# Techmath - Math Seminars in Israel

*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:*

I shall attempt to describe the following four notions in quantum information: Qubit, Quantum key distribution, The GHZ game and Teleportation. The talk is aimed at non-specialists.

*Abstract:*

Our first theorem is a hierarchy theorem for the query complexity of testing graph properties with one-sided error; more precisely, we show that for every sufficiently fast-growing function f from (0,1) to the natural numbers, there is a graph property whose one-sided-error query complexity is precisely f(\Theta(\epsilon)). No result of this type was previously known for any f which is super-polynomial. Goldreich [ECCC 2005] asked to exhibit a graph property whose query complexity is exponential in 1/\epsilon. Our hierarchy theorem partially resolves this problem by exhibiting a property whose one-sided-error query complexity is exponential in 1/\epsilon. We also use our hierarchy theorem in order to resolve a problem raised by Alon and Shapira [STOC 2005] regarding testing relaxed versions of bipartiteness. Our second theorem states that for any function f there is a graph property whose one-sided-error query complexity is at least f(\epsilon) while its two-sided-error query complexity is only polynomial in 1/\epsilon. This is the first indication of the surprising power that two-sided-error testing algorithms have over one-sided-error ones, even when restricted to properties that are testable with one-sided error. Again, no result of this type was previously known for any f that is super-polynomial. The above theorems are derived from a graph theoretic result which we think is of independent interest, and might have further applications. Alon and Shikhelman [JCTB 2016] introduced the following generalized Turan problem: for fixed graphs H and T, and an integer n, what is the maximum number of copies of T, denoted by ex(n,T,H), that can appear in an n-vertex H-free graph? This problem received a lot of attention recently, with an emphasis on T = C_3, H=C_{2m+1}. Our third theorem gives tight bounds for ex(n,C_k,C_m) for all the remaining values of k and m. Joint work with Asaf Shapira.

*Abstract:*

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

organized by the CMS.

*Announcement:*

Lecture 1: April 23, 2018 at 15:30

Lecture 2: April 25, 2018 at 15:30

Lecture 3: April 26, 2018 at 15:30

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

*Abstract:*

Let X be a smooth compact algebraic curve defined by equations with rational coefficients. If the genus of X is not less than 2, then by Mordel's conjecture = Faltings' theorem, the set X(Q) of rational points of X is finite. This gives rise to the problem of computing such sets algorithmically. This quest for an ``effective Mordel's conjecture'' is regarded as a central goal of arithmetic geometry. An approach pioneered by Minhyong Kim revolves around a certain conjecture; in Kim's approach, rational points face an obstruction coming from the prounipotent completion of the fundamental group, and the conjecture asserts that this obstruction completely determines the set of integral points inside the set of p-adic points for an auxiliary prime p of good reduction. In joint work with Tomer Schlank, we divide Kim's conjecture into a series of smaller conjectures with a homotopical flavor.

*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:*

TBA

*Announcement:*

Lecture 1: April 23, 2018 at 15:30

Lecture 2: April 25, 2018 at 15:30

Lecture 3: April 26, 2018 at 15:30

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

*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.

*Announcement:*

Lecture 1: April 23, 2018 at 15:30

Lecture 2: April 25, 2018 at 15:30

Lecture 3: April 26, 2018 at 15:30

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

*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.

*Announcement:*

Title of lectures: *Billiard paths on polygons: Where do they lead?*

LECTURE 1: Monday, April 30, 2018 at 15:30

LECTURE 2: Tuesday, May 1, 2018 at 15:30

LECTURE 3: Thursday, May 3, 2018 at 15:30

*Abstract:*

T.B.A.

*Announcement:*

Title of lectures: *Billiard paths on polygons: Where do they lead?*

LECTURE 1: Monday, April 30, 2018 at 15:30

LECTURE 2: Tuesday, May 1, 2018 at 15:30

LECTURE 3: Thursday, May 3, 2018 at 15:30

*Announcement:*

Title of lectures: *Billiard paths on polygons: Where do they lead?*

LECTURE 1: Monday, April 30, 2018 at 15:30

LECTURE 2: Tuesday, May 1, 2018 at 15:30

LECTURE 3: Thursday, May 3, 2018 at 15:30

*Abstract:*

We study the problem of identifying correlations in multivariate data, under information constraints: Either on the amount of memory that can be used by the algorithm, or the amount of communi- cation when the data is distributed across several machines. We prove a tight trade-off between the memory/communication complexity and the sample complexity, implying (for example) that to detect pairwise correlations with optimal sample complexity, the number of required mem-ory/communication bits is at least quadratic in the dimension. Our results substantially improve those of Shamir (2014), which studied a similar question in a much more restricted setting. To the best of our knowledge, these are the first provable sample/memory/communication trade-offs for a practical estimation problem, using standard distributions, and in the natural regime where the memory/communication budget is larger than the size of a single data point. To derive our theorems, we prove a new information-theoretic result, which may be relevant for studying other information-constrained learning problems. Joint work with Ohad Shamir

*Abstract:*

T.B.A.

*Abstract:*

TBA

*Abstract:*

In digital geometry, Euclidean objects are represented by their discrete approximations e.g., subsets of the lattice of integers. Rigid motions of such sets have to be defined as maps from and onto a given discrete space. One way to design such motions is to combine continuous rigid motions defined on Euclidean space with a digitization operator. However, digitized rigid motions often no longer satisfy properties of their continuous siblings. Indeed, due to digitization, such transformations do not preserve distances, furthermore bijectivity and point connectivity are generally lost. In the context of digitized rigid motions on the 3D integer lattice we first focus on the open problem of determining whether a 3D digitized rotation is bijective. In our approach, we explore arithmetic properties of Lipschitz quaternions. This leads to an algorithm which answers the question whether a given digitized rotation—related to a Lipschitz quaternion—is bijective. Finally, we study at a local scale geometric and topological defects induced by digitized rigid motions. Such an analysis consists of generating all the images of a finite digital set under digitized rigid motions. This problem amounts to computing an arrangement of hypersurfaces in a 6D parameter space. The dimensionality makes the problem practically unsolvable for state-of-the-art techniques such as cylindrical algebraic decomposition. We propose an ad hoc solution, which mainly relies on parameter uncoupling, and an algorithm for computing sample points of 3D connected components in an arrangement of second degree polynomials. http://www.cs.technion.ac.il/~cggc/seminar.html

*Abstract:*

T.B.A.

*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.

*Announcement:*

Technion–Israel Institute of Technology

Center for Mathematical Sciences

Supported by the Mallat Family Fund for Research in Mathematics

invite you to a workshop on the topic of

**Nonpositively Curved Groups on the Mediterranean**

**Nahsholim, 23-29.5.18**

**Organizers:**

Kim Ruane,Tufts University

Michah Sageev, Technion–Israel Institute of Technology

Daniel Wise, McGill University

**For more information:**

** http://cms-math.net.technion.ac.il/nonpositively-curved-groups-on-the-mediterranean/ **

*Abstract:*

PLENARY SPEAKERS: ZEEV RUDNICK (TAU) AND SERGIU HART (HUJI) The Erdős and Nessyahu Prizes will be awarded Poster session Sessions and organizers: Analysis (Ram Band) Algebra (Chen Meiri and Danny Neftin) Applied mathematics (Yaniv Almog) Discrete mathematics (Yuval Filmus) Dynamical systems (Zemer Kosslof) Education (Orit Hazzan) Game theory (Ron Holzman) Non-linear analysis and optimization (Simeon Reich and Alex Zaslavski) Number theory (Moshe Baruch) Probability theory (Oren Louidor) Please register online.

*Abstract:*

Announcement We are happy to announce the Conference "Perspectives in Modern Analysis" to be held on May 28-31, 2018, at the Holon Institute of Technology. The event will honor the distinguished Israeli analysts Dov Aharonov, Samuel Krushkal, Simeon Reich, and Lawrence Zalcman. The meeting will provide a forum for discussions and exchange of new ideas, perspectives and recent developments in the broad field of Modern Analysis. The topics to be addressed include (but are not restricted to) * Complex Analysis * Operator Theory and Nonlinear Analysis * Harmonic Analysis and PDE * Quasiconformal Mappings and Geometry The following institutions have contributed to the organization of this conference: Bar-Ilan University, Holon Institute of Technology, Israel Mathematical Union, ORT Braude College of Engineering and the Technion -- Israel Institute of Technology.

*Abstract:*

TBA

*Abstract:*

TBA

*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).

*Announcement:*

**TBA....**

For further information please click the link below:

http://cms-math.net.technion.ac.il/summer-school-the-complex-math-of-the-real-world/

*Announcement:*

**SUMMER PROJECTS IN MATHEMATICS AT THE TECHNION**

**Sunday-Friday, September 2-7, 2018**

**PLEASE CLICK HERE FOR FURTHER DETAILS**

**Organizers: Ram Band, Tali Pinsky, Ron Rosenthal**