# Techmath - Math Seminars in Israel

*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:*

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 consider the formal Schroedinger operator $$Hu=(-\Delta+W+W_\Gamma) u$$ on $R^n$ with a real-valued regular potential $W\in L^\infty(R^n)$ and a singular potential $W_\Gamma$ which is an operator of multiplication by a distribution $W_\Gamma\in \mathcal D(R^n)$ of the first order of singularity with a support on unbounded smooth hypersurface $\Gamma\subset R^n.$ We associate with the operator $H$ an unbounded operator $\mathcal H$ in L^2(R^n)$ of the diffraction problem on $R^n$ defined by the regular Schroedinger operator $$(-\Delta+W) u(x) = 0; x \in R^n\setminus\Gamma$$ with some diffraction conditions on $\Gamma.$ We formulate conditions for $\mathcal H$ to be a self-adjoint operator in $L^2(R^n)$ and describe the location of the essential spectrum of the operator $\mathcal H.$ The Schroedinger operators with $\delta$-type potentials supported on hypersurfaces are important in Quantum Physics and have attracted a lot of attention: for instance they are used for a description of quantum particles interacting with charged hypersurfaces, in approximations of Hamiltonians of the propagation of electrons through thin barriers, etc.

*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:*

In convex optimization, quantum information theory and real convex algebraic geometry, many practical and theoretical questions are related to containment problems between convex sets defined by a linear matrix inequality (LMI domains for short). One difficulty when we wish to check for containment of the n-dimensional cube inside some other LMI domain, is that this is computationally hard (NP-hard complexity). These sort of problems become computationally tractable when we relax them to containment problems between *matrix* LMI domains. In fact, this relaxation of the problem enables the use of a semidefinite program to check for matrix LMI domain containment. In this talk we will survey some of the geometric aspects of these relaxations. We will explain how to move the original problem to the relaxed problem, the connections with the existence of quantum channels, and how in some cases we can get an estimate (which is sometimes sharp) for the error of passing from the original LMI containment problem to the relaxed matrix LMI containment problem. *Joint work with Kenneth R. Davidson, Orr Moshe Shalit and Baruch Solel.

*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:*

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:*

Dear All, We will have a one-day miniworkshop, "Geometry of Singularities" in Ben Gurion University, April 9, 2018. Some details are here: https://www.math.bgu.ac.il/~kernerdm/GeoS.pdf There is no special registration procedure, but if you would like to participate, please inform me well in advance. ---------------------------- Best regards, Dmitry Kerner <kernerdm@math.bgu.ac.il> http://www.math.bgu.ac.il/~kernerdm/

*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:*

Initial developments in the theory of (topological) quantum groups were motivated on one hand by the desire to extend the classical Pontryagin duality for locally compact abelian groups to a wider class of objects and on the other by the idea of replacing the study of a space by the investigation of the algebra of functions on it. In 1980s the theory was given a big boost by the discovery of a big class of examples arising as deformations of classical compact Lie groups and the resulting conceptual progress, mainly due to Woronowicz. In this talk we will describe this background and present two approaches to constructions of quantum groups developed in the last decade, leading to so-called quantum symmetry groups and liberated quantum groups. They turn out to produce very interesting examples and offer connections to noncommutative geometry and free probability.

*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:*

(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:*

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.

*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.

*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.

*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:*

T.B.A.

*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:*

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:*

* 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**