# Faculty Activities[ Edit ]

*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:*

**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:*

**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:*

**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:*

**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:*

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:*

**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:*

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:*

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:*

**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:*

**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:*

**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:*

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:*

**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:*

** 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.

*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:*

**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:*

**Advisor: **Assistant Professor Barak Fishbain

**Abstract:** Air pollution is a significant risk factor for multiple health situations. In addition, it causes many negative effects on the environment. Thus, arises the need for assessing air-quality. Air quality modeling is an essential tool this task and is in use in many studies such as air quality management and control, epidemiological studies and public health. Today, most of air-pollution modeling is based on data acquired from Air Quality Monitoring (AQM) stations. AQM provides continuous measurements and considered to be accurate; however, they are expansive to build and operate, therefore scattered sparingly. As the number of measuring sites is limited, the information obtained from those measurements is generalized with mathematical methods.Here we introduce two methods to improve the spatio-temporal coverage. The first method, a new interpolation scheme, will expand the scope of the spatial coverage in order to infer the pollution levels in the entire study area. The second is a long-term forecasting method, to implement a better and wide perspective of the temporal coverage. Many researches in air quality modeling uses interpolation schemes such as IDW or Ordinary Kriging. Yet, the mathematical basis of those schemes defines that the extremum value obtained at the measuring places (without considering edge effects). In addition, they are not considering the location of pollution source or any physicochemical characteristics of pollution, hence does not reveal the real spatial coverage. Our interpolation scheme takes into account patterns of dispersion and source location. Source detection is achieved through a novel Hough Transform-like technique.Extending the temporal coverage of the measuring array is achieved through long-term forecasting. Nowadays there are only short-term forecasting methods (24-72 hours ahead), no method exists for long-term (e.g. a year) forecasting. Discrete Time Markov Model is a well-known probabilistic model used to describe and analyze stochastic processes. Here we first define and introduce a method for long-term forecasting based on Discrete-time Markov model for a better temporal coverage.These building blocks which, will be presented in the talk, facilitate the future study of spatio-temporal interpolation methods, which improve the current state-of-the-art by devising new source-location based interpolation methods.

*Abstract:*

**Adviser**: Prof. Eddy Meir-Wolf

**Abstract: **The Onsager-Machlup functional of a Cameron-Martin path relates to the probability that the solution of a stochastic differential equation lies in a small ball (or “tube”) around the path. Its computation is typically dependent of “approximatelimits” of Wiener functionals with respect to a given measurable norm.

We will discuss certain stochastic differential equations driven by fractional Brownian motion and the paths near which their solutions typically reside

*Abstract:*

**Adviser**: Prof. Amy Novick-Cohen

**Abstract**: If we look at most materials under a microscope, we will see a network of grains and grain boundaries as well as holes, cracks, cavities and additional various defects. These features determine the microstructure of the material, whose properties are crucial in determining the various mechanical, electric, magnetic, and optical properties of the material. The microstructure is in turn influenced by the evolution of the exterior surface via the grain boundaries. To describe the evolution we assume that the grain boundaries evolve according to mean curvature motion and the exterior surfaces evolve according to surface diffusion motion. The resultant description for the motion of the grain boundaries, exterior surfaces, quadruple junctions and thermal grooves in thin/thick specimen of triangular geometry containing three grains yields a PDAE system, namely a system of partial differential algebraic equations, which we then solve numerically using an implicit finite difference scheme on staggered grids with a partially parallelized algorithm. Using the program, we identified new physical instabilities numerically. For example, we found that either annihilation of the smallest grain or hole formation at the quadruple junction could occur, depending on the model parameters. A variant algorithm for wetting/dewetting isolated a new grain-hole dewetting instability.

*Abstract:*

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

**Abstract**:

We establish an error estimate for counting lattice points in Euclidean norm balls (associated to an arbitrary irreducible linear representation) for lattices in simple Lie groups of real rank at least two. Our approach utilizes refined spectral estimates based on the existence of universal pointwise bounds for spherical functions on the groups involved. In the talk I will present the principles of our method. Moreover I will give a natural example in which we found improvement of the best current bound established by Duke, Rudnick and Sarnak in 1991. The group in the example will be SL(n+1, R) for n > 2 with any lattice, and with the adjoint representation

*Abstract:*

**Adviser**: Prof. Naama Brenner

**Abstract: **Phenotypic variability is a hallmark of cell populations, even when clonal and grown under uniform conditions. This variability appears in many measured cellular properties, such as cell-size, protein content, organelle copy number and more. Cells in a population constantly grow and divide, stochastically inheriting their cellular properties to the next generation. Thus, phenotypic variability is tightly connected to long-term cellular growth and division dynamics.

Of special interest and biological relevance are highly abundant proteins, which have recently been found to exhibit properties of a global cellular variable. In particular, they accumulate smoothly throughout the entire cell cycle with a rate correlated to that of cell-size accumulation; this accumulation appears to be negatively regulated similar to cell size control. In addition, both protein and cell-size distributions across a population, as well as across generations in a

single cell, are highly non-Gaussian and display a universal shape.

We propose a modeling approach which describes the multiple interacting components of cellular phenotype and reconstructs the subtle measured properties of phenotypic variability.

These include correlations among phenotype components and across time, and the universal and non-universal statistical properties of phenotype components.

*Abstract:*

Advisor: Nir Gavish

Abstract: The non-local Cahn-Hilliard (Ohta-Kawasaki) equation manifests spatio-temporal behaviors driven by competing short-range forces and long-range Coulombic interactions. These models

are often being employed to study di-block copolymers, and for renewable energy applications that are based on complex nano-materials, such as ionic liquids and polyelectrolyte membranes. Asymmetric properties between different materials, e.g, phase-dependent permittivity and tilted free energy potential, are included in extended Ohta-Kawasaki model.

Using perturbation methods and numerical continuation methods, we study the distinct solution families of Ohta-Kawasaki equations. Specifically, we focus on spatially localized states in 1-space dimensions, and show that in gradient coupled parabolic and elliptic PDEs (phase separation coupled to electrostatics), 1D homoclinic snaking appears as not-slanted and describe the dependence of localized stripes vs. hexagons, on the domain size.

*Abstract:*

The 22nd Amitsur Memorial Symposium will be held at the University of Haifa on June 20-21.

Speakers:

A. Giambruno (Palermo)

Y. Ginosar (Haifa)

B. Kunyavski (Bar Ilan)

D. Neftin (Technion)

C. Procesi (Rome)

A. Regev (Weizmann)

E. Sayag (Ben Gurion)

T. Weigel (Milan)

S. Westreich (Bar Ilan)

Please let us know if you wish to participate in the festive dinner at the end of the first day.

email: ginosar@math.haifa.ac.il

Please forward this email to anyone who might be interested.

Hoping to see you here,

The organizing committee:

E. Aljadeff

A. Braun

Y. Ginosar

*Abstract:*

**Advisor: **Tobias Hartnick

**Abstract: **The Out(G)-action on the group cohomology H^n(G) of a group G is an important object of study in group theory. On the contrary, almost nothing is known about the corresponding Out(G)-action on the bounded group cohomology H^n_b(G). This talk will introduce bounded group cohomology and then look at the case of G=F_2 and n=2. There the dynamics of the unipotent elements in Out(F_2) on a dense subset B(F_2) of H^{^2}_b(F_2) will be presented concretely and visualized. In particular we will show that no element of B(F_2) is fixed by the Out(F_2)-action, partly answering a question of Miklós Abért.

*Abstract:*

**יום בית פתוח לתארים מתקדמים**

**09:00-09:15 דברי פתיחה / פרופ' אלי אלחדף, דיקן הפקולטה**

**09:15-09:30 פרופ' מיכאל פוליאק, מרכז הוועדה לתארים מתקדמים**

**09:30-09:45 פרופ' יהודה עגנון, מרכז התכנית הבין יחידתית במתמטיקה שימושית**

**09:45-09:55 הפסקה**

**09:55-10:55 פרופ'מ ניר גביש**

** פרופ'מ אור שליט**

** פרופ"ח אמיר יהודיוף**

** פרופ'מ אורי שפירא**

**10:55-11:10 הפסקה**

**11:10-12:45 פאנל בהשתתפות הפרופסורים: יעקב רובינשטיין, מיכה שגיב ונציגי הסטודנטים **

** לתארים מתקדמים**

**13:00 ארוחת צהריים**

*Abstract:*

**Supervisor: **Prof. Shlomo Gelaki and Prof. Emeritus Arye Johasz

**Abstract: **Let W be a purely odd finite dimensional supervector space over an algebraically closed field with characteristic zero. The category sRep(W) consisting of /\W-supermodules with even morphisms is a non-semisimple symmetric finite tensor category.

We classify braided finite tensor categories containing sRep(W) as a Lagrangian subcategory (=maximal symmetric subcategory).

*Abstract:*

**מתמטיקה: חומר הלימוד שלא נמצא בסילבוס / ד"ר נתן לוי**

לימודי המתמטיקה מפתחים כישורים חשובים אשר אינם ברי מדידה אך חיוניים בתחום המחקר והפיתוח.בהרצאה זאת נתאר מספר דוגמאות לכישורים מתמטיים הנרכשים בלימודים ויישומם בתעשייה.

**כיצד ניתן להשתמש במתמטיקה תיאורטית בעולם התעשייה? / ד"ר יונתן אפללו**

חשיבותם של כלי המתמטיקה השימושית בתעשייה היא עובדה ידועה ומוכחת.למרות זאת, לימודי המתמטיקה אינם מהווים כלים לפתרון בעיות בלבד אלא גם מאפשרים דרך לפיתוח שיטות פתרון באופן מקורי, יעיל ופרקטי.

בהרצאה, ניתן דוגמאות לבעיות הניתנות לפתרון בעזרת כלים מתמטיים ולבעיות אחרות אשר פתרונן מורכב אך אינו דורש ידע מתמטי נרחב.

*Abstract:*

This lecture deals with various recent developments concerning the old and very classical concept of topological degree for continuous maps from the circle into itself (also called winding number or index).

I will first explain how it can be extended beyond the class of continuous maps.

This led to the "accidental"discovery of a simple, but intriguing formula connecting the degree of a map to its Fourier coefficients. The relation is easily justified when the map is smooth. However, the situation turns out to be extremely delicate if one assumes only continuity, or even Holder continuity. This "marriage" is more difficult than expected and there are many difficulties in this couple such as the following question I raised:

" Can you hear the degree of a map from the circle into itself?"

I will also present estimates for the degree leading to the question :

" How much energy do you need to produce a map of given degree?".

Many simple looking problems remain open.

The initial motivation for this research came from the analysis of the Ginzburg-Landau model in Physics.

The lecture will be accessible to a wide audience, including undergraduate students

*Abstract:*

**Supervisor**: Eli Aljadeff

**Abstract: **When studying noncommutative f.d. algebras, the building blocks, in a sense, are the matrix algebras over division algebras (e.g. the real quaternions). This led to the idea of a generic division algebra such that all the division algebras are just specializations of it. In particular, many properties satisfied by the generic division algebra are inherited by all other division algebras. The generic crossed product arises in a similar manner, when we consider division algebras with a crossed product structure. In this lecture, I will talk about the place of division algebras and crossed products in the study of f.d. algebras, and how to construct their generic versions. Moreover, I will show why the center of these generic objects play such a central role, and how to compute it using field invariants

*Abstract:*

The talk will concern the distance between two polytopes defined for every hypergraph: CP, the covering polytope, which is the convex hull of the characteristic vectors of the covers, and FCP, the fractional covering polytope, which is the set of all fractional covers. Clearly, the first is contained in the second. Given a direction u in space, we can measure the distance between CP and FCP in two ways. One is the ratio t/t*, where t (resp. t*) is smallest such that t u \in CP (resp. FCP). The (better known) second distance measure is the ratio s/s*, where s (resp. s*) is smallest such that a hyperplane perpendicular to u of distance s/|u| (resp. s*/|u|) from the origin meets CP (resp. FCP).

Partially joint work with Ron Aharoni and Ron Holzman

*Abstract:*

**Supervisor**: Professor Emeritus Raphael Loewy

**Abstract: **Nonnegative matrices are important in many areas. Of particular importance are the spectral properties of square nonnegative matrices. Some spectral properties are given by the well-known Perron-Frobenius theory, which is about 100 years old. One of the most difficult problems in matrix theory is to determine the lists of $n$ complex numbers (respectively real numbers) which are the spectra of $ n \times n $ nonnegative (respectively symmetric nonnegative) matrices. In fact, this problem is open for any $ n \geq 5 $. Our work deals with the first open case, that is $ n = 5 $, for a list of real numbers. We made a significant progress towards the solution of this case. In particular, we obtain the solution when the sum of the five given numbers is zero or at least half of the largest one.

*Abstract:*

Thermoacoustics is a highly promising technology for the upcoming decades. Based on the heat transfer between a temperature gradient stack and an oscillating acoustic wave, it is expected to replace current heat pumps or heat engines, such as solar panels or air conditioners, due to its improved efficiency compared to conventional methods. However, it is yet a young research topic with many improvements and tests to come. One of the most recent improvements is the addition of a reactive component to the inert media as it creates a coating layer over the stack pore's walls and forms an additional concentration gradient as a result of its phase-exchange interaction with the active component, encouraging the oscillating amplitudes for a more desirable performance and efficiency. This thesis develops the mathematical non-dimensional approach established by the physical parameters of the system building it from the conservation of momentum, concentration and energy equations applying the required boundary conditions to solve for the velocity, concentration and temperature fluctuations, respectively. Then plugging the results into the continuity equation in order to arrive to the wave equation, which describes the pressure fluctuations along the stack. At this point, it is possible to find the acoustic work flux, which determines the efficiency of the system based on all the physical parameters involved. The results evaluation is focused on the analysis of limiting cases and approximations, such as the inviscid limit or the boundary layer approximation, and their theoretic impact on the performance. Finally, the required setup for a concentration and a temperature gradient onset is found independently in the pursuit of triggering an instable oscillation.

*Abstract:*

Following the work of Linial and Meshulam on random simplicial complexes, we introduce a model for random complexes with bounded degree and study its topology. We use spectral gap theory, in particular, Garland's method, to show homological connectivity of these random complexes. We also show an upper bound on Betti numbers of complexes with highly connected links, in the spiritof Garland's method.

*Abstract:*

Given a random Cayley graph we wish to study its limiting distribution. Marklof and Str¨ombersson used the limit distribution for Frobenius numbers in m+1 variables to prove that the diameter of a random Cayley graphs of Z/kZ with a generating set of fixed size m>1 has a limit in distribution and found that limit. In this talk we survey their result and expand the discussion to Cayley graphs of Z^n/Sigma, with a generating set of fixed size m>n, where Sigma is a sublattice of Z^n

*Abstract:*

The relation between continued fraction expansions and the geodesic flow on the quotient space SL2(R)/SL2(Z) is well studied and understood and dates back to Artin. In this talk we will discuss its positive characteristic analogue, for its similarities to the real case, and its surprising differences. Time permitting, we shall discuss some recent results.

*Abstract:*

**ACTION NOW WANDERING SEMINAR**

First meeting, Technion, 1.12.15,Butler Auditorium (near Forscheimer Faculty club)

• 9:30 cookies, small pastries and fruit from Israel's northern foothills, and refreshing drinks

• 10:00 **Uri Bader**, Weizmann

**Ozawa's proof of Gromov's polynomial growth theorem. **

Gromov's polynomial growth theorem from the early 1980's, stating that every group of polynomial growth is virtually nilpotent, is a milestone in Geometric Group Theory. In this talk I will present the entropy based, two pages proof given recently by Narutaka Ozawa.

• 11:15 **Uri Shapira**, Technion

**Stationary measures on homogeneous spaces.**

Let mu be a compactly supported measure on SL_3(R) generating a group whose Zariski closure is semi-simple. Let X be the space of all rank-2 discrete subgroups of R^3 (identified up to dilation). We describe a classification of the mu-stationary measures on X. This is part of an ongoing project with Oliver Sargent in which we classify stationary measures in situations similar to the above. The proof is an adaptation of the Benoist-Quint approach for classifying stationary measures on homogeneous spaces obtained by quotienting by a discrete subgroup. As an application we show that if v in Z^3 varies along a the quadratic surface x^2 + y^2 - z^2 then the the shapes of the 2-lattices obtained by intersecting Z^3 with the orthocomplement of v is dense in the space of shapes.

• 12:15 Lunch and informal discussions

• 14:00 **Tom Meyerovich**, Ben Gurion University

**Sofic groups, sofic entropy, stabilizers and invariant random subgroups **

Sofic groups where introduced by Gromov (under a different name), and Weiss towards the end of the millennium. This is a class of groups retaining some finiteness properties, a common generalization of amenable and residually finite groups. Entropy theory for sofic groups, initiated by L. Bowen, is developing rapidly. After recalling the concepts of soficity and entropy, I will explain some results relating them to invariant random subgroups, or equivalently stabilizer groups for measure preserving actions.

• 15:15 **Brandon Seward**, Hebrew University

**Positive entropy actions of countable groups factor onto Bernoulli shifts**

I will prove that if a free ergodic action of a countable group has positive Rokhlin entropy (or, less generally, positive sofic entropy) then it factors onto all Bernoulli shifts of lesser or equal entropy. This extends to all countable groups the well-known Sinai factor theorem from classical entropy theory. As an application, I will show that for a large class of non-amenable groups, every positive entropy free ergodic action satisfies the measurable von Neumann conjecture.

*Announcement:*

**Summer Projects in Mathematics at the Technion**

The Mathematics Department at the Technion is inviting advanced undergraduate students to experience research level mathematics in a week of projects (Sunday-Thursday, September 6-10, 2015). The projects will be mentored by members, postdocs and graduate students from the department.

Please visit the Summer Projects website for more details:

http://mathweek.net.technion.ac.il/

Organizers: Ram Band, Michah Sageev and Amir Yehudayoff.

*Abstract:*

**Supervisor: **Associate Professor Uri Bader

**Abstract: **We shall study a variant of property (EH) (defined by Bader-Finkelshtein), called weak property (EH), and its relation to triviality of certain reduced horoboundary actions. We will use this property to show that generalized Heisenberg groups act trivially on their horoboundary, both in the discrete and in the smooth case, extending the results of Bader-Finkelshtein.

*Abstract:*

**Supervisor:** Associate Professor Uri Bader

**Abstract: **It is a well known question of Gromov whether there exist groups with no fixed point free action on CAT(0) spaces. Gromov conjectured that random groups have this property. In this talk we will present groups that have no fix point free isometric actions on Hadamard manifolds. These are the Steinberg groups defined over the ring R = Fp [t] In the talk we will define the Steinberg groups and show some nice properties regarding to them. We will also describe Hadamard manifolds which are complete simply connected non-positively curved Riemaniann manifolds. In particular we will study fat triangles in these manifolds and show how to deduce fix point properties** **

*Announcement:*

**Supervisor:**

Professor Amy Novick-Cohen

**Abstract:**

If we look at most materials under a microscope, we will see a network of grains and grain boundaries as well as holes, cracks, cavities and additional various defects. These features determine the microstructure of the material, whose properties are crucial in determining the various mechanical, electric, magnetic, and optical properties of the material. The microstructure is in turn influenced by the evolution of the exterior surface via the grain boundaries.

In my lecture I shall report on 3D numerical studies of the motion of quadruple junctions and thermal grooves in thin polycrystalline films where the mean curvature motion of the grain boundaries and the surface diffusion evolution of the exterior surfaces couple along the thermal grooves. Our algorithms could also be used to study hole evolution in thin monocrystalline and polycrystalline films, where only the motion of the exterior surface needs to be considered.

To describe the physical models and their motion, we used a system of partial differential algebraic equations with boundary and initial conditions. Our numerical approach used a finite difference scheme on a staggered grid with partially parallelized numerical algorithms, the backward Euler method, and Newton’s method.. Simulations, written in MATLAB and ”C”, were able to indicate some new instabilities.

*Abstract:*

Expander graphs have played, in the last few decades, an important role in computer science, and in the last decade, also in pure mathematics. In recent years a theory of "high-dimensional expanders" is starting to emerge - i.e., simplical complexes which generalize various properties of expander graphs. This has some geometric motivations (led by Gromov) and combinatorial ones (started by Linial and Meshulam). The talk will survey the various directions of research and their applications, as well as potential applications in math and CS. Some of these lead to questions about buildings and representation theory of p-adic groups.

We will survey the work of a number of people. The works of the speaker in this direction are mainly with Tali Kaufman, David Kazhdan and Roy Meshulam.

*Abstract:*

**Abstract**: If one seeks an associative algebra which corresponds canonically to a Euclidean space E (or to any vector space with a quadratic form Q) - canonically means that we refrain from "choosing", say a basis/coordinate system - an option is the Clifford algebra of the space, defined as the associative algebra generated by E, with the relations that the square in the algebra of each vector v in E equals Q(v)1. This algebra contains a plethora of interesting members and structures.Focusing mainly on the Euclidean 4-space, we shall describe its basics and try to stroll in its garden and pick some flowers. In particular, we shall encounter objects mentioned in Maria Elena Luna-Elizarraras' lecture: the 2-dim extension of the reals by a k satisfying k^2=1, and bi-quaternions.

*Abstract:*

**Prof. Martin R. Bridson**

Whitehead Professor of Pure Mathematics Mathematical Institute

**University of Oxford**

Lecture 1: **The ubiquity and wonder of non-positive curvature**

**Time: **Monday, November 10, 2014 at 17:00

**Place: **Lecture will be held at Segoe Building, Benjamin hall (Segoe 1)

Lecture 2: **The universe of groups: Geometry, Complexity and Imposters**

**Time: **Tuesday, November 11, 2014 at 17:00

**Place:** Lecture will be held at Segoe Building, Benjamin hall (Segoe 1)

Lecture 3: **Complexity, and finite shadows of infinite groups**

**Time: **Thursday, November 13, 2014 at 17:00

**Place: **Lecture will be held at Segoe Building, Benjamin hall (Segoe 1)

Refreshments will be served before the lectures