of the results of F. Maucourant to subgroups of infinite co-volume.

*Abstract:*

This will be the second of two talks in which we will study the recent preprint "Non-commutative peaking phenomena and a local version of the hyperrigidity conjecture", by Raphael Clouatre. Link:

https://arxiv.org/pdf/1709.01649.pdf

*Abstract:*

We derive sharp eigenvalue asymptotics for Dirichlet-to-Neumann operator in the domain with edges and discuss obstacles for deriving the second term.

*Abstract:*

The group ring first emerged as an auxiliary tool in grouptheory and representation theory at the end of the 19th century and becamean object of interest in itself some decades later. It can be seen as a structurejoining in an elegant manor the algebraic theories on rings and groups and, inthe case of the coefficient ring being the ring of integers, also number theoryenters the picture.

Denoting the group ring of a group G over a ring R by RG, in particularthe group of units of RG and its connection to the structure of G inspired alot of research. The coefficient ring keeping the closest connection to G arethe integers, since they keep the arithmetic information which would be lostwhen one is allowed to divide by some primes.

In this talk I will present basic results and questions about the unit groupof a group ring with special emphasis on finite subgroups of the unit groupof the integral group ring ZG, such as: Is G determined by the group ring? Are the orders of units determined by G? How close are the finite subgroupsof units in ZG to being subgroups of G?

*Abstract:*

A classical problem in geometry goes as follows. Suppose we are given two sets of $D$ dimensional data, that is, sets of points in $R^D$. The data sets are indexed by the same set, and we know that pairwise distances between corresponding points are equal in the two data sets. In other words, the sets are isometric. Can this correspondence be extended to an isometry of the ambient Euclidean space? In this form the question is not terribly interesting; the answer has long been known to be yes (see [Wells and Williams 1975], for example). But a related question is actually fundamental in data analysis: here the known points are samples from larger, unknown sets -- say, manifolds in $R^D$-- and we seek to know what can be said about the manifolds themselves. A typical example might be a face recognition problem, where all we have is multiple finite images of people's faces from various views. An added complication is that in general we are not given exact distances. We have noise and so we need to demand that instead of the pairwise distances being equal, they should be close in some reasonable metric. Some results on almost isometries in Euclidean spaces can be found in [John 1961; Alestalo et al. 2003]. This talk will consist of two parts. I will discuss various works in progress re this problem with Michael Werman (Hebrew U), Kai Diethelm (Braunschweig) and Charles Fefferman (Princeton). As it turns out the problem relates to the problem of Whitney extensions, interpolation in $R^D$ and bounds for Hilbert transforms. Moreover, for practical algorithms there is a natural deep learning framework as well for both labeled and unlabeled data.

*Abstract:*

One of the mainstream and modern tools in the study of non abelian groups are quasi-morphisms. These are functions from a group to the reals which satisfy homomorphism condition up to a bounded error. Nowadays they are used in many fields of mathematics. For instance, they are related to bounded cohomology, stable commutator length, metrics on diffeomorphism groups, displacement of sets in symplectic topology, dynamics, knot theory, orderability, and the study of mapping class groups and of concordance group of knots.

Let S be a compact oriented surface. In this talk I will discuss several invariant metrics and quasi-morphisms on the identity component Diff_0(S, area) of the group of area preserving diffeomorphisms of S. In particular, I will show that some quasi-morphisms on Diff_0(S, area) are related to the topological entropy. More precisely, I will discuss a construction of infinitely many linearly independent quasi-morphisms on Diff_0(S, area) whose absolute values bound from below the topological entropy. If time permits, I will define a bi-invariant metric on this group, called the entropy metric, and show that it is unbounded. Based on a joint work with M. Marcinkowski.

*Abstract:*

Abstract : A group defined by a presentation with only one relator is called a one-relator group.Various conjectures and questions suggested that a one-relator with no subgroup isomorphic to a Baumslag--Solitar group $BS(m, n)$ for $m \neq \pm n$ would enjoy various nice geometric properties, such as automaticity and acting freely on CAT(0) cube complexes.In this talk I will introduce examples showing that this is not the case.Joint work with Daniel Woodhouse.

*Abstract:*

Given an automorphism of the free group, we consider the mapping torus defined with respect to the automorphism. One seeks to understand the relationship between geometric properties of the resulting free-by-cyclic group and algebraic properties of the automorphism. For example, under certain natural conditions on the automorphism, Kapovich--Kleiner prove the visual boundary of the free-by-cyclic group is homeomorphic to the Menger curve. However, their proof is very general and gives no tools to further study the boundary and large-scale geometry of these groups. In this talk, I will explain how to construct explicit embeddings of non-planar graphs into the boundary of these groups whenever the group is hyperbolic. This is joint work with Yael Algom-Kfir and Arnaud Hilion.

*Announcement:*

**דר' טלי פינסקי**

הפקולטה למתמטיקה

טכניון

**Dr. Tali Pinsky**

The Faculty of Mathematics

Technion

**Math Club 5.12.17**

**אנא שימו לב לשעת ההרצאה הלא שגרתית**

אחרי ההרצאה יתקיים טקס הענקת פרסים של התחרות ע"ש גרוסמן

**P****OSTE****R**

**מחזור שלוש גורר כאוס**

בהרצאה נסתכל על מודל מתמטי להתרבות של אוכלוסיות.

נניח שאם ברגע מסויים מספר החיידקים בצלחת מעבדה הוא x אז מספר החיידקים כעבור דקה הוא f(x). האם אפשר לצפות את מספר החיידקים בצלחת כעבור זמן?

נראה, לפי מאמר של לי ויורק בשם "מחזור שלוש גורר כאוס", שאם מספר החיידקים קופץ בין שלושה ערכים שונים אז התרבות החיידקים היא כאוטית ולא ניתן לצפות אותה.

**Period three implies chaos**

We will consider a mathematical model for population growth.

Suppose that if we have x germs in a sample, the number of germs after one minute is given by f(x).

Is it possible to predict the number of germs as time passes?

Following a paper by Li and York called "period three implies chaos", we will show that if the number of germs oscillates between three different values then the growth is chaotic and the number of germs cannot be predicted

ההרצאה תהיה בעברית

The lecture will be in Hebrew

*Abstract:*

This will be the first of two talks in which we will study the recent preprint "Non-commutative peaking phenomena and a local version of the hyperrigidity conjecture", by Raphael Clouatre. Link:

https://arxiv.org/pdf/1709.01649.pdf

*Abstract:*

It is known that the essential spectrum of a Schrödinger operator H on l^2(N) is equal to the union of the spectra of right limits of H. The natural generalization of this relation to Z^n is known to hold as well. In this talk we study the possibility of generalizing this characterization of \sigma_{ess}(H) to trees. We give indications for the failure of the general statement in this case, while presenting a natural family of models where it still holds. This is a joint work with Jonathan Breuer. (see abstract pdf).

*Abstract:*

The deep connection between the Monge optimal transport problem and the foundations of geomtrical optics will be presented. This connection will be applied to classify all the solutions to the phase-from-intensity problem, and even to the construction of an actual phase detector.

Joint work with Gershon Wolansky.

*Abstract:*

The primitive equations are a fundamental model for many geophysical flows. They are derived from the Navier-Stokes equations by assuming a hydrostatic balance for the pressure term. These equations are known to be globally and strongly well-posed in the three-dimensional setting for arbitrarily large data belonging to $H^1$ by the seminal result of Cao and Titi. Here, I would like to consider the primitive equations in $L^p$-spaces using an evolution equation method. This yields several new results on global strong well-posedness for rough initial data as well as on the regularity of solutions. More precisely, one obtains well-posedness for anisotropic initial values in the scaling invariant space $L^{infty}(R^2;L^1(-h,0))$ and the analyticity of solutions in time and space.

*Abstract:*

We decompose any object in the wrapped Fukaya category of a 2n-dimensional Weinstein manifold as a twisted complex built from the cocores of the n-dimensional handles in a Weinstein handle decomposition. If time permits, we will also discuss how to generalize this result to Weinstein sectors.

This is joint work with Baptiste Chantraine, Georgios Dimitroglou Rizell and Paolo Ghiggini.

*Abstract:*

When the first Betti number $b_{1}(M)$ of a 3-manifold $M$ is greater than one, it follows from Thurston norm theory that if $M$ fibers over the circle, it fibers in infinitely many ways. This talk studies fiberings that are extremal in the sense that the Betti number of the fiber realises the lower bound $b_{1}(M)-1$. It is shown that in hyperbolic manifolds, such fiberings are unique up to isotopy, and can be characterised as having monodromy in a specific normal subgroup of the mapping class group.

***Double feature: please note the special time***

*Abstract:*

The talk is a special Geometry and Topology seminar.

Abstract :

When the first Betti number $b_{1}(M)$ of a 3-manifold $M$ is greater than one, it follows from Thurston norm theory that if $M$ fibers over the circle, it fibers in infinitely many ways. This talk studies fiberings that are extremal in the sense that the Betti number of the fiber realises the lower bound $b_{1}(M)-1$. It is shown that in hyperbolic manifolds, such fiberings are unique up to isotopy, and can be characterised as having monodromy in a specific normal subgroup ofthe mapping class group.

*Abstract:*

TBA

*Abstract:*

We propose a high-order compact method for the approximation of the biharmonic and Navier-Stokes equations in planar irregular geometry. This is based on a fourth order Cartesian Embedded scheme for the biharmonic problem, where a bidimensional Lagrange-Hermite polynomial was introduced. A variety of numerical results assure fourth-order convergence rates. In addition, a purely one dimensional procedure was designed for the Navier-Stokes equations. Numerical results demonstrate fourth-order convergence rates. Joint work with M. Ben-Artzi and Jean-Pierre Croisille

*Abstract:*

We revisit the old construction of Gromov and Lawson that yields a Riemannian metric of positivescalar curvature on a connected sum of manifolds admitting such metrics. This is joint workwith C. Sormani and J. Basilio. Our refinement is to show that the "tunnel" constructed betweenthe two summands can be made to have arbitrarily small length and volume. We use this tocreate examples of sequences of compact manifolds with positive scalar curvature whose Gromov-Hausdorff limits do not have positive scalar curvature in a certain generalized sense.

*Abstract:*

Let $(M,d)$ be a metric space and let $Y$ be a Banach space. Suppose that for each point $x$ of $M$ we are given a compact convex subset $F(x)$ in $Y$ of dimension at most $m$. A ``Lipschitz selection'' for the family $\{F(x): x\in M\}$ is a Lipschitz map $f$ from $M$ into $Y$ such that $f(x)$ belongs to $F(x)$ for each $x\in M$. The talk explains how one can decide whether a Lipschitz selection exists. We discuss the following ``Finiteness Principle'' for the existence of a Lipschitz selection: Suppose that on every subset $M'$ of $M$ consisting of at most $2^{m+1}$ points, $F$ has a Lipschitz selection with Lipschitz constant at most $1$. Then $F$ has a Lipschitz selection on all of $M$. Furthermore, the Lipschitz constant of this selection is bounded by a certain constant depending only on $m$. The result is joint work with Charles Fefferman.

*Abstract:*

We introduce an intersection theory problem for maps into a smooth manifold equipped with a stratification. We investigate the problem in the special case when the target is the unitary group and the domain is a circle. The first main result is an index theorem that equates a global intersection index with a finite sum of locally defined intersection indices. The local indices are integers arising from the geometry of the stratification.

The result is used to study a well-known problem in chemical physics, namely, the problem of enumerating the electronic excitations (excitons) of a molecule equipped with scattering data. We provide a lower bound for this number. The bound is shown to be sharp in a limiting case.

*Abstract:*

Abstract: The aim of the talk is to explain the concept of a minimal representative of a dynamical system: A system possessing only periodic orbits that exist in any system in its isotopy class.This concept allows one to use topological methods to study dynamical systems in low dimensions. We'll review the use of minimal representatives in dimensions one and two, and discuss some new ideas that may allow one to apply this concept in dimension three.

*Abstract:*

A number of methods of the algebraic graph theory were influenced by the spectral theory of Riemann surfaces. We pay it back, and take some classical results for graphs to the continuous setting. In particular, I will talk about colorings, average distance and discrete random walks on surfaces. Based on joint works with E. DeCorte and A. Kamber.

*Abstract:*

TBA

*Abstract:*

Statistical Learning Theory is centred on finding ways in which random data can be used to approximate an unknown random variable. At the heart of the area is the following question: Let F be a class of functions defined on a probability space (\Omega,\mu) and let Y be an unknown random variable. Find some function that is (almost) as 'close' to Y as the 'best function' in F. A crucial facet of the problem is the information one has: both Y and the underlying probability measure \mu are not known. Instead, the given data is an independent sample (X_i,Y_i)_{i=1}^N, selected according to the joint distribution of \mu and Y. One has to design a procedure that receives as input the sample (and the identity of the class F) and returns an approximating function. The success of the procedure is measured by the tradeoff between the accuracy (level of approximation) and the confidence (probability) with which that accuracy is achieved. In the talk I explore some surprising connections the problem has with high-dimensional geometry. Specifically, I explain how geometric considerations played an instrumental role in the problem's recent solution-leading to the introduction of a prediction procedure that is optimal in a very strong sense and under minimal assumptions.

*Abstract:*

This talk will investigate a certain class of continuous time Markov processes using machinery from algebraic topology. To each such process, we will associate a homological observable, the average current, which is a measurement of the net flow of probability of the system. We show that the average current quantizes in the low temperature limit. We also explain how the quantized version admits a topological description.

*Abstract:*

We describe Witten's conjectures (now theorems) on intersection theories on moduli spaces of curves and r-spin curves, and their relations to reductions of the KP integrable hierarchy (everything will be defined). We then describe their open analogs (proven in genus 0). Based on joint works with Pandharipande-Solomon and with Buryak-Clader.

*Abstract:*

Abstract: In this talk we will present some results on the first order theory of higher rank arithmetic lattices. The main result is that if G is an irreducible non-uniform higher-rank characteristic zero arithmetic lattice (e.g., SL_n(Z) for n > 2) and H is a finitely generated group that satisfies the same first order sentences as G, then H is isomorphic to G.

*Abstract:*

TBA

*Abstract:*

Accessibility is an important concept in the study of groups and manifolds as it helps decomposing the object in question into simpler pieces. In my talk I will survey some accessibility results of groups and manifolds, and explain how to relate the two. I will then discuss a joint work with Benjamin Beeker on a higher dimensional version of these ideas using CAT(0) cube complexes.

*Abstract:*

In this talk we present a certain extrapolation technique which we apply to some well-known projection, subgradient projection and other fixed point algorithms. All of them can be considered within the general string averaging framework. The analytical results show that under certain assumptions, the convergence can be linear, which is known to be the case for the extrapolated simultaneous projection method. This is joint work with Christian Bargetz, Victor I. Kolobov and Simeon Reich.

*Abstract:*

It is known that for most translation surfaces the number of saddle connections whose length is less than T grows asymptotically like T^2 by works of Eskin and Masur.One main idea in their proof is to use ergodicity of the SL(2,R) action on the space of translation surfaces. We will review the dynamically part of their proof. It is nowknown that this action also exhibits a spectral gap which allows one to conclude an additional error term. This effectivization is joint work with Amos Nevo and Barak Weiss.

*Abstract:*

I will give a review of the subject. I will present the steps of the classification of surfaces, using very nice methods and techniques, such as: degeneration of surfaces, braid monodromy, calculations of fundamental groups and Coxeter groups. We will see interesting examples of classification of known and significant surfaces, such as Hirzebruch surfaces.

***Please note special date/time***

*Abstract:*

**Advisor: **Prof. Roy Meshulam

**Abstract: **Let X be a simplicial complex on n vertices without missing faces of dimension larger than d. Let L_k denote the k-Laplacian acting on real k-cochains of X and let μ_k(X) denote its minimal eigenvalue. We study the connection between the spectral gaps μ_k(X) for k ≥ d and μ_{d-1}(X). As an application we prove a fractional extension of a Hall type theorem of Holmsen, Martinez-Sandoval and Montejano for general position sets in matroids.

*Abstract:*

The Landau-de Gennes model is a widely used continuum description of nematic liquid crystals, in which liquid crystal configurations are described by fields taking values in the space of real, symmetric traceless $3\times 3$ matrices (called $Q$-tensors in this context). The model is an extension of the simpler $S^2$- or $RP^2$-valued Oseen-Frank theory, and provides a relaxation of an ${\mathbb R}P^2-$, $S^2-$ or $S^3$-valued harmonic map problem on two- and three-dimensional domains. There are similarities as well as differences with the $\mathbb{C}$-valued Ginzburg-Landau model.There is current interest in understanding the structure and disposition of defects in the Landau-de Gennes model. After introducing and motivating the model, I will discuss some recent and current work on defects in two-dimensional domains, in the harmonic-map limit as well as perturbations therefrom This is joint work with G di Fratta, V Slastikov and A Zarnescu.

*Abstract:*

We will survey recent developments in the symplectictopology that lead to various notions of distance on the category ofLagrangian submanifolds of a symplectic manifold. We will explain boththe algebraic as well as geometric sides of the story and outline someapplications.

*Abstract:*

A Kleinian group is convex cocompact if its orbit in hyperbolic 3-space is quasi-convex or, equivalently, that it acts cocompactly on the convex hull of its limit set in in hyperbolic 3-space.

Subgroup stability is a strong quasi-convexity condition in finitely generated groups which is intrinsic to the geometry of the ambient group and generalizes the classical quasi-convexity condition above. Importantly, it coincides with quasi-convexity in hyperbolic groups and the notion of convex cocompactness in mapping class groups which was developed by Farb-Mosher, Kent-Leininger, and Hamenstädt.

Using the Morse boundary, I will describe an equivalent characterization of subgroup stability which generalizes the above boundary characterization from Kleinian groups. Along the way I will discuss some known results about stable subgroups of various groups, including the mapping class group and right-angled Artin groups. The talk will include joint work with Matthew Gentry Durham and joint work with David Hume.

*Abstract:*

Dirichlet's Theorem states that for a real mxn matrix A, ||Aq+p||^m ≤ t, ||q||^n < t has nontrivial integer solutions for all t > 1. Davenport and Schmidt have observed that if 1/t is replaced with c/t, c<1, almost no A has the property that there exist solutions for all sufficiently large t. Replacing c/t with an arbitrary function, it's natural to ask when precisely does the set of such A drop to a null set. In the case m=1=n, we give an answer using dynamics of continued fractions. We then discuss an approach to higher dimensions based on dynamics on the space of lattices. Where this approach meets an obstruction, a similar approach to the analogous inhomogeneous approximation problem will succeed. Joint work with Dmitry Kleinbock.

*Abstract:*

(This is is the second of two lectures on this subject)

We shall present the background of Arveson-Douglas conjecture on essential normality, and discuss two papers by Ron Douglas and Yi Wang on the subject:

1) "Geometric Arveson-Douglas Conjecture and Holomorphic Extension"

link: https://arxiv.org/pdf/1511.00782.pdf

2) "Geometric Arveson-Douglas Conjecture - Decomposition of Varieties"

*Abstract:*

In tame geometry, a cell (or cylinder) is defined as follows. A onedimensional cell is an interval; a two-dimensional cell is the domainbounded between the graphs of two functions on a one-dimensional cell;and so on. Cellular decomposition (covering or subdiving a set intocells) and preparation theorems (decomposing the domain of a functioninto cells where the function has a simple form) are two of the keytechnical tools in semialgebraic, subanalytic and o-minimal geometry.

Cells are normally seen as intrinsically real objects, defined interms of the order relation on $\mathbb R$. We (joint with Novikov)introduce the notion of \emph{complex cells}, a complexification ofreal cells where real intervals are replaced by complexannuli. Complex cells are naturally endowed with a notion of analyticextension to a neighborhood, called $\delta$-extension. It turns outthat complex cells carry a rich hyperbolic-geometric structure, andthe geometry of a complex cell embedded in its $\delta$-extensionoffers powerful new tools from geometric function theory that areinaccessible in the real setting. Using these tools we show that thereal cells of the subanalytic cellular decomposition and preparationtheorems can be analytically continued to complex cells.

Complex cells are closely related to uniformization and resolution ofsingularities constructions in local complex analytic geometry. Inparticular we will see that using complex cells, these constructionscan be carried out uniformly over families (which is impossible in theclassical setting). If time permits I will also discuss how thisrelates to the Yomdin-Gromov theorem on $C^k$-smooth resolutions andsome modern variations.

*Abstract:*

I intend to sketch well-known facts about ellipsoids, viewed as a particular case of symmetric convex sets, giving some background on the latter. The ambient spaces will be (finite or infinite dimensional) real linear spaces (some notions not depending on specifying a topology there).

*Abstract:*

Under the assumption of the GRH(Generalized Riemann Hypothesis), we show that there is a real quadratic field K such that the étale fundamental group of the spectrum of the ring of integers of K is isomorphic to A5. To the best of the author's knowledge, this is the first example of a nonabelian simple étale fundamental group in the literature under the assumption of the GRH. (The talk will be basic and tha above notions will be defined).

**Note that there is another algebra seminar talk, right before. **

*Abstract:*

By Quantum Matrix algebras one usually means the algebras defined via braidings,i.e. solutions to the Quantum Yang-Baxter equation. I plan to discuss the problemof classification of braidings. Also, I plan to introduce some Quantum Matrixalgebras and exhibit their properties. In particular, I plan to definequantum analogs of basic symmetric polynomials (elementary, full, Schur...)and to present a quantum version of the Cayley-Hamilton identity.The talk is supposed to be introductory.

**Note that there is another algebra seminar talk, right after.**

*Abstract:*

We shall present the background of Arveson-Douglas conjecture on essential normality, and discuss two papers by Ron Douglas and Yi Wang on the subject:

1) "Geometric Arveson-Douglas Conjecture and Holomorphic Extension"

link: https://arxiv.org/pdf/1511.00782.pdf

2) "Geometric Arveson-Douglas Conjecture - Decomposition of Varieties"

*Abstract:*

**Advisor: **Eli Aljadeff

**Abstract:**

For a Galois extension $K/k$ we consider the question of classifying

the $K/k$-forms of a finite dimensional path algebra $A=k\Gamma$, i.e., find

up to $k$-isomorphism all the $k$-algebras $B$ such

that $A\otimes_{k}K\cong B\otimes_{k}K$. Here $\Gamma$ is an acyclic

quiver. By Galois descent, we show that when $char\left(k\right)=0$

the $K/k$-forms of $A$ are classified by the cohomology pointed

set $H^{1}\left(Gal\left(K/k\right),\,S_{\Gamma}\right)$, where $S_{\Gamma}$

is a certain finite subgroup of automorphisms of the quiver. This

translates the classification of $K/k$-forms of the algebra $k\Gamma$

into a combinatorial problem. We define the notion of combinatoric

forms of a quiver $\Gamma$ and develop a combinatoric descent for

classifing these forms. We equip the combinatoric forms with algebraic

structures (which are certain tensor type path algebras), and show

that the $K/k$-forms of $k\Gamma$ are classified by evaluations

of combinatorial forms of $\Gamma$.

*Abstract:*

**Supervisor: **Assistant Professor Ram Band

**Abstract: **The Laplacian eigenvalue problem on a bounded domain admits an increasing sequence of eigenvalues and a basis of eigenfunctions. The nodal domains of an eigenfunction are the connected components on which the function has a fixed sign. Courant's theorem asserts that the number of nodal domains of the n'th eigenfunction is bounded by n. In this work, we determine the eigenfunctions and eigenvalues which attain Courant's bound in some specific domains in R^d. Our analysis involves interesting symmetry properties of the eigenfunctions and surprising lattice counting arguments.

*Abstract:*

Special MSc Seminar

The Laplacian eigenvalue problem on a bounded domain admits an increasing sequence of eigenvalues and a basis of eigenfunctions. The nodal domains of an eigenfunction are the connected components on which the function has a fixed sign. Courant's theorem asserts that the number of nodal domains of the n'th eigenfunction is bounded by n. In this work, we determine the eigenfunctions and eigenvalues which attain Courant's bound in some specific domains in R^d. Our analysis involves interesting symmetry properties of the eigenfunctions and surprising lattice counting arguments.

Supervisor: Assistant Professor Ram Band

*Abstract:*

**Adviser: **Assistant Professor Danny Neftin

**Abstract: **Let K be a number field and f ∈ K [X] . Carney, Horts h and Zieve proved that the induced map f : K −→ K is at most N to 1 outside of a finite set where N is the largest integer such that cos (2π/N) f ∈ K. In particular every f ∈ Q [X] is at most 6 to 1 outside of a finite set. They conjectured that for every rational map X → Y between d dimensional varieties over a number field the map X (K) → X (K) is at most N (d) to 1 outside of a Zariski losed subvariety. The most difficult remaining open case for curves is rational functions f : P 1 → P 1 . That is, that for every number field K there exists a constant N (K) such that for any rational function f ∈ K (X) the induced map f : P 1 (K) → P 1 (K) is at most N (K) to 1 outside of a finite set. We shall discuss advancements towards proving this conjecture.

*Abstract:*

**Abstract: ** I will consider deterministic and random perturbations of dynamical systems and stochastic processes. Under certain assumptions, the long-time evolution of the perturbed system can be described by a motion on the simplex of invariant measures of the non-perturbed system. If we have a de- scription of the simplex, the motion on it is dened by either an averaging principle, or by large deviations, or by a diusion approximation. Various classes of problems will be considered from this point of view: nite Markov chains, random perturbations of dynamical systems with multiple stable attractors, perturbations of incompressible 3D- ows with a conservation law, wave fronts in reaction diusion equations, elliptic PDEs with a small parameter, homogenization.

*Abstract:*

Given two disjoint convex polyhedra, we look for a pair of points, one in each polyhedron, attaining the minimum distance between the sets. We propose a process based on projections onto the half-spaces defining the two polyhedra.

*Abstract:*

A well-known result says that the Euclidean unit ball is the unique fixed point of the polarity operator. This result implies that the only norm which can be defined on a finite-dimensional real vector space so that its induced unit ball be equal to the unit ball of the dual (polar) norm is the Euclidean norm. Motivated by these results and by relatively recent results in convex analysis and convex geometry regarding various properties of order reversing operators, we consider, in a real Hilbert space setting, a more general fixed point equation in which the polarity operator is composed with a continuous invertible linear operator. We show that if the linear operator is positive definite, then the considered equation is uniquely solvable by an ellipsoid. Otherwise, the equation can have several (possibly infinitely many) solutions or no solution at all. Our analysis yields a few by-products of possibly independent interest, among them results related to positive definite operators, to coercive bilinear forms and hence to partial differential equations, to infinite- dimensional convex geometry, and to a new class of linear operators (semi-skew operators) which is introduced here. This is joint work with Simeon Reich.

*Abstract:*

The validity, and invalidity, of the Entropy Method in Kac's many-particle model is a prominent problem in the field of Kinetic Theory. At its heart, it is an attempt to find a functional inequality, which is independent of the number of particles in the model, that will demonstrate an exponential rate of convergence to equilibrium. Surprisingly enough, a resolution of this method is still unavailable, and while the master equation for the process is simple, its reliance on the number of particles and the geometry of the appropriate sphere is remarkably strong. It seems that any significant advance in this problem always involves an interdisciplinary approach. In this talk I will present recent work with Eric Carlen and Maria Carvalho, where we have introduced new functional properties, and a notion of chaoticity, with which we have managed to considerably improve what is known about the entropy-entropy production ratio on Kac's sphere. Moreover, with that in hand, I will show how Kac's original hope to deduce a rate of decay for his model's limit equation from the many-particle model itself, is achieved.

*Abstract:*

There are subsets N of R^n for which one can find a real-valued Lipschitz function f defined on the whole of R^n but non-differentiable at every point of N. Of course, by the Rademacher theorem any such set N is Lebesgue null. However, due to a celebrated result of Preiss from 1990 not every Lebesgue null subset of R^n gives rise to such a Lipschitz function f.

In this talk I explain that a sufficient condition on a set N for such f to exist is being locally unrectifiable with respect to curves in a cone of directions. In particular, every purely unrectifiable set U possesses a Lipschitz function non-differentiable on U in the strongest possible sense. I also give an example of a universal differentiability set unrectifiable with respect to a fixed cone of directions, showing that one cannot relax the conditions.

This is joint work with David Preiss.

*Abstract:*

**Advisor**: Prof. Amos Nevo

**Abstract**: We show that averages on geometrically finite Fuchsian groups, when embedded via a representation into a space of matrices, have a homogeneous asymptotic limit when properly rescaled. This generalizes some of the results of F. Maucourant to subgroups of infinite co-volume.

*Abstract:*

In 1989, Pansu introduced the notion of the conformal dimension of the boundary at infinity of a negatively curved manifold. This notion, applied to the boundary at infinity of a Gromov hyperbolic group, gives a natural quasi-isometric invariant of the group. In these talks I'll survey some of what is known about conformal dimension and the challenge of calculating or even estimating its value.

Third and final lecture.

*Abstract:*

In 1989, Pansu introduced the notion of the conformal dimension of the boundary at infinity of a negatively curved manifold. This notion, applied to the boundary at infinity of a Gromov hyperbolic group, gives a natural quasi-isometric invariant of the group. In these talks I'll survey some of what is known about conformal dimension and the challenge of calculating or even estimating its value.

Second in a series of three lectures.

*Abstract:*

In 1989, Pansu introduced the notion of the conformal dimension of the boundary at infinity of a negatively curved manifold. This notion, applied to the boundary at infinity of a Gromov hyperbolic group, gives a natural quasi-isometric invariant of the group. In these talks I'll survey some of what is known about conformal dimension and the challenge of calculating or even estimating its value.

First in a series of three lectures.

*Abstract:*

A common mechanism for intramembrane cavitation bioeffects is presented and possible bioeffects, both delicate and reversible or destructive and irreversible, are discussed. Two conditions are required for creating intramembrane cavitation in a bi-layer sonophore (BLS) *in vivo*: low peak pressure of a pressure wave and an elastic wave of liquid removal from its surroundings. Such elastic waves may be generated by a shock wave, by motion of a free surface, by radiation pressure, by a moving beam of focused ultrasound or any other source of localized distortion of the elastic structure. Soft, cell laden tissues such as the liver, brain and the lung, are more susceptible to irreversible damage. Here, we show the similarity between ultrasound, explosion and impact, where the driving force is negative pressure, and decompression, induced by imbalance of gas concentration. Based on this unified mechanism, one can develop a set of safety criteria for cases where the above driving forces act separately or in tandem, (e.g., ultrasound and decompression). Supporting histological evidence is provided to show locations prone to IMC-related damage; where the damaging forces are relatively high and the localized mechanical strength is relatively poor.

*Abstract:*

This informal talk will review the notion of simple Harnack curve, in particular, the proof of rigid uniqueness of such curves (a theory developed about 15-20 years ago) from the viewpoint of quantum indices of real algebraic curves in the plane (discovered in the last couple of years). NOTE THE UNUSIAL DAY, TIME, AND LOCATION!!

*Abstract:*

A recent result characterizes the fully order reversing operators acting on the class of lower semicontinuous proper convex functions in a real Banach space as certain linear deformations of the Legendre-Fenchel transform. Motivated by the Hilbert space version of this result and by the well-known result saying that this convex conjugate transform has a unique fixed point (namely, the normalized energy function), we investigate the fixed point equation in which the involved operator is fully order reversing and acts on the above-mentioned class of functions. It turns out that this nonlinear equation is very sensitive to the involved parameters and can have no solution, a unique solution, or infinitely many ones. Our analysis yields a few byproducts, such as results related to positive semi-definite operators and to functional equations and inclusions involving monotone operators. The talk is based on joint work with Alfredo N. Iusem (IMPA) and Simeon Reich (The Technion).

*Abstract:*

joint with Yair Hartman, Kate Juschenko and Pooya Vahidi-Ferdowsi.

The notion of a proximal topological action was introduced by Glasner in the 1970's, together with the related notion of a strongly amenable group. Only a handful of new insights have been gained since then, and much remains mysterious. For example, it is known that all virtually nilpotent groups are strongly amenable, but it is not known if all strongly amenable groups are virtually nilpotent (within the class of discrete groups). We will introduce the definitions, survey what is known, and show that Thompson's infamous group F is not strongly amenable.

*Abstract:*

==== NOTE THE SPECIAL TIME ===

Let M be a compact complex manifold. Consider the action of the diffeomorphism group Diff(M) on the (infinite-dimensional) space Comp(M) of complex structures. A complex structure is called ergodic if its Diff(M)-orbit is dense in the connected component of Comp(M). I will show that on a hyperkaehler manifold or a compact torus, a generic complex structure is ergodic. If time permits, I would explain geometric applications of these results to hyperbolicity. I would try to make the talk accessible to non-specialists.

*Abstract:*

Bidding games are extensive form games, where in each turn players bid in order to determine who will play next. Zero-sum bidding games like Bidding Tic-Tac-Toe (also known as Richman games) have been extensively studied [Lazarus et al.'99, Develin and Payne '10]. We extend the theory of bidding games to general-sum two player games, showing the existence of pure subgame-perfect Nash equilibria (PSPE), and studying their properties. In particular, we show that the set of all PSPEs forms a semilattice, whose bottom point is unique. Our main result shows that if the underlying game has the form of a binary tree (only two actions available to the players in each node), then the Bottom PSPE is monotone in the budget, Pareto-efficient, and fair. In addition, we discuss applications of bidding games to combinatorial bargaining, and provide a polynomial-time algorithm to compute the Bottom PSPE. Joint work with Gil Kalai and Moshe Tennenholtz

*Abstract:*

Typically, when semi-discrete approximations to time-dependent partial differential equations (PDE) or explicit multistep schemes for ordinary differential equation (ODE) are constructed they are derived such that they are stable and have a specified truncation error $\tau$. Under these conditions, the Lax--Richtmyer equivalence theorem assures that the scheme converges and that the error is, at most, of the order of $||\tau||$. In most cases, the error is in indeed of the order of $||\tau||$.

We demonstrate that schemes can be constructed, whose truncation errors are $\tau$, however, the actual errors are much smaller. This error reduction is done by constructing the schemes such that they inhibit the accumulation of the local errors, therefore they are called Error Inhibiting Schemes (EIS).

ADI DITKOWSKI, School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel. email: adid@post.tau.ac.il

*Abstract:*

Please see the attached file.

*Abstract:*

We describe the asymptotic behavior of critical points of $\int_\Omega [(1/2)|\nabla u|^2+W(u)/\varepsilon^2]$ when $\varepsilon \to 0$. Here $W$ is a Ginzburg-Landau type potential vanishing on a simple closed curve $\Gamma$. Unlike the case of the standard Ginzburg-Landau potential $W(u)=(1-|u|^2)^2/4$, studied by Bethuel, Brezis and H\'elein, we do not assume any symmetry of $W$ or $\Gamma$. This is a joint work with Petru Mironescu (Lyon I).

*Abstract:*

Let X be a uniformly distributed binary sequence of length n. Let Y be a noisy version of X, obtained by flipping each coordinate of X independently with probability epsilon. We want to come up with a one-bit function of Y which provides as much information as possible about X. Courtade and Kumar conjectured that the best one can do is to choose a coordinate function f(Y) = Y_i, for some i between 1 and n. We prove the conjecture for large values of epsilon (epsilon > 1/2 - delta, for some absolute constant delta). The main new technical ingredient in the proof is the claim that if F is a real-valued function on the boolean cube, and G is a noisy version of F, then the entropy Ent(G) is upper-bounded by the expected entropy of a projection of F on a random coordinate subset of a certain size.

*Abstract:*

A nonlocal nonlinear Schrödinger (NLS) equation was recently introduced in Phys.Rev.Lett. 110, 064105 (2013) and shown to be an integrable infinite dimensional Hamiltonian evolution equation. In this talk we present a detailed study of the inverse scattering transform of this nonlocal NLS equation. The direct and inverse scattering problems are analyzed. Key symmetries of the eigenfunctions and scattering data and conserved quantities are discussed. The inverse scattering theory is developed by using a novel left-right Riemann–Hilbert problem. The Cauchy problem for the nonlocal NLS equation is formulated and methods to find pure soliton solutions are presented; this leads to explicit time-periodic one and two soliton solutions. A detailed comparison with the classical NLS equation is given and brief remarks about nonlocal versions of the modified Korteweg–de Vries and sine-Gordon equations are made.

*Abstract:*

One of the main characteristics of infinite-dimensional dissipative evolution equations, such as the Navier-Stokes equations and reaction-diffusion systems, is that their long-time dynamics is determined by finitely many parameters -- finite number of determining modes, nodes, volume elements and other determining interpolants. In this talk I will show how to explore this finite-dimensional feature of the long-time behavior of infinite-dimensional dissipative systems to design finite-dimensional feedback control for stabilizing their solutions. Notably, it is observed that this very same approach can be implemented for designing data assimilation algorithms of weather prediction based on discrete measurements. In addition, I will also show that the long-time dynamics of the Navier-Stokes equations can be imbedded in an infinite-dimensional dynamical system that is induced by an ordinary differential equations, named *determining form*, which is governed by a globally Lipschitz vector field. Remarkably, as a result of this machinery I will eventually show that the global dynamics of the Navier-Stokes equations is be determining by only one parameter that is governed by an ODE. The Navier-Stokes equations are used as an illustrative example, and all the above mentioned results equally hold to other dissipative evolution PDEs, in particular to various dissipative reaction-diffusion systems and geophysical models.

*Abstract:*

The Choquet order on measures is used to establish that states on a function system always have a representing measure supported on the set of extreme points of the state space (in a technical sense). We introduce a new operator-theoretic order on measures, and prove that it is equivalent to the Choquet order. This leads to some improvements in the classical theory, but more importantly it leads to some new operator-theoretic consequences. In particular, we establish Arveson’s hyperrigidity conjecture for function systems. This yields a significant strengthening of the classical approximation theorems of Korovkin and Saskin. This is joint work with Matthew Kennedy.

The lecture will take place in Amado 233 (NOTE THE UNUSUAL ROOM).

*Abstract:*

====== NOTE THE SPECIAL TIME ====

A subset S of a group G invariably generates G if for every choice of g(s) \in G,s \in S the set {s^g(s):s\in S} is a generating set of G. We say that a group G is invariably generated if such S exists, or equivalently if S=G invariably generates G. In this talk, we study invariable generation of Thompson groups. We show that Thompson group F is invariable generated by a finite set, whereas Thompson groups T and V are not invariable generated. This is joint work with Tsachik Gelander and Kate Juschenko.

*Abstract:*

In the theory of Diophantine approximations, singular points are ones for which Dirichlet’s theorem can be infinitely improved. It is easy to see that all rational points are singular. In the special case of dimension one, the only singular points are the rational ones. In higher dimensions, points lying on a rational hyperplane are also obviously singular. However, in this case there are additional singular points. In the dynamical setting the singular points are related to divergent trajectories. In the talk I will define obvious divergent trajectories and explain the relation to rational points. In addition, I will present the more general setting involving Q-algebraic groups. Lastly I will discuss results concerning classification of divergent trajectories in Q-algebraic groups.

*Abstract:*

We discuss the question of global regularity for a general class of Eulerian dynamics driven by a forcing with a commutator structure.

The study of such systems is motivated by the hydrodynamic description of agent-based models for flocking driven by alignment.

For commutators involving bounded kernels, existence of strong solutions follows for initial data which are sub-critical, namely -- the initial divergence is “not too negative” and the initial spectral gap is “not too large”. Singular kernels, corresponding to fractional Laplacian of order 0<s<1, behave better: global regularity persists and flocking follows. Singularity helps! A similar role of the spectral gap is found in our study of two-dimensional pressure-less equations, corresponding to the formal limit s=0. Here, we develop a new BV framework to prove the existence of weak dual solutions for the 2D pressure-less Euler equations as vanishing viscosity limits.

*Abstract:*

In 1964, Arnold constructed an example of a nearly integrable deterministic system exhibiting instabilities. In the 1970s, Chirikov, a physicist, coined the term “Arnold diffusion” for this phenomenon, where diffusion refers to the stochastic nature of instability.One of the most famous examples of stochastic instabilities for nearly integrable systems,discovered numerically by Wisdom, an astronomer, is the dynamics of Asteroids in Kirkwood gaps in the Asteroid belt. In the talk we will describe a class of nearly integrable deterministic systems, where we prove stochastic diffusive behavior. Namely, we show that distributions given by a deterministic evolution of certain random initial conditions weakly converge to a diffusion process.This result is conceptually different from known mathematical results, where the existence of “diffusing orbits” is shown. This work is based on joint papers with Castejon, Guardia, J.Zhang, and K.Zhang.

*Abstract:*

We establish metric convergence theorems for infinite products of possibly discontinuous operators defined on Hadamard spaces. This is joint work with Zuly Salinas.

*Abstract:*

**NOTICE THE SPECIAL DATE AND TIME!**

In 1975 George Mackey pointed out an analogy between certain unitary representations of a semisimple Lie group and its Cartan Motion group. Recently this analogy was proven to be a part of a bijection between the tempered dual of a real reductive group and the tempered dual of its Cartan Motion group.

In this talk, I will state a conjecture characterizing the Mackey bijection as an algebraic isomorphism between the admissible duals. This will be done in terms of certain algebraic families of Harish-Chandra modules. We shall see that the conjecture hold in the case of SL(2,R).

*Abstract:*

Suppose that for each point x of a metric space X we are given a compact convex set K(x) in R^D. A "Lipschitz selection" for the family (K(x):x\in X} is a Lipschitz map F:X->R^D such that F(x) belongs to K(x) for each x in X.The talk explains how one can decide whether a Lipschitz selection exists. The result is joint work with P. Shvartsman.

*Abstract:*

**Abstract**: Suppose that for each point 𝑥 of a metric space 𝑋 we are given a compact convex set 𝐾(𝑥) in ℝ𝐷. A "Lipschitz selection" for the family {𝐾(𝑥)∶𝑥∈𝑋} is a Lipschitz map 𝐹:𝑋→ℝ𝐷 such that 𝐹(𝑥) belongs to 𝐾(𝑥) for each 𝑥 in 𝑋. The talk explains how one can decide whether a Lipschitz selection exists. The result is joint work with P. Shvartsman.

Light refreshments will be given before the talk in the lounge of the Faculty of Mathematics on the 8th floor.

*Announcement:*

We are pleased to invite you to our annual Elisha Netanyahu Memorial Lecture on the 7th of June at 17:00 in Sego 1 auditorium at Sego building. The lecturer this year is Professor Gil Kalai from the Hebrew University of Jerusalem. The title of his talk is *"Puzzles** about trees, high dimensions, elections, errors and computation". *

* *Light refreshments will be given before the talk in Faculty Lounge on the 8th floor.

Attached is the poster of the talk.

*Abstract:*

In his famous 1900 ICM address Hilbert proposed his famous list of problems for the 20th century. Among these was his 6th problem which was less clearly formulated than the others but dealt with a rigorous derivation of the macroscopic equations of continuum mechanics from the available microscopic theory of his time, i.e. statistical mechanics and specifically Boltzmann's kinetic theory of gases. The problem has drawn attention from analysts over the years and even Hilbert himself made a contribution. In this talk I will note how an exact summation of the Chapman-Enskog expansion for the Boltzmann equation due to Ilya Karlin ( ETH) and Alexander Gorban (Leicester) can be used to represent solutions of the Boltzmann equation and then show that these solutions CANNOT converge the classical balance laws of mass, momentum, and energy associated the Euler equation of compressible gas dynamics. Hence alas Hilbert's program (at least with respect to gas dynamics) has a negative outcome.

Some references:

1) Gorban, Alexander N.; Karlin, Ilya Hilbert's 6th problem: exact and approximate hydrodynamic manifolds for kinetic equations. *Bull. Amer. Math. Soc. (N.S.)* 51 (2014), no. 2, 187–246.

2) Famous Fluid Equations Are Incomplete, in Quanta Magazine, https://www.quantamagazine.org/20150721-famous-fluid-equations-are-incomplete/

3) A.N. Gorban, I.V. Karlin Beyond Navier–Stokes equations: capillarity of ideal gas, Contemporary Physics, 58(1) (2016), 70-90.

4)The Mathematician's Shiva by Stuart Rojstaczer

*Abstract:*

The mathematical problem of group synchronization deals with the question of how to estimate unknown group elements from a set of their mutual relations. This problem appears as an important step in solving many real-world problems in vision, robotics, tomography, and more. In this talk, we present a novel solution for synchronization over the class of Cartan motion groups, which includes the special important case of rigid motions. Our method is based on the idea of group contraction, an algebraic notion origin in relativistic mechanics.

*Abstract:*

COMPLEX AND HARMONIC ANALYSIS III

In memory of

PROFESSOR URI SREBRO (Z"L)

June 4 – 8, 2017

TECHNION – Israel Institute of Technology HIT – Holon Institute of Technology

The Conference will provide a forum for discussions and exchange of new ideas, concepts and recent developments in the broad field of Modern Analysis. The topics to be addressed include (but not restricted to)

* Complex Analysis

* Harmonic Analysis and PDE

* Quasi-Conformal Mappings and Geometry

The event will take place on June 4 – 8, 2017 in the TECHNION on June 7 and in HIT June 4,5,8 in HIT.

For registration and information please contact Anaoly Goldberg at golberga@hit.ac.il

On behalf of the Organizing Committee

,

Sincerely,

Anatoly Golberg

Holon Institute of Technology

*Abstract:*

In the first part of this talk we study sections of B = {x : |x_1| + ... + |x_n| < 1} with (n-1)-dimensional subspaces of R^n and present a new method of determining sections of maximal and minimal (n-1)-dimensional volume, using probabilistic methods. This part is based on joint work with A. Eskenazis and T. Tkocz. In the second part a similar problem for projections is studied using Fourier analytic methods on the discrete cube. This task boils down to the study of the optimal constant in the so-called Khinchine inequality. This part is based on articles of K. Ball and S. Szarek.

*Abstract:*

We describe a higher dimensional analogue of the Stallings folding sequence for group actions on CAT(0) cube complexes. We use it to give a characterization of quasiconvex subgroups of hyperbolic groups which act properly co-compactly on CAT(0) cube complexes via finiteness properties of their hyperplane stabilizers. Joint work with Benjamin Beeker.

*Abstract:*

Continued fraction expansion (CFE) is a presentation of numbers which is closely related to Diophantine approximation and other number theoretic concepts. It is well known that for almost every x in (0,1), the coefficients appearing in the CFE of x obey the Gauss-Kuzmin statistics. This claim is not true for all x, and in particular it is not true for rational numbers which have finite CFE. In order to still have some statistical law, we instead group together the rationals p/q in (0,1) for q fixed and (p,q)=1 and ask whether their combined statistics converges as q goes to infinity. In this talk I will show how this equidistribution problem can be reformulated and solved using the language of dynamics of lattices in SL_2(Z)\SL_2(R) (and given time, how it extends naturally to the Adelic setting). This will in turn imply a stronger equidistribution of the CFE of rational numbers. This is a joint work with Uri Shapira.

*Abstract:*

Sample constructions of two algebras, both with the ideal of relations defined by a finite Groebner basis will be presented. For the first algebra the question whether a given element is nilpotent is algorithmically unsolvable, for the second the question whether a given element is a zero divisor is algorithmically unsolvable. This gives a negative answer to questions raised by Latyshev.

Joint work with Ilya Ivanov-Pogodaev.

*Abstract:*

Legendre duality is prominent in mathematics, physics, and elsewhere. In recent joint work with Berndtsson, Cordero-Erausquin, and Klartag, we introduce a complex analogue of the classical Legendre transform. This turns out to have ties to several foundational works in interpolation theory going back to Calderon, Coifman, Cwikel, Rochberg, Sagher, and Weiss, as well as in complex analysis/geometry going back to Alexander--Wermer, Slodkowski, Moriyon, Lempert, Mabuchi, Semmes, and Donaldson.

*Abstract:*

**The First Joint IMU-INdAM Conference in Analysis**

**May 29 - June 1, 2017**

**Grand Beach Hotel, Tel Aviv, Israel **

We are pleased to announce on the **First Joint Conference in Analysis** of the Israel Mathematical Union and the Istituto Nazionale di Alta Matematica "F.Severi", in cooperation with Tel Aviv University, the Technion - Israel Institute of Technology and the Galilee Research Center for Applied Mathematics, ORT Braude Academic College of Engineering, which will be held in the Grand Beach Hotel, Tel Aviv from May 29 (arrival May 28) to June 1, 2017. On May 31 there will be an excursion for the Italian guests.

We would like to ask kindly to distribute this announcement among your friends, colleagues and anyone of interest. If you have any queries please do not hesitate to contact the Organizing Committee. We are looking forward to seeing you in Tel Aviv.

*Abstract:*

The 2017 annual meeting in Akko – Israel Mathematical Union

#### 25-28/5/2017

#### Registration (mandatory)

**Schedule and Program**

https://imudotorgdotil.wordpress.com/annual-meeting/

**Plenary speakers:**

Amos Nevo (Technion-IIT)Edriss S. Titi (Weizmann Institute and Texas A&M)

**The Erdős, Nessyahu and Levitzki Prizes will be awarded**

**Zeev@80: Zeev Schuss 80 Birthday**

**Sessions and organizers:**

- Analysis – Emanuel Milman and Baptiste Devyver
- Algebra – Chen Meiri and Danny Neftin
- Applied mathematics – Nir Gavish
- Discrete mathematics – Gil Kalai and Nathan Keller
- Dynamical systems – Uri Bader and Tobias Hartnick
- Education* – Alon Pinto (*discussions in Hebrew)
- Non-linear analysis and optimization – Simeon Reich and Alexander Zaslavski
- Probability theory – Ron Ronsenthal and Nick Crawford
- Topology – Yoav Moriah and Michah Sageev

The IMU offers a limited number of discount rooms (PhD students and postdoctoral fellows: free rooms, two students/fellows in a room. Members of the IMU: 50% discount) to those who register early

For more details contact imu@imu.org.il

Organizing committee: Yehuda Pinchover, Koby Rubisntein, Amir Yehudayoff

*Abstract:*

The question of finding an epsilon-biased set with close to optimal support size, or, equivalently, finding an explicit binary code with distance $\frac{1-\eps}{2}$ and rate close to the Gilbert-Varshamov bound, attracted a lot of attention in recent decades. In this paper we solve the problem almost optimally and show an explicit $\eps$-biased set over $k$ bits with support size $O(\frac{k}{\eps^{2+o(1)}})$. This improves upon all previous explicit constructions which were in the order of $\frac{k^2}{\eps^2}$, $\frac{k}{\eps^3}$ or $\frac{k^{5/4}}{\eps^{5/2}}$. The result is close to the Gilbert-Varshamov bound which is $O(\frac{k}{\eps^2})$ and the lower bound which is $\Omega(\frac{k}{\eps^2 \logeps})$. The main technical tool we use is bias amplification with the $s$-wide replacement product. The sum of two independent samples from an $\eps$-biased set is $\eps^2$ biased. Rozenman and Wigderson showed how to amplify the bias more economically by choosing two samples with an expander. Based on that they suggested a recursive construction that achieves sample size $O(\frac{k}{\eps^4})$. We show that amplification with a long random walk over the $s$-wide replacement product reduces the bias almost optimally.

*Abstract:*

In this talk I will discuss a model for auto-ignition of fully developed free round turbulent jets consisting of oxidizing and chemically reacting components.I will present the derivation of the model and present results of its mathematical analysis.

The derivation of the model is based on well established experimental fact that the fully developed free round turbulent jets, in a first approximation, have the shape

of a conical frustum. Moreover, the velocity as well as concentrations fields within such jets, prior to auto-ignition, assume self-similar profiles and can be viewed as prescribed. Using these facts as well as appropriately modified

Semenov-Frank-Kamenetskii theory of thermal explosion I will derive an equation that describes initial stage of evolution of the temperature field within the jet.

The resulting model falls into a general class of Gelfand type problems.

The detailed analysis of the model results in a sharp condition for auto-ignition of free round turbulent jets in terms of principal physical and geometric parameters involved in this problem. This is a joint work with M.C. Hicks and U.G. Hegde of NASA Glenn Research Center.

*Abstract:*

Given a closed smooth Riemannian manifold M, the Laplace operator is known to possess a discrete spectrum of eigenvalues going to infinity. We are interested in the properties of the nodal sets and nodal domains of corresponding eigenfunctions in the high energy limit. We focus on some recent results on the size of nodal domains and tubular neighbourhoods of nodal sets of such high energy eigenfunctions. (joint work with Bogdan Georgiev)

*Abstract:*

Let G be a group and let r(n,G) denote the number of isomorphism classes of n-dimensional complex irreducible representations of G. Representation growth is a branch of asymptotic group theory that studies the asymptotic and arithmetic properties of the sequence (r(n,G)). In 2008 Larsen and Lubotzky conjectured that all irreducible lattices in a high rank semisimple Lie group have the same polynomial growth rate. In this talk I will explain the conjecture and describe the ideas around the proof of a variant of the conjecture: if the lattices have polynomial representation growth (which is known to be true in most cases) then they have the same polynomial growth rate. This is a joint work with Nir Avni, Benjamin Klopsch and Christopher Voll.

*Abstract:*

This talk is devoted to inequalities for best approximations and moduli of smoothness of functions and their derivatives in the spaces $L_p, p > 0.$ Namely, we consider the so-called direct inequalities (upper estimates of a best approximation (modulus of smoothness) of a function via the best approximation (modulus of smoothness) of the derivatives of the function) and the corresponding (weak) inverse inequalities. In the spaces $L_p, p \ge 1,$ both inequalities are well studied. In contrast, in the spaces $L_p, 0 < p < 1,$ there are only some partial positive results related to the inverse inequalities and some examples of functions for which the standard direct inequalities in $L_p, 0 < p < 1,$ are impossible. In my talk, first positive results related to the direct inequalities in the spaces $L_p, 0 < p < 1,$ will be presented. New (weak) inverse inequalities will also be discussed. These results are obtained for the approximation of functions by trigonometric polynomials, algebraic polynomials, and splines, as well as for periodic and non-periodic moduli of smoothness.

*Abstract:*

When time-narrow wave-packets scatter by complex target, the field is trapped for some time, and emerges as a time broadened pulse, whose shape reflects the distribution of the delay (trapping)-times. I shall present a comprehensive framework for the computation of the delay-time distribution, and its dependence on the scattering dynamics, the wave-packet envelope (profile) and the dispersion relation. I shall then show how the well-known Wigner-Smith mean delay time and the semi-classical approximation emerge as limiting cases, valid only under special circumstances. For scattering on random media, localization has a drastic effect on the delay-time distribution. I shall demonstrate it for a particular one-dimensional system which can be analytically solved.

*Abstract:*

The Hilbert scheme of points on the plane is one of the central objects of modern geometry. We will review some of the interesting connections of this space with representation theory and the theory of symmetric functions, and we will present some recent geometric results motivated by knot theory.

*Abstract:*

Haglund showed that given an isometry of a CAT(0) cube complex that doesn't fix a 0-cube, there exists a biinfinite combinatorial geodesic axis.

I will explain how to generalize this theorem to show that given a proper action of Z^n on a CAT(0) cube complex, there is a nice subcomplex that embeds isometrically in the combinatorial metric and is stabilized by Z^n.

The motivation from group theory will also be given.

*Abstract:*

The spectral gap conjecture for compact semisimple Lie groups stipulates that any adapted random walk on such a group equidistributes at exponential speed. In the first part of the talk, we shall review results of Bourgain and Gamburd, which relate this conjecture to diophantine properties of subgroups in Lie groups. Then, we shall study this diophantine problem in nilpotent Lie groups.

*Abstract:*

Contramodules are module-like algebraic structures endowed with infinite summation or, occasionally, integration operations understood algebraically as infinitary linear operations subject to natural axioms.For about every abelian category of torsion, discrete, or smooth modules there is a no less interesting, but much less familiar, dual analogous abelian category of contramodules. So there are many kinds of contramodule categories, including contramodules over coalgebras and corings, associative rings with a fixed centrally generated ideal, topological rings, topological Lie algebras, topological groups, etc. The comodule-contramodule correspondence is a covariant equivalence between additive subcategories in or (conventional or exotic) derived categories of the abelian categories of comodules and contramodules. Several examples of contramodule categories will be defined in the talk, and various versions of the comodule-contramodule correspondence discussed.

*Abstract:*

Adoption of new products that mainly spread through word-of-mouth is a classical problem in Marketing. In this talk, I will use agent-based models to study spatial (network) effects, temporal effects, and the role of heterogeneity, in the adoption of solar PV systems**. **

*Abstract:*

See the attached file.

*Abstract:*

We will prove that for any finite solvable group G, there exists a cyclic extension K/Q and a Galois extension M/Q such that the Galois group Gal(M/K) is isomorphic to G and M/K is unramified.

We will apply the theory of embedding problem of Galois extensions to this problem and gives a recursive procedure to construct such extensions.

*Abstract:*

We propose a methodology for constructing decision rules for integer and continuous decision variables in multiperiod robust linear optimization problems. This type of problem finds application in, for example, inventory management, lot sizing, and manpower management. We show that by iteratively splitting the uncertainty set into subsets, one can differentiate the later-period decisions based on the revealed uncertain parameters. At the same time, the problem's computational complexity stays at the same level as for the static robust problem. This also holds in the nonfixed recourse situation. In the fixed recourse situation our approach can be combined with linear decision rules for the continuous decision variables. We provide theoretical results on how to split the uncertainty set. Based on this theory, we propose several heuristics. Joint work with Dick den Hertog (Tilburg University).

*Abstract:*

We prove that if a knot or link has a sufficiently complicated plat projection, then that plat projection is unique. More precisely, if a knot or link has a 2m-plat projection, where m is at least 3, each twist region of the plat contains at least three crossings, and n, the length of the plat, satisfies n > 4m(m − 2), then such a projection is unique up to obvious rotations. In particular, this projection gives a canonical form for such knots and links, and thus provides a classification of these links. This is joint work with Jessica S. Purcell.

*Abstract:*

A geodesic conjugacy between two Riemannian manifolds is a diffeomorphism of the unit tangent bundles which commutes with the respective geodesic flows. A natural question to ask is whether a conjugacy determines a manifold up to isometry. In this talk we shall briefly explain the development of the geodesic conjugacy problem and describe some recent results.

*Abstract:*

The u-invariant of a field is the maximal dimension of a nonsingular anisotropic quadratic form over that field, whose order in the Witt group of the field is finite. By a classical theorem of Elman and Lam, the u-invariant of a linked field of characteristic different from 2 can be either 0,1,2,4 or 8. The analogous question in the case of characteristic 2 remained open for a long time. We will discuss the proof of the equivalent statement in characteristic 2, recently obtained in a joint work by Andrew Dolphin and the speaker.

*Abstract:*

Please see event no. 428.

*Abstract:*

A bicycle is a fascinating object, from many points of views, both practical and theoretical. In this talk I will concentrate, mostly, on the geometry of bicycle tracks. At first sight, the pair of front and back wheel tracks left by a passing bike on a sandy or muddy terrain seems like a random pair of curves. This is not the case. For example, one can usually distinguish between the front and back wheel tracks, and even the direction at which these were traversed, based solely on their shapes. Another example: If the front wheel traverses a closed path, then, typically, the back track does not closes up, by an amount related to the area enclosed by the front track and the bicycle length (this fact was used to build an area measuring device, now obsolete, called the Hatchet planimeter). Recently, the subject has attracted attention due to newly discovered relations with the theory of completely integrable systems.

*Abstract:*

Abstract within link...

*Abstract:*

In the last 15 years, there has been much progress on higher dimensional solutions to the Einstein equation, much of it from the physics community. They are particularly interesting as, unlike 4 dimensional spacetimes, the horizon is no longer restricted to being diffeomorphic to the sphere, as demonstrated by the celebrated black ring solution of Emparan and Reall. Using the Weyl-Papapetrou coordinates and harmonic map, we show the existence of stationary solutions to the 5 dimensional vacuum Einstein equation, which are bi-axisymmetric solutions with lens space horizons. This is a joint project with Marcus Khuri and Sumio Yamada.

*Abstract:*

Diffeology, introduced around 1980 by Jean-Marie Souriaufollowing earlier work of Kuo-Tsai Chen, gives a wayto generalize differential calculus beyond Euclidean spaces.Examples include (possibly non-Hausdorff) quotients of manifoldsand spaces of smooth mappings between (possibly non-compact) manifolds.A diffeology on a set declares which maps from open subsetsof Euclidean spaces to the set are "smooth". In spite of its simplicity, diffeology often captures surprisingly rich information.I will present the subject through a sample of examples, results,and questions.

*Abstract:*

09:00-09:10 פרופ' אלי אלחדף, דיקן הפקולטה ומרכז לימודים מתקדמים

09:15-09:25 פרופ' יהודה עגנון, מרכז התכנית הבין יחידתית במתמטיקה שימושית

09:30 הרצאות

ד"ר דני נפטין

פרופ"מ רמי בנד

ד"ר רון רוזנטל

פרופ"מ גיא רמון

פרופ"מ בני צ'וקורל

11:10 הצגת פוסטרים ותחומי מחקר

12:00 פאנל בהשתתפות: פרופ' אלי אלחדף, פרופ' מיכה שגיב, פרופ"מ עומרי ברק ונציגי הסטודנטים לתארים מתקדמים

13:00 ארוחת צהריים

*Abstract:*

Frankl and Furedi conjectured in 1989 that the maximum Lagrangian of all r-uniform hypergraphs of given size m is realised by the initial segment of the colexicographic order. For r=3 this was partially solved by Talbot, but for r\geq 4 the conjecture was widely open. We verify the conjecture for all r\geq 4, whenever $\binom{t-1}{r} \leq m \leq \binom{t}{r}- \gamma_r t^{r-2}$ for a constant $\gamma_r>0$. This range includes the principal case $m=\binom{t}{r}$ for large enough $t$.

*Abstract:*

We consider a general class of sparse graphs which includes for example graphs that satisfy a strong isoperimetric inequality. First, we characterize these graphs in a functional analytic way by means of the form domain of Schrödinger operators. Furthermore, we study spectral bounds and characterize discreteness of the spectrum. As a particular consequence we obtain estimates on the eigenvalue asymptotics in this case. (This is joint work with Michel Bonnefont and Sylvain Golénia.)

*Abstract:*

Euclidean tilings, and especially quasiperiodic ones, such as Penrose tilings, are not only beautiful but crucially important in crystallography. A very powerful tool to study such tilings is cohomology. In order to define it, the first approach is to define a metric on the set of tilings and then define the hull of a tiling as the closure of its orbit under translations. The cohomology of a tiling is then defined as the Cech cohomology of its hull. A more direct (and recent) definition involves treating a tiling as a CW-structure and considering the "pattern-equivariant" subcomplex of the cellular cochain complex. These two definitions yield isomorphic results (J. Kellendonk, 2002) We'll also see some applications of tiling cohomology to the study of shape deformations, and compute some examples.

*Abstract:*

One-dimensional Toeplitz words generalize periodic sequences and are therefore used as model for quasicrystals. They are constructed from periodic words with "holes" (that is, undetermined positions) by successively filling the holes with other periodic words. In this talk, the subclass of so called simple Toeplitz words is considered. We will discuss combinatorial properties of subshifts associated them. In addition to describing certain aspects of how ordered the word is, these properties are important tools for other questions as well. We will apply them to answer questions concerning the spectrum of Schrödinger operators and Jacobi operators on the subshift.

*Abstract:*

In this talk, we will study optimization problems with ambiguous stochastic constraints where only partial information consisting of means and dispersion measures of the underlying random parameter is available. Whereas the past literature used the variance as the dispersion measure, here we use the mean absolute deviation from the mean (MAD). The approach is based on the availability of tight upper and lower bounds on the expectation of a convex function of a random variable, first discovered in 1972. We then use these bounds to derive exact robust counterparts of expected feasibility of convex constraints and to construct new safe tractable approximations of chance constraints. We test the applicability of the theoretical results numerically on various practical problems in Operations Research and Engineering.

*Abstract:*

In this talk we find the optimal error bound (smallest possible estimate, independent of the starting point) for the linear convergence rate of the simultaneous projection method applied to closed linear subspaces in a real Hilbert space. We achieve this by computing the norm of an error operator which we also express in terms of the Friedrichs number. We compare our estimate with the optimal one provided for the alternating projection method by Kayalar and Weinert (1988). Moreover, we relate our result to the alternating projection formalization of Pierra (1984) in a product space. Finally, we adjust our results to closed affine subspaces and put them in context with recent dichotomy theorems. This is joint work with Simeon Reich.

*Abstract:*

There are two interesting norms on free groups and surface groups which are invariant under the group of all automorphisms:

A) For free groups we have the primitive norm, i.e., |g|_p = the minimal number of primitive elements one has to multiply to get g.

B) For fundamental group of genus g surface we have the simple curves norm, i.e., |g|_s = the minimal number of simple closed curves one need to concatenate to get g.

We prove the following dichotomy: either |g^n| is bounded or growths linearly with n. For free groups and surface groups we give an explicit characterisation of (un)bounded elements. It follows for example, that if g is a simple separating curve on a surface, then |g^n| growths linearly. However, if g is a simple non-separating curve, then |g^n| <= 2 for every n. This answers a question of D. Calegari.

The main idea of the proof is to construct appropriate quasimorphisms. M. Abert asked if there are Aut-invariant nontrivial homogeneous quasimorphisms on free groups. As a by-product of our technique we answer this question in the positive for rank 2. This is a joint work with M. Brandenbursky.

*Abstract:*

We will study n-dimensional badly approximable points on curves. Given an analytic non-degenerate curve in R^n, we will show that any countable intersection of the sets of weighted badly approximable points on the curve has full Hausdorff dimension. This strengthens a previous result of Beresnevich by removing the condition on weights. Compared with the work of Beresnevich, we study the problem through homogeneous dynamics. It turns out that the problem is closely related to the study of distribution of long pieces of unipotent orbits in homogeneous spaces.

*Abstract:*

Given two permutations A and B which "almost" commute, are they "close" to permutations A' and B' which really commute? This can be seen as a question about a property the equation XY=YX. Studying analogous problems for more general equations (or systems of equations) leads to the notion of "locally testable groups" (aka "stable groups").

We will take the opportunity to say something about "local testability" in general, which is an important subject in computer science. We will then describe some results and methods developed (in a work in progress), together with Alex Lubotzky, to decide whether various groups are locally testable or not.This will bring in some important notions in group theory, such as amenability, Kazhdan's Property (T) and sofic groups.

*Abstract:*

We study global solutions $u:{\mathbb R}^3\to{\mathbb R}^2$ of the Ginzburg-Landau equation $-\Delta u=(1-|u|^2)u$ which are local minimizers in the sense of De Giorgi. We prove that a local minimizer satisfying the condition $\liminf_{R\to\infty}\frac{E(u;B_R)}{R\ln R}<2\pi$ must be constant. The main tool is a new sharp $\eta$-ellipticity result for minimizers in dimension three that might be of independent interest. This is a joint work with Etienne Sandier (Universit\'e Paris-Est).

*Abstract:*

For almost every real number x, the inequality |x-p/q|<1/q^a has finitely many solutions if and only if a>2. By Roth's theorem, any irrational algebraic number x also satisfies this property, so that from that point of view, algebraic numbers and random numbers behave similarly.We will present some generalizations of this phenomenon, for which we will use ideas of Kleinbock and Margulis on analysis on the space of lattices in R^d, as well as Schmidt's subspace theorem.

*Abstract:*

I will outline how one starts with a symplectic manifold and defines a category enriched in local systems (up to homotopy) on this manifold. The construction relies on deformation quantization and is related to other methods of constructing a category from a symplectic manifolds, such as the Fukaya category and the sheaf-theoretical microlocal category of Tamarkin. The talk will be accessible, with main examples being the plane, the cylinder, and the two-torus.

*Abstract:*

Abstract: We provide explicit Diophantine conditions on the coefficients of degree 2 polynomials under which the limit of an averaged pair correlation density is consistent with the Poisson distribution, using a recent effective Ratner equidistribution result on the space of affine lattices due to Strömbergsson. This is joint work with Jens Marklof.

*Abstract:*

This is a special seminar in Mathematical Physics, please note the special time and place.

We consider a quantum mechanical system, which is modeled by a Hamiltonian acting on a finite dimensional space with degenerate eigenvalues interacting with a field of relativistic bosons. Provided a mild infrared assumption holds, we prove the existence of the ground state eigenvalues and ground state eigenvectors using an operator theoretic renormalization. We show that the eigenvectors and eigenvalues are analytic functions of the coupling constant in a cone with apex at the origin.

*Abstract:*

Milnor-Witt K-groups of fields have been discovered by Morel and Hopkins within the framework of A^1 homotopy theory. These groups play a role in the classification of vector bundles over smooth schemes via Euler classes and oriented Chow groups. Together with Stephen Scully and Changlong Zhong we have generalized these groups to (semi-)local rings and shown that they have the same relation to quadratic forms and Milnor K-groups as in the field case. An applications of this result is that the unramified Milnor-Witt K-groups are a birational invariant of smooth proper schemes over a field.

(joint work with Stephen Scully and Changlong Zhong)

*Abstract:*

In this talk we discuss asymptotic relations between sharp constants of approximation theory in a general setting. We first present a general model that includes a circle of problems of finding sharp or asymptotically sharp constants in some areas of univariate and multivariate approximation theory, such as inequalities for approximating elements, approximation of individual elements, and approximation on classes of elements. Next we discuss sufficient conditions that imply limit inequalities and equalities between various sharp constants. Finally, we present applications of these results to sharp constants in Bernstein-V. A. Markov type inequalities of different metrics for univariate and multivariate trigonometric and algebraic polynomials and entire functions of exponential type.

*Abstract:*

Earlier and recent one-dimensional estimates and asymptotic relations for the cosine and sine Fourier transform of a function of bounded variation are refined in such a way that become applicable for obtaining multidimensional asymptotic relations for the Fourier transform of a function with bounded Hardy variation.

*Abstract:*

The sloshing problem is a Steklov type eigenvalue problem describing small oscillations of an ideal fluid. We will give an overview of some latest advances in the study of Steklov and sloshing spectral asymptotics, highlighting the effects arising from corners, which appear naturally in the context of sloshing. In particular, we will outline an approach towards proving the conjectures posed by Fox and Kuttler back in 1983 on the asymptotics of sloshing frequencies in two dimensions. The talk is based on a joint work in progress with M. Levitin, L. Parnovski and D. Sher.

*Abstract:*

*Abstract:*

In plain words chaos refers to extreme dynamical instability and unpredictability.Yet in spite of such inherent instability, quantum systems with classically chaotic dynamics exhibit remarkable universality. In particular, their energy levels often display the universal statistical properties which can be effectively described by Random Matrix Theory. From the semiclassical point of view this remarkable phenomenon can be attributed to the existence of pairs of classical periodic orbits with small action differences. So far, however, the scope of this theory has, by and large, been restricted to low dimensional systems. I will discuss recent efforts to extend this program to hyperbolic coupled map lattices with a large number of sites. The crucial ingredient of our approach are two-dimensional symbolic dynamics which allow an effective representation of periodic orbits and their pairings. I will illustrate the theory with a specific model of coupled cat maps, where such symbolic dynamics can be constructed explicitly.

*Abstract:*

Lecture Series : Coffee 9:30, L1 10:00-10:50 (intro), L2 11:00-11:40, L3 10:50-12:30. In equilibrium systems there is a long tradition of modelling systems by postulating an energy and identifying stable states with local or global minimizers of this energy. In recent years, with the discovery of Wasserstein and related gradient flows, there is the potential to do the same for time-evolving systems with overdamped (non-inertial, viscosity-dominated) dynamics. Such a modelling route, however, requires an understanding of which energies (or entropies) drive a given system, which dissipation mechanisms are present, and how these two interact. Especially for the Wasserstein-based dissipations this was unclear until rather recently. In these talks I will discuss some of the modelling arguments that underlie the use of energies, entropies, and the Wasserstein gradient flows. This understanding springs from the common connection between large deviations for stochastic particle processes on one hand, and energies, entropies, and gradient flows on the other. In the first talk I will describe the variational structure of gradient flows, introduce generalized gradient flows, and give examples. In the second talk I will enter more deeply into the connection between gradient flows on one hand and stochastic processes on the other, in order to explain `where the gradient-flow structures come from. Organizers: Amy Novick-Cohen and Nir Gavish

*Abstract:*

In equilibrium systems there is a long tradition of modelling systems by postulating an energy and identifying stable states with local or global minimizers of this energy. In recent years, with the discovery of Wasserstein and related gradient flows, there is the potential to do the same for time-evolving systems with overdamped (non-inertial, viscosity-dominated) dynamics. Such a modelling route, however, requires an understanding of which energies (or entropies) drive a given system, which dissipation mechanisms are present, and how these two interact. Especially for the Wasserstein-based dissipations this was unclear until rather recently.

In these talks I will discuss some of the modelling arguments that underlie the use of energies, entropies, and the Wasserstein gradient flows. This understanding springs from the common connection between large deviations for stochastic particle processes on one hand, and energies, entropies, and gradient flows on the other.

In the first talk I will describe the variational structure of gradient flows, introduce generalized gradient flows, and give examples. In the second talk I will enter more deeply into the connection between gradient flows on one hand and stochastic processes on the other, in order to explain ׳where the gradient-flow structures come from׳.

-------------------

This mini-lecture series will be held 9:30-12:30 on Mon, Feb 27.

9:30 - Coffee

10:00-10:50 Lecture I (at an introductory level)

11:00-11:40 Lecture II

10:50-12:30 Lecture III

Organizers: Amy Novick-Cohen and Nir Gavish

*Abstract:*

**Advisor: **Prof. Jacob Rubinstein

**Abstract:** One of the fundamental problems in optical design is *perfect *imaging of a given set of objects or wave fronts by an optical system. An optical system is defined as a finite number of refractive and reflective surfaces and considered to be *perfect* if all the light rays from the object on one side of the system arrive to a single image on the other side of the system. In the case of a single point object we can easily solve the problem using a single optical surface called Cartesian oval. However, in the general case we need to find a set of optical surfaces that map a given set of n objects onto n respective images. In our work we study the problem for n=2 objects in two-dimensional geometry. We discuss a method of designing an optical system with two free-form surfaces that provides a –solution. We then consider a way to construct a solution with minimal degrees of freedom and extend it to wave front imaging. We will also show an application for calculating a multi-surface customized eye model by generating two twice differentiable refractive curves from wave front refraction data.

*Abstract:*

The reality of the zeros of the product and cross-product of Bessel and modified Bessel functions of the first kind is studied. As a consequence, the reality of the zeros of two hypergeometric polynomials is obtained together with the number of the Fourier critical points of the normalized forms of the product and cross-product of Bessel functions. Moreover, the interlacing properties of the real zeros of these products of Bessel functions and their derivatives are also obtained. As an application some geometric properties of the normalized forms of the cross-product and product of Bessel and modified Bessel functions of the first kind are studied. For the cross-product and the product three different kinds of normalization are investigated and for each of the six functions the radii of starlikeness and convexity are precisely determined by using their Hadamard factorization. For these radii of starlikeness and convexity tight lower and upper bounds are given via Euler-Rayleigh inequalities. Necessary and sufficient conditions are also given for the parameters such that the six normalized functions are starlike and convex in the open unit disk. The properties and the characterization of real entire functions from the Laguerre-Polya class via hyperbolic polynomials play an important role. Some open problems are also stated, which may be of interest for further research.

*Abstract:*

The fourth lecture in the series.

*Abstract:*

**Advisor: **Prof. Gershon Elber, CS dept

**Abstract:** Algebraic constraints arise in various applications, across domains in science and engineering. Polynomial and piece-wise polynomial (B-Spline) constraints are an important class, frequently arising in geometric modeling, computer graphics and computer aided design, due to the useful NURBs representation of the involved geometries. Subdivision based solvers use properties of the NURBs representation, enabling, under proper assumptions, to solve non-linear, multi-variate algebraic constraints - globally in a given domain, while focusing on the real roots. In this talk, we present three research results addressing problems in the field of subdivision based solvers.

The first presents a topologically guaranteed solver for algebraic problems with two degrees of freedom. The main contribution of this work is a topologically guaranteed subdivision termination criterion, enabling to terminate the subdivision process when the (yet unknown) solution in the tested sub-domain is homeomorphic to a two dimensional disk. Sufficient conditions for the disk-topology are tested via inspection of the univariate solution curve(s) on the sub-domain’s boundary, together with a condition for the injective projection on a two dimensional plane, based on the underlying implicit function and its gradients.

The second result provides a subdivision based method for detecting critical points of a given algebraic system. To find critical points, we formulate an additional algebraic system, with the semantics of searching for locations where the gradients of the input problem are linearly dependent. We formulate the new problem using function valued determinants, representing the maximal minors of the input problem’s Jacobian matrix, searching for locations where they simultaneously vanish. Consequently, an over-constrained system is obtained, involving only the original parameters. The over-constrained system is then solved as a minimization problem, such that all constrains are accounted for in a balanced manner.

The third result applies the subdivision method to the specific problem of Minkowski sum computation of free-form surfaces. As a first step, a two-DOF algebraic system is formulated, searching for parameter locations that correspond to parallel (or anti-parallel) normal vectors on the input surfaces. Only such locations can contribute to the Minkowski sum envelope surface – which is the required representation for the (typically) volumetric object given by the Minkowski sum. A purging algorithm is then executed, to further refine redundant solution locations: surface patches that admit matched normal directions, but cannot contribute to the envelope. The talk summarizes the research towards PhD in applied mathematics, under supervision of Prof. Gershon Elber.

*Abstract:*

NOTE: The series continues to January 29th and February 5th.

**Abstract: **

In algebraic topology, the Borsuk-Ulam theorem and its extensions place restrictions on maps between compact spaces. Essential to this story is the antipodal action on the sphere, which sends each point x to -x, so equivariant maps are commonly called "odd". The original Borsuk-Ulam theorem then says that there is no odd map from a sphere of high dimension to a sphere of low dimension. This may be extended to more general compact spaces with free actions of finite groups by considering connectivity of a domain X and dimension of a codomain Y.

I will present my work on extending this theorem and similar results to C*-algebras, as motivated by the results and conjectures of other researchers (Yamashita, Taghavi, Baum-Dabrowski-Hajac). Along the way, we will see how this point of view may be used to improve topological results, and how the noncommutative setting differs from the commutative setting.

*Abstract:*

Geometric group theory arose from the study of periodic tilings of proper geodesic metric spaces, or equivalently the study of uniform lattices in isometry groups of such spaces. It provides a way to study finitely-generated infinite groups geometrically.

In joint work with Michael Björklund we propose a framework to study aperiodic tilings of proper geodesic metric spaces. This framework is based on three main ingredients:

1) Tao's notion of approximate subgroups (generalizing Meyer's notion of a model set in R^n)

2) Delone sets in locally compact groups

3) Classical geometric group theory

In this talk I will define the central notions of uniform and non-uniform approximate lattices arising in this framework, and explain some first steps towards a "geometric approximate group theory", i.e. a geometric theory of finitely generated (uniform) approximate lattice.

*Abstract:*

We present a new approach (joint with M. Bjorklund (Chalmers)) for finding new patterns in difference sets E-E, where E has a positive density in Z^d, through measure rigidity of associated action.

By use of measure rigidity results of Bourgain-Furman-Lindenstrauss-Mozes and Benoist-Quint for algebraic actions on homogeneous spaces, we prove that for every set E of positive density inside traceless square matrices with integer values, there exists positive k such that the set of characteristic polynomials of matrices in E - E contains ALL characteristic polynomials of traceless matrices divisible by k.

By use of this approach Bjorklund and Bulinski (Sydney), recently showed that for any quadratic form Q in d variables (d >=3) of a mixed signature, and any set E in Z^d of positive density the set Q(E-E) contains kZ for some positive k. Another corollary of our approach is the following result due to Bjorklund-Bulinski-Fish: the discriminants D = {xy-z^2 , x,y,z in B} over a Bohr-zero non-periodic set B covers all the integers.

*Abstract:*

Abstract The Graph Isomorphism problem is the algorithmic problem to decide whether or not two given finite graphs are isomorphic. Recent work by the speaker has brought the worst-case complexity of this problem down from exp(\sqrt{n log n}) (Luks, 1983) to quasipolynomial (exp((log n)^c )), where n is the number of vertices.

In the first talk we state a core group theoretic lemma and sketch its role in the algorithm: the construction of global automorphisms out of local information.

The focus of the second and third talks will be the development of the main combinatorial “divide-and-conquer” tool, centered around the concept of coherent configurations. These highly regular structures, going back to Schur (1933), are a common generalization of strongly regular graphs and the more general distance-regular graphs and association schemes arising in the study of block designs on the one hand and the orbital structure of permutation groups on the other hand. Johnson graphs are examples of distance-regular graphs with a very high degree of symmetry.

Informally, the main combinatorial lemma says that any finite relational structure of small arity either has a measurable (say 10%) hidden irregularity or has a large degree of hidden symmetry manifested in a canonically embedded Johnson graph on more than 90% of the underlying set.

*Abstract:*

**Abstract**: We propose a variation of the classical isomorphism problem for group rings in the context of projective representations. We formulate several weaker conditions following from our notion and give all logical connections between these condition by studying concrete examples. We introduce methods to study the problem and provide results for various classes of groups, including abelian groups, groups of central type, $p$-groups of order $p^4$ and groups of order $p^2q^2$, where $p$ and $q$ denote different primes. Joint work with Leo Margolis.

*Abstract:*

Ben Passer will give the second lecture in his series of lectures on Noncommutative Borsuk Ulam theorems.

**Abstract: **

In algebraic topology, the Borsuk-Ulam theorem and its extensions place restrictions on maps between compact spaces. Essential to this story is the antipodal action on the sphere, which sends each point x to -x, so equivariant maps are commonly called "odd". The original Borsuk-Ulam theorem then says that there is no odd map from a sphere of high dimension to a sphere of low dimension. This may be extended to more general compact spaces with free actions of finite groups by considering connectivity of a domain X and dimension of a codomain Y.

I will present my work on extending this theorem and similar results to C*-algebras, as motivated by the results and conjectures of other researchers (Yamashita, Taghavi, Baum-Dabrowski-Hajac). Along the way, we will see how this point of view may be used to improve topological results, and how the noncommutative setting differs from the commutative setting.

*Abstract:*

Attached.

*Abstract:*

I will describe the abstract commensurability classification within a class of hyperbolic right-angled Coxeter groups. I will explain the relationship between these groups and a related class of geometric amalgams of free groups, and I will highlight the differences between the quasi-isometry classification and abstract commensurability classification in this setting. This is joint work with Pallavi Dani and Anne Thomas.

*Abstract:*

Let $b$ be a positive integer larger than or equal to two. A real number $x$ is called normal to base $b$, if in its base-$b$ expansion all finite blocks of digits occur with the expected frequency. Equivalently, $x$ is normal to base $b$ if the orbit of $x$ under the multiplication-by-$b$ map is uniformly distributed in the unit interval with respect to Lebesgue measure.While there are many explicit constructions of normal numbers to a single base it remains an open problem going back to Borel in 1909 to exhibit an easy example of an absolutely normal number (i.e. a real number that is normal to all integer bases simultaneously). In this talk I will explain algorithms by Sierpinski and Becher-Heiber-Slaman that produce absolutely normal numbers one digit after the other. I will show how these algorithms can be extended to give computable constructions of absolutely normal numbers that also have a normal continued fraction expansion, or are normal with respect to expansions to non-integer bases. Some ideas from ergodic theory will occur, but the proofs are based on large deviation theorems from probability theory for sums of dependent random variables. This allows to make certain constants implied by the Shannon-McMillan-Breimann theorem in special cases explicit so we can in fact avoid ergodic theory. If time permits, I will also say something about the trade-off between time-complexity and speed of convergence to normality for normal numbers.

*Announcement:*

**מרצה בכיר רון רוזנטל**

**הפקולטה למתמטיקה**

**טכניון**

**Assistant Professor Ron Rosenthal**

**The Faculty of Mathematics**

**Technion**

**Math Club 17.1.17**

**מודלי גידול**

מודלי גידול מתארים תופעות טבעיות רבות, כגון גידול של גבישים, מושבות של בקטריות, אצות, אלמוגים ועוד. דרך טבעית לתיאור מודלי גידול היא באמצעות תהליכים הסתברותיים בהם חלקיקים מצטברים באופן אקראי סביב נקודה התחלתית. אנו נתאר מספר מודלי גידול ואת ההתנהגות שלהם.

**Growth models**

Growth models describe many natural phenomena such as the growth of crystals, colonies of bacteria, algae, corals, etc. One method for constructing growth models is to use random processes in which particles aggregate around an initial cluster. We will describe several growth models and their behavior.

ההרצאה תהיה בעברית

The lecture will be in Hebrew

*Abstract:*

We discuss the main ideas in the derivation of two-sided estimates of Green functions for a class of Schroedinger operators defined on Lipschitz bounded domains. An important ingredient is the Boundary Harnack Principle which in smooth domains is closely related to Hopf's lemma. Except for some special cases, these estimates seem to be new even in the case of smooth domains. In Lipschitz domains the estimates are known for the Laplacian and for Schroedinger equations provided that the potential has no strong singularity.

*Abstract:*

The purpose of this talk is to introduce a new concept, the "radius" of elements in arbitrary finite-dimensional power-associative algebras over the field of real or complex numbers. It is an extension of the well known notion of spectral radius.

As examples, we shall discuss this new kind of radius in the setting of matrix algebras, where it indeed reduces to the spectral radius, and then in the Cayley-Dickson algebras, where it is something quite different.

We shall also describe two applications of this new concept, which are related, respectively, to the Gelfand formula, and to the stability of norms and subnorms.

*Abstract:*

We study convex bi-level optimization problems for which the inner level consists of minimization of the sum of smooth and nonsmooth functions. The outer level aims at minimizing a smooth and strongly convex function over the optimal solution set of the inner problem. We analyze two first order methods and global sublinear rate of convergence of the methods is established in terms of the inner objective function values. The talk is based on two works: one with Amir Beck (Technion) and one with Shimrit Shtern (MIT).

*Abstract:*

On the 15, 22 and perhaps also 29 of January, Ben Passer will give a series of lectures on Noncommutative Borsuk-Ulam Theorems.

**Abstract: **

In algebraic topology, the Borsuk-Ulam theorem and its extensions place restrictions on maps between compact spaces. Essential to this story is the antipodal action on the sphere, which sends each point x to -x, so equivariant maps are commonly called "odd". The original Borsuk-Ulam theorem then says that there is no odd map from a sphere of high dimension to a sphere of low dimension. This may be extended to more general compact spaces with free actions of finite groups by considering connectivity of a domain X and dimension of a codomain Y.

I will present my work on extending this theorem and similar results to C*-algebras, as motivated by the results and conjectures of other researchers (Yamashita, Taghavi, Baum-Dabrowski-Hajac). Along the way, we will see how this point of view may be used to improve topological results, and how the noncommutative setting differs from the commutative setting.

*Abstract:*

I will discuss a convolution operator associated with Harish-Chandra’s Schwartz space of discrete groups of any semisimple Lie group. I will show that the latter space carries a natural structure of convolution algebra. Besides, a control of the l^2 convolutor norm by the norm of this space holds. I will explain how this inequality is related to property RD and I will make a connection with the Baum-Connes conjecture.

*Abstract:*

The past couple of years have seen several major results in the study of Latin squares. A transversal in an order-n Latin square is a set of n elements, one from each row and column and one of each symbol. Let T(n) denote the maximal number of transversals that an order-n Latin square can have. In a joint work with Roman Glebov, we proved asymptotically tight upper and lower bounds on T(n), using probabilistic methods. More recent developments include an algebraic construction of Latin squares that achieve the lower bound. It was also shown that Keevash's recent construction of designs could be used to show that whp random Latin squares attain the lower bound. The expander mixing lemma is concerned with the discrepancy of regular graphs. One can consider this parameter in higher dimensions as well, and in particular for Latin squares. In a joint work with Nati Linial, we conjectured that a typical Latin square has low discrepancy, and proved a related result. More recently, Kwan and Sudakov showed that a breakthrough result of Liebenau and Wormald on the enumeration of regular graphs implies our conjecture for Latin squares up to a multiplicative factor of log^2(n). Many open questions remain.

*Abstract:*

The first quasicrystals where discovered by D. Shechtman in the year 1984. From the mathematical point of view, the study of the associated Schrödinger operators turns out to be a challenging question. Up to know, we can mainly analyze one-dimensional systems by using the method of transfer matrices. In 1987, A. Tsai et al. discovered a quasicrystalline structure in an Aluminum-Copper-Iron composition. By changing the concentration of the chemical elements, they produce a stable quasicrystaline structure by an approximation process of periodic crystals. In light of that it is natural to ask whether Schrödinger operators related to aperiodic structures can be approximated by periodic ones while preserving spectral properties. The aim of the talk is to provide a mathematical foundation for such approximations.

In the talk, we develop a theory for the continuous variation of the associated spectra in the Hausdorff metric meaning the continuous behavior of the spectral gaps. We show that the convergence of the spectra is characterized by the convergence of the underlying dynamics. Hence, periodic approximations of Schrödinger operators can be constructed by periodic approximations of the dynamical systems which we will describe along the lines of an example.

*Abstract:*

I will discuss isoperimetric problems and their generalizations and applications. The generalization will involve more global notions of boundary as well as partitions into more than 2 parts.

*Abstract:*

Realistic physical models represented by elliptic boundary value problems are of immense importance in predictive science and engineering applications. Effective solution of such problems, essentially, requires accurate numerical discretization that take into account complexities such as irregular geometries and unstable interfaces. This typically leads to large-scale (1M unknowns or more) ill-conditioned linear systems, that can only be resolved by iterative methods combined with multilevel preconditioning schemes. The class of hierarchical matrix approximations is a multilevel scheme which offers unique advantages over other traditional multilevel methods, e.g., multigrid. Essentially, a hierarchical matrix is a perturbed version of the input linear system. Thus, in principle, the magnitude of the perturbation needs to be smaller than the smallest modulus eigenvalue of the system matrix. For many problems, the perturbation may have to be chosen quite small, generally, leading to less efficient preconditioners. In this talk we will present a new strong hierarchical preconditioning scheme that overcomes the perturbation limit. We will start with an overview on hierarchical matrices, and continue with theoretical results on optimal preconditioning in the symmetric positive definite case. The effectiveness of the new method which outperforms other classical techniques will be illustrated through numerical experiments. In the final part of the talk we will also suggest directions towards extending the theory to indefinite and non-symmetric linear systems.

*Abstract:*

We study the properties of the set S of non-differentiable points of viscosity solutions of the Hamilton-Jacobi equation, for a Tonelli Hamiltonian.The main surprise is the fact that this set is locally arc connected—it is even locally contractible. This last property is far from generic in the class of semi-concave functions.We also “identify” the connected components of this set S. This work relies on the idea of Cannarsa and Cheng to use the positive Lax-Oleinik operator to construct a global propagation of singularities (without necessarily obtaining uniqueness of the propagation).

This is a joint work with Piermarco Cannarsa and Wei Cheng.

*Abstract:*

Metallic nanoparticles are optically extraordinary in that they support resonances at wavelengths that greatly exceed their own size. These “surface-plasmon” resonances are normally in the visible range, the (roughly scale-invariant) “colours” sensitively depending on material and shape. In creating the dichroic glass of the Lycurgus cup, the ancient Romans had exploited the phenomenon, probably unknowingly, already in the 4th Century. Nowadays, surface-plasmon resonance is fundamental to the field of nanophotonics, where the goal is to manipulate light on small scales below the so-called diffraction limit. Numerous emerging applications rely on the ability to design and realise compound nanostructures that support tunable and strongly localised resonances.

In this talk I will focus on the misleadingly simple-looking eigenvalue problem governing the colours of plasmonic nanostructures. I’ll present new asymptotic solutions that describe the resonances of the multiple-scale structures ubiquitous in applications: dimers of nearly touching nanowires (2D) and spheres (3D), elongated nano-rods, particles nearly touching a mirror etc. The plasmonic spectrum of these structures can be quite rich. For example, the spectrum of a sphere dimer is compound of three families of modes, each behaving differently in the near-contact limit; moreover, these asymptotic trends mutate at moderately high mode numbers (and again at yet larger mode numbers). This non-commutativity of limits will lead me to a discussion of the convergence in 2D and 3D of the spectrum to a universal accumulation point (the “surface-plasmon frequency”) as the mode number tends to infinity. Time permitting, I will also discuss the asymptotic renormalisation of the singular eigenvalues of closely separated dimer configurations owing to “nonlocal” effects (with Richard V. Craster, Vincenzo Giannini and Stefan A. Maier).

*Abstract:*

**Abstract**: Let R be a discrete valuation ring with fraction field K. It is a classical result that two nondegenerate quadratic forms over R that become isomorphic over K are already isomorphic over R. [Here, a quadratic form over R is a map q:R^n->R of the form q(x)=x^{T}Mx with M a symmetric matrix, and q is nondegenerate if M is invertible over R.] This result is a special case of a conjecture of Grothendieck and Serre concerning the etale cohomology of reductive group schemes over local regular rings. Much progress has been made recently in proving the conjecture, mostly due to Panin.I will discuss a generalization of the aforementioned result to certain degenerate quadratic and also to hermitian forms over certain (non-commutative) R-algebras. This generalization suggests that the conjecture of Grothedieck and Serre may apply to certain families of non-reductive groups arising from Bruhat-Tits theory. Certain cases of this extended conjecture were already verified and others are currently under investigation.

*Abstract:*

The mean curvature flow appears naturally in the motion of interfaces in material science, physics and biology. It also arises in geometry and has found its applications in topological classification of surfaces. In this talk I will discuss recent results on formation of singularities under this flow. In particular, I will describe the 'spectral' picture of singularity formation and sketch the proof of the neck pinching results obtained jointly with Zhou Gang and Dan Knopf.

*Abstract:*

I will describe a new approach to chaotic flows in dimension three, using knot theory. I'll use this to show that one can get rid of the singularities in the famous Lorenz flow on R^3, and obtain a flow on a trefoil knot complement. The flow can then be related to the geodesic flow on the modular surface. When changing the parameters, we find other knots for the Lorenz system and so this uncovers certain topological phases in the Lorenz system.

*Abstract:*

Let $K$ be a commutative ring. Consider the groups $GL_n(K)$. Bernstein and Zelevinsky have studied the representations of the general linear groups in case the ring $K$ is a finite field. Instead of studying the representations of $GL_n(K)$ for each $n$ separately, they have studied all the representations of all the groups $GL_n(K)$ simultaneously. They considered on $R:=\oplus_n R(GL_n(K))$ structures called parabolic (or Harish-Chandra) induction and restriction, and showed that they enrich $R$ with a structure of a so called positive self adjoint Hopf algebra (or PSH algebra). They use this structure to reduce the study of representations of the groups $GL_n(K)$ to the following two tasks:

1. Study a special family of representations of $GL_n(K)$, called "cuspidal representations''. These are representations which do not arise as direct summands of parabolic induction of smaller representations.

2. Study representations of the symmetric groups. These representation also has a nice combinatorial description, using partitions.

In this talk I will discuss the study of representations of $GL_n(K)$ where $K$ is a finite quotient of a discrete valuation ring (such as $\Z/p^r$ or $k[x]/x^r$, where $k$ is a finite field). One reason to study such representation is that all continuous complex representations of the groups $GL_n(\Z_p)$ and $GL_n(k[[x]])$ (where $\Z_p$ denotes the $p$-adic integers) arise from these finite quotients. I will explain why the natural generalization of the Harish-Chandra functors do not furnish a PSH algebra in this case,and how is this related to the Bruhat decomposition and Gauss elimination.

In order to overcome this issue we have constructed a generalization of the Harish-Chandra functors. I will explain this generalization, describe some of the new functors properties, and explain how can they be applied to studying complex representations.

The talk will be based on a joint work with Tyrone Crisp and Uri Onn.

*Abstract:*

I will overview how tubular groups have been studied over the past 30-40 years in geometric group theory before explaining recent results relating to the cubulation of tubular groups including my own work classifying which tubular groups are virtually special.

*Abstract:*

We consider the orbits {pu(n^{1+r})} in Γ∖PSL(2,R), where r>0, Γ is a non-uniform lattice in PSL(2,R) and u(t) is the standard unipotent group in PSL(2,R). Under a Diophantine condition on the intial point p, we prove that such an orbit is equidistributed in Γ∖PSL(2,R) for small r>0, which generalizes a result of Venkatesh. Also we generalize this Diophantine condition to any finite-volume homogeneous space G/Γ, and compute Hausdorff dimensions of Diophantine points of various types in a rank one homogeneous space G/Γ. In particular, this gives a Jarnik-Besicovitch theorem on Diophantine approximation in Heisenberg groups.

*Abstract:*

The Operator Scaling problem asks whether a set of complex matrices can be jointly moved to a certain canonical (isotropic) position. This problem has a remarkable number of myriad incarnations: non-commutative algebra, invariant theory, arithmetic complexity, quantum information theory, analytic inequalities and more. We will describe an efficient algorithm solving all these related problems, and explain how their analysis combines ideas from all these areas. Through these connections, the algorithm can be shown to solve some non-convex optimization problems, some systems of quadratic equations, and some linear programs with exponentially many inequalities. All these, and concrete examples we will give, suggest that it might be a powerful algorithmic tool via reductions to these problems. No special background will be assumed! Joint on two joint works with Ankit Garg, Leonid Gurvits and Rafael Olivera. This talk is longer than usual and has a two-hour slot.

*Abstract:*

The space of smoothly embedded n-spheres in R^{n+1} is the quotient space M_n:=Emb(S^n,R^{n+1})/Diff(S^n). In 1959 Smale proved that M_1 is contractible and conjectured that M_2 is contractible as well, a fact that was proved by Hatcher in 1983.For n\geq 3, even the simplest questions regarding M_n are both open and central. For instance, whether or not M_3 is path connected is an equivalent form of one of the most important open questions in differential topology - the smooth Schoenflies conjecture. In particular, if M_3 is not path connected, the smooth 4 dimensional Poincare conjecture can not be true. In this talk, I will explain how mean curvature flow, a geometric analogue of the heat equation, can assist in studying the topology of geometric relatives of M_n.I will first illustrate how the theory of 1-d mean curvature flow (aka curve shortening flow) yields a very simple proof of Smale's theorem about the contractibility of M_1.I will then describe a recent joint work with Reto Buzano and Robert Haslhofer, utilizing mean curvature flow with surgery to prove that the space of 2-convex embedded spheres is path connected.

*Abstract:*

Amazon lets clients bid for coputing resources and publishes the uniform prices that result from this auction. Analyzing these prices and reverse engineering them revealed that prices were usually set artificially and not market driven, in contransr to Amazon's declaration.

***This lecture is intended for undergraduate students **

*Abstract:*

Liouville's rigidity theorem (1850) states that a map $f:\Omega\subset R^d\to R^d$ that satisfies $Df \in SO(d)$ is an affine map. Reshetnyak (1967) generalized this result and showed that if a sequence $f_n$ satisfies $Df_n \to SO(d)$ in $L^p$, then $f_n$ converges to an affine map.

In this talk I will discuss generalizations of these theorems to mappings between manifolds, present some open questions, and describe how these rigidity questions arise in the theory of elasticity of pre-stressed materials (non-Euclidean elasticity).

If time permits, I will sketch the main ideas of the proof, using Young measures and harmonic analysis techniques, adapted to Riemannian settings.

Based on a joint work with Asaf Shachar and Raz Kupferman.

*Abstract:*

While the topic of geometric incidences has existed for several decades, in recent years it has been experiencing a renaissance due to the introduction of new polynomial methods. This progress involves a variety of new results and techniques, and also interactions with fields such as algebraic geometry and harmonic analysis.

A simple example of an incidences problem: Given a set of n points and set of n lines, both in R^2, what is the maximum number of point-line pairs such that the point is on the line. Studying incidence problems often involves the uncovering of hidden structure and symmetries.

In this talk we introduce and survey the topic of geometric incidences, focusing on the recent polynomial techniques and results (some by the speaker). We will see how various algebraic and analysis tools can be used to solve such combinatorial problems.

*Abstract:*

Character rings of Lie superalgebras have a nice presentation as rings of supersymmetric Laurent polynomials as was shown by Sergeev and Veselov. The Duflo-Serganova functor is a useful tool for studying the category of finite-dimensional modules over a Lie superalgebra, however this functor is not exact. We have shown that the Duflo-Serganova functor induces a ring homomorphism on a natural quotient of the Grothendieck ring, which is isomorphic to the character ring. We can realize this homomorphism as a certain evaluation of functions related to the supersymmetry property defining the character ring, and we used this realization to describe its kernel and image. Joint with Reif.

*Abstract:*

It is an old conjecture that closed (even dimensional) manifolds with nonzero Euler characteristic admit no flat structure. Although it turns out that there do exist manifolds with nonzero Euler characteristic that admit a flat structure, for closed aspherical manifolds this conjecture is still widely open. In 1958 Milnor proved the conjecture for surfaces through his celebrated inequality. Gromov naturally put Milnor’s inequality in the context of bounded cohomology, relating it to the simplicial volume.

I will show how to find upper and lower bounds for the simplicial volume of complex hyperbolic surfaces. The upper bound naturally leads to so-called Milnor-Wood inequalities strong enough to exclude the existence of flat structures on these manifolds.

*Abstract:*

For the abstract see the attached .pdf

*Abstract:*

**Advisor:** Prof. Yoav Moriah

**Abstract:** Every closed orientable 3-dimensional manifold M admits a Heegaard splitting, i.e. a decomposition into two handlebodies which meet along their boundary. This common boundary is called a Heegaard surface in M, and is usually considered only up to isotopy in M. The genus g of the Heegaard surface is said to be the genus of the handlebodies. A Heegaard splitting gives us the Heegaard distance, which is defined using the curve complex. The fact that a Heegaard splitting is high distance has important consequences for the geometry of the 3-manifold determined by it. We will discuss two previously introduced combinatorial conditions on the Heegaard distance - the rectangle condition and the double rectangle condition - and their affect on the Heegaard distance, and hence on the geometry of the 3-manifold.

*Abstract:*

In his 1947 paper that inaugurated the probabilistic method, Erdős proved the existence of (2+o(1))log(n)-Ramsey graphs on n vertices. Matching Erdős' result with a constructive proof is an intriguing problem in combinatorics that has gained significant attention in the literature. In this talk, we will present recent works towards this goal.

*Announcement:*

**פרופ' רון הולצמן**

הפקולטה למתמטיקה

טכניון

**Prof. Ron Holzman**

The Faculty of Mathematics

Technion

**Math Club 13.12.16**

**אנא שימו לב לשעת ההרצאה הלא שגרתית. **

**אחרי ההרצאה יתקיים טקס הענקת הפרסים של התחרות ע"ש גרוסמן**

**דינמיקת הרוב על גרפים**

נתון גרף, שכל אחד מקדקודיו צבוע באדום או בירוק. הצביעה מתעדכנת מדי יום, כאשר כל קדקוד מאמץ לעצמו את הצבע שבו נצבעו רוב שכניו ביום הקודם (במקרה של תיקו – הוא שומר על צבעו). מה יקרה ברבות הימים?

**ההרצאה מוקדשת לזכרו של פרופ' גדי מורן ז"ל.**

**Majority dynamics on graphs **

A graph is given, with every vertex colored red or green. The coloring is updated every day, each vertex adopting the color of the majority of its neighbors on the previous day (in case of a tie - it keeps its own color). What will happen eventually?

**The lecture is dedicated to the memory of Prof. Gadi Moran. **

ההרצאה תהיה בעברית

The lecture will be in Hebrew

*Abstract:*

Atomic systems are regularly studied as large sets of point-like particles, and so understanding how particles can be arranged in such systems is a very natural problem. However, aside from perfect crystals and ideal gases, describing this kind of “structure” in an insightful yet tractable manner can be challenging. Analysis of the configuration space of local arrangements of neighbors, with some help from the Borsuk-Ulam theorem, helps explain limitations of continuous metric approaches to this problem, and motivates the use of Voronoi cell topology. Several short examples from materials research help illustrate strengths of this approach.

*Abstract:*

I will give a very personal overview of the evolution of mainstream applied mathematics from the early 60's onwards. This era started pre computer with mostly analytic techniques, followed by linear stability analysis for finite difference approximations, to shock waves, to image processing, to the motion of fronts and interfaces, to compressive sensing and the associated optimization challenges, to the use of sparsity in Schrodinger's equation and other PDE's, to overcoming the curse of dimensionality in parts of control theory and in solving the associated high dimensional Hamilton-Jacobi equations.

*Abstract:*

Given a finite group G, we consider in this talk ``parametric sets'' (over $\mathbb{Q}$), {\it{i.e.}}, sets $S$ of (regular) Galois extensions of $\mathbb{Q}(T)$ with Galois group $G$ whose specializations provide all the Galois extensions of $\mathbb{Q}$ with Galois group $G$. This relates to the Beckmann-Black Problem (which asks whether the strategy by specialization to solve the Inverse Galois Problem is optimal) which can be formulated as follows: does a given finite group $G$ have a parametric set over $\mathbb{Q}$?

We show that many finite groups $G$ have no finite parametric set over $\mathbb{Q}$. We also provide a similar conclusion for some infinite sets, under a conjectural ``uniform Faltings theorem''.

This is a joint work with Joachim K\"onig.

*Abstract:*

Assuming that the absence of perturbations guarantees either weak or strong convergence to a common fixed point, we study the behavior of perturbed products of an infinite family of nonexpansive operators. Our main result indicates that the convergence rate of unperturbed products is essentially preserved in the presence of perturbations. This, in particular, applies to the linear convergence rate of dynamic string averaging projection methods, which we establish here as well. This is joint work with Christian Bargetz and Simeon Reich.

*Abstract:*

Stable subgroups and the Morse boundary are two systematic approaches to collect and study the hyperbolic aspects of finitely generated groups. I will introduce a new quasi-isometry invariant of geodesic metric spaces which generalizes these strategies: the stable dimension. In the case of a proper Gromov hyperbolic space the stable dimension is the asymptotic dimension. Time permitting I will also discuss the stable dimension in the cases of right-angled Artin groups, mapping class groups, and Teichm¨uller space. This is joint work with David Hume.

*Abstract:*

A lattice is topologically locally rigid (t.l.r) if small deformations of it are isomorphic lattices. Uniform lattices in Lie groups were shown to be t.l.r by Weil [60']. We show that uniform lattices are t.l.r in any compactly generated topological group.

A lattice is locally rigid (l.r) is small deformations arise from conjugation. It is a classical fact due to Weil [62'] that lattices in semi-simple Lie groups are l.r. Relying on our t.l.r results and on recent work by Caprace-Monod we prove l.r for uniform lattices in the isometry groups of proper geodesically complete CAT(0) spaces, with the exception of SL_2(\R) factors which occurs already in the classical case.

Moreover we are able to extend certain finiteness results due to Wang to this more general context of CAT(0) groups.

In the talk I will explain the above notions and results, and present some ideas from the proofs.

This is a joint work with Tsachik Gelander.

*Abstract:*

** Advisor: **Professor Alexander Nepomnyashchy.

**Abstract**: The transport induced by hydrodynamical flows often reveals anomalous properties. The anomalous transport in the case of a chaotic advection is a well-developed field. However, the anomalous properties of the advection by viscous flows in the absence of the Lagrangian chaos are much less explored.

In my talk I will introduce some basic concepts about the hydrodynamical problem and its interpretation in the framework of the theory of dynamical systems. The conception of Special Flow Introduced formerly in Ergodic Theory will help to understand the mechanisms behind the anomalous properties of the transport. I will describe the corresponding statistical properties induced by such a flow in order to draw conclusions on the original system.

*Announcement:*

**מרצה בכיר מיכאל חנבסקי**

**הפקולטה למתמטיקה**

**טכניון**

**Assistant Professor Michael Khanevsky**

**The Faculty of Mathematics**

**Technion**

**Math Club 6.12.16**

**ריצופים של המישור**

ריצוף הוא כיסוי של משטח באריחים בלי חיתוכים וללא פערים. הם מופיעים בכל מקום מסביב: מרצפות פרקט וחלות דבש עד התמונות של אשר. אנו נדבר על המתמטיקה מאחורי ריצופים אלו.

**Tessellations of the plane**

Tessellation is a tiling by geometric shapes with no gaps and no overlaps. They appear everywhere around us: from parquet floors and honeycombs to paintings by Escher. We will discuss the mathematics standing behind such tilings.

ההרצאה תהיה בעברית

The lecture will be in Hebrew

*Abstract:*

Brezis raised the question of uniqueness of positive radial solutions for critical exponent problems in a ball. Long back this was affirmatively solved in dimensions greater than two using the clever use of Pohozaev's identity. In dimension two, the critical nonlinearity is of exponential nature and the Pohozaev's identity is not effective. Using the Asymptotic analysis, I would like to show that Large solutions are unique.

*Abstract:*

**Supervisors**: Assoc. Prof. Alexander M. Leshansky and Dr. Konstantin I. Morozov, in the Faculty of Chemical Engineering

**Abstract**: Recent technological progress in micro- and nanoscale fabrication techniques allows for the construction and development of micron-scale robotic swimmers that can be potentially used for biomedical applications, such as targeted drug delivery and minimally invasive surgical procedures. An efficient technique for controlled steering of robotic microswimmers is by applying time-varying external magnetic fields. Recently, a general theory explaining the dynamics of arbitrary-shaped rigid objects in a rotating magnetic field was developed. Based on this theory, the genetic algorithm approach is applied in this study to optimize the shape of microrobots of certain symmetries. In addition, a numerical model of elastic magnetic microrobots will be presented.

*Abstract:*

Studying the regular part and shock curves for the entropy solutions for scalar conservation laws is a major research in this field. Assume that the initial date is constant in the connected components outside a compact interval, T.P Liu and Dafermos-Shearer obtained an interesting criterion when the solution admits one shock after a finite time for the uniformly convex flux. In this talk I will talk about the same phenomena for the flux which has finitely many inflection points. The proof relies on the structure theorem for entropy solutions for convex flux.

*Abstract:*

I will briefly recall the theory of Hurwitz spaces and their relevance to the Inverse Galois Problem. I will then describe techniques for explicit computation. Finally I will give a survey of problems to which these techniques can be applied. The focus will be on producing "nice" polynomials for nice groups, rather than providing exhaustive theoretical results.

*Abstract:*

We study variational inequalities in a real Hilbert space, which are governed by a strongly monotone and Lipschitz continuous operator $F$ over a closed and convex set $C$. We assume that the set C can be outerly approximated by the fixed point sets of a sequence of certain quasi-nonexpansive operators called cutters. We propose an iterative method the main idea of which is to project at each step onto a particular super-half-space constructed by using the input data. Our approach is based on a method presented by Fukushima in 1986, which has recently been extended by several authors. We establish strong convergence in Hilbert space. To the best of our knowledge, Fukushima's method has so far been considered only in the Euclidean setting with different conditions on $F$. We also provide numerical illustrations of our theoretical results. This talk is based on joint work with Aviv Gibali and Rafal Zalas.

*Abstract:*

It is well known that uniform spaces are inverse limits of pseudo-metric and, dually, that coarse spaces are direct limits of infinity-metric spaces. Usually, if, for example, a uniform space has some really nice covering property, then one can expect each of the metric spaces in the inverse approximation to also have that property. The dual statement for coarse spaces is also true. In a strong sense, both uniform spaces and coarse spaces are just special cases of groupoids. In a joint project with Joav Orovitz, we are employing an inverse approximation technique to topological groupoids that generalizes both of the above cases and we apply our metric approximations to extend the classical disintegration theorem of groupoid representations of John Renault. I plan on giving an overview of how the approximations work and how we use them in our proof of the aforementioned theorem.

*Abstract:*

Manifolds of negative sectional curvature are an object of interest and their study goes back to Cartan and Hadamard.

It is well known that the topology of such manifolds is controlled, to some extent, by their volume. This is best illustrated in dimension 2: the homemorphism type of a compact orientable surface is determined by its volume (suitably normalized) - this follows from the celebrated Gauss-Bonnet theorem. Gromov proved in 1978 that the Betti numbers of negatively curved manifolds are bounded by means of the volume in every dimension, but also provided an example of a sequence of negatively curved 3-manifolds of uniformly bounded volume and pairwise different first integral homology. A crucial tool in Gromov's proof is the famous "thin-thick decomposition" of a manifold.

In my talk I will report on a joint work with Gelander and Sauer, in which we introduce a modification of this decomposition that gives a better model for the topology of a manifold: a negatively curved manifold is homotopic to a simplicial complex with handles, where the number of simplices is bounded by means of the volume of the manifold. This shows in particular that Gromov's 3d example could not be given in higher dimensions and that in dimension 5 and more the number of homeomorphism types of manifolds is bounded by means of the volume.

*Abstract:*

In this talk we study short edge-disjoint paths in expander graphs(here it mean: graph with constant mixing time). We use the Lovasz Local Lemma to prove the following result: Given a d-regular

*Abstract:*

In this talk we study short edge-disjoint paths in expander graphs (here it means: graph with constant mixing time). We use the Lovasz Local Lemma to prove the following result: Given a d-regular expander graph G and a set L={(s_i,t_i)} such that each vertex of G appears at most O(d) times in the list, there exist a set of edge disjoint paths of constant length connecting each s_i to t_i. This result has applications to multi-party computations performed over networks in the presence of random noise. Based on work with Noga Alon, Mark Braverman, Ran Gelles, Bernhard Haeupler.

*Abstract:*

The celebrated Faber-Krahn inequality yields that, among all domainsof a fixed volume, the ball minimizes the lowest eigenvalue of the Dirichlet Laplacian. This result can be viewed as a spectral counterpart of the well known geometric isoperimetric inequality. The aim of this talk is to discuss generalizations of the Faber-Krahn inequality for optimization of the lowest eigenvalues for:

- Schrodinger operators with $\delta$-interactions supported on conical surfaces and open arcs [1,3];
- Robin Laplacians on exterior domains and planes with slits [2,3].

Beyond a physical relevance of $\delta$-interactions and Robin Laplacians,a purely mathematical motivation to consider these optimization problems stems from the fact that standard methods, going back to the papers of Faber and Krahn, are not applicable anymore. Another interesting novel aspect is that in some cases the shape of the optimizer bifurcates as the boundary parameter varies while in the othercases no optimizer exists. The results in the talk are obtained in collaboration with P. Exner and D. Krejcirik.

Bibliography

- P. Exner and V. Lotoreichik, A spectral isoperimetric inequality for cones, \emph{to appear in Lett. Math. Phys., arXiv:1512.01970.
- D. Krejcirik and V.Lotoreichik, Optimisation of the lowest Robin eigenvalue in the exterior of a compact set, submitted, arXiv:1608.04896.
- V. Lotoreichik, Spectral isoperimetric inequalities for $\delta$-interactions on open arcs and for the Robin Laplacian on planes with slits, submitted, arXiv:1609.07598.

*Abstract:*

Erdős-Ko-Rado type problems' have been widely studied in Combinatorics and Theoretical Computer Science over the last fifty years. In general, these ask for the maximum possible size of a family of objects, subject to the constraint that any two (or three…) of the objects `intersect' or `agree' in some way. A classical example is the so-called Erdős-Ko-Rado theorem, which gives the maximum possible size of a family of k-element subsets of an n-element set, subject to the constraint that any two sets in the family have nonempty intersection. As well as families of sets, one may consider families of more highly structured objects, such as graphs or permutations; one may also consider what happens when additional `symmetry' requirements are imposed on the families. A surprisingly rich variety of techniques from different areas of mathematics have been used successfully in this area: combinatorial, probabilistic, analytic and algebraic. For example, Fourier analysis and representation theory have recently proved useful. I will discuss some results and open problems in the area, some of the techniques used, and some links with other areas.

*Abstract:*

In this talk we study new algorithmic structures with Douglas--Rachford (DR) operators to solve convex feasibility problems. We propose to embed the basic two-set-DR algorithmic operator into the string-averaging projections and into the block-iterative projection algorithmic structures, thereby creating new DR algorithmic schemes that include the recently proposed cyclic DR algorithm and the averaged DR algorithm as special cases. We further propose and investigate a new multiple-set-DR algorithmic operator. Convergence of all these algorithmic schemes is studied by using properties of strongly quasi-nonexpansive operators and firmly nonexpansive operators. This is joint work with Yair Censor.

*Abstract:*

We will study the preprint by Fritz, Netzer and Thom with the title as above, available on the arxiv:

https://arxiv.org/abs/1609.07908

*Abstract:*

In my talk I will give a review of the subject. I will present the steps of the classification of surfaces, using very nice methods and techniques, such as: degeneration of surfaces, braid monodromy, calculations of fundamental groups and Coxeter groups. We will see interesting examples of classification of known and significant surfaces, such as Hirzebruch surfaces.

*Abstract:*

We present a new approach (joint with M. Bjorklund (Chalmers)) for finding new patterns in difference sets E-E, where E has a positive density in Z^d, through measure rigidity of associated action.

By use of measure rigidity results of Bourgain-Furman-Lindenstrauss-Mozes and Benoist-Quint for algebraic actions on homogeneous spaces, we prove that for every set E of positive density inside traceless square matrices with integer values, there exists positive k such that the set of characteristic polynomials of matrices in E - E contains ALL characteristic polynomials of traceless matrices divisible by k.

By use of this approach Bjorklund and Bulinski (Sydney), recently showed that for any quadratic form Q in d variables (d >=3) of a mixed signature, and any set E in Z^d of positive density the set Q(E-E) contains kZ for some positive k. Another corollary of our approach is the following result due to Bjorklund-Bulinski-Fish: the discriminants D = {xy-z^2 , x,y,z in B} over a Bohr-zero non-periodic set B covers all the integers.

*Abstract:*

**Adviser**: Prof. Dan Givoli from** t**he Interdisciplinary Program for Applied Mathematics

**Abstract: **The need to reduce the size of large discrete models is a reoccurring theme in computational mechanics in recent years. One situation which calls for such a reduction is that where the solution in some region in a high-dimensional computational domain behaves in a low-dimensional way. Typically, this situation occurs when the LowD (Low-Dimensional) model is employed as an approximation to the HighD (High-Dimensional) model in a partial region of the spatial domain. Then, one has to couple the two models on the interface between them. Fields of application where the scenario of LowD-HighD coupling is of special interest include, among others, blood-flow analysis, hydrological and geophysical flow models and elastic structures, where slender members behave in a 1D way, while joints connecting these members possess a 3D behavior. The hybrid HighD-LowD model, if designed properly, is much more efficient than the standard HighD model taken for the entire problem.

This work focuses on the coupling of two-dimensional (2D) and one-dimensional (1D) models in time-harmonic elasticity. The 2D and 1D structural regions are discretized by using 2D and 1D Finite Element (FE) formulations. Two important issues related to such hybrid 2D-1D models are: (a) the design of the hybrid model and its validation (with respect to the original problem), and (b) the way the 2D-1D coupling is done, and the coupling error generated. This research focuses on the second issue.

Several methods are adapted to the 2D-1D coupling scenario, implemented and compared numerically through a specially designed benchmark problem , as well as some more advanced problems.** **

*Abstract:*

I present a new approach to classify the asymptotic behavior of certain types of wave equations, supercritical and others, with large initial data. In some cases, as for Nirenberg type equations, a fairly complete classification of the solutions (finite time blowup or global existence and scattering) is proved.

New results are obtained for the well known monomials wave equations in the sub/critical/super critical cases.

This approach, developed jointly with M. Beceanu, is based on a new decomposition into incoming and outgoing waves for the wave equation, and the positivity of the fundamental solution of the wave equation in three dimensions.

*Abstract:*

The concept of measurable entropy goes back to Kolmogorov and Sinai who in the late 50ies defined an isomorphism invariant for measure preserving Z-actions. While a similar theory can be developed in an analogous manner for abelian or even amenable groups, the situation gets far more complicated when dealing with groups which are "very" non-commutative, such as free groups. We start the talk with a warm-up about the classical Kolmogorov-Sinai entropy. Using the free group on two generators as an illustrative example, we show how to define cocycle entropy as a new isomorphism invariant for measure preserving actions of quite general countable groups. Further, we draw connections to other notions of entropy and to open problems in the field. We conclude the talk by clarifying pointwise almost sure approximation of cocycle entropy values. To this end, we present a first Shannon-McMillan-Breiman theorem for actions of non-amenable groups. Joint work with Amos Nevo.

*Abstract:*

The interpretation of Einstein's equations as a geometric flow (the Einstein flow) allows to study the evolution of spacetimes from a dynamical point of view. Two types of initial data are mainly considered: Firstly, asymptotically flat data describing initial states of isolated self-gravitating systems and secondly, data on closed manifolds describing initial states for cosmological spacetimes. Studying the evolution of data under the flow we aim to understand its long-time behavior and the global geometry of its time-development. We are interested in the construction of static solutions (or static up to a time-rescaling) as potential attractors of the flow and their nonlinear stability, completeness and incompleteness properties of spacetimes and singularity formation. We present new methods to construct and study solutions by geometric and analytical tools as well as several results in the directions mentioned above. We consider in particular the case of matter models coupled to the Einstein equations, which turns out to provide several interesting phenomena and new classes of solutions.

*Abstract:*

We continue studying the paper of Alexeev, Netzer and Thom discussed in the first two lectures.