# Techmath - Math Seminars in Israel

*Abstract:*

In this talk I will present Nevanlinna-type tight bounds on the minimalIn this talk I will present Nevanlinna-type tight bounds on the minimal possible growth of subharmonic functions with a large zero set. We use a technique inspired by a paper of Jones and Makarov.

*Abstract:*

In this talk, I will present some of my results in multiple translationalIn this talk, I will present some of my results in multiple translational tiling in the Euclidean plane. For examples, besides parallelograms and centrally symmetric hexagons, there is no other convex domain which can form any two-, three- or four-fold translative tiling in the plane. However, there are two types of octagons and one type of decagons which can form nontrivial five-fold translative tilings. Furthermore, a convex domain can form a five-fold translative tiling of the plane if and only if it can form a five-fold lattice tiling, a multiple translational tile in the plane is a multiple lattice tile. This talk is based on a joint work with Professor Chuanming Zong.

*Abstract:*

For any element w in the free group on k generators and a group G, there is a map G^k --> G is defined by substitution. These so-called word maps are the analogs of polynomial maps in the category of groups. I will talk about the images of word maps in arithmetic groups, and then segue into the model theory of these groups ending with new rigidity phenomenon for them. The non-survey parts of this work are joint with Alex Lubotzky and Chen Meiri.

*Abstract:*

A long-standing open problem, known as Hadwiger’s covering problem, asks what is the smallest natural number $N(n)$ such that every convex body in ${\mathbb R}^n$ can be covered by a union of the interiors of at most $N(n)$ of its translates. In this talk, I will present a recent work in which we prove a new upper bound for $N(n)$. This bound improves Rogers' previous best bound, which is of the order of ${2n \choose n} n\ln n$, by a sub-exponential factor. As a key step, we use thin-shell estimates for isotropic log-concave measures to prove a new lower bound for the maximum volume of the intersection of a convex body $K$ with a translate of $-K$. We further show that the same bound holds for the volume of $K\cap(-K)$ if the center of mass of $K$ is at the origin.

*Abstract:*

Semigroup identities determine characteristic properties of various algebraic objects. While matrices over an infinite field do not admit semigroup identities, tropical matrices do satisfy (nontrivial) semigroup identities. An explicit construction of such semigroup identities is obtained by utilizing the algebraic-combinatorial setup of tropical matrices. Tropical representations deliver these identities to other combinatorial objects, including weighted digraphs and Young Tableaux.

*Abstract:*

Dispersion of particles in chaotic, turbulent or random flows has beenstudied for a long time. It is known that the action of advection on largespatial and temporal scales typically can be described as an (anisotropic)normal diffusion process. In random but strongly correlated velocityfields, an anomalous diffusion is possible. Anomalous diffusion ispossible also in spatially regular velocity fields in the presence ofLagrangian chaos.It is less known that an anomalous transport can take place in steadytwo-dimensional flows, in the absence of any kind of chaos. In the presenttalk, we discuss two examples of such behavior.The first example is the deterministic advection in spatially periodic,steady two-dimensional velocity fields, which include stagnation pointsor solid obstacles, so that the passage time is infinite along somestreamlines. The large-time asymptotics of the dispersion law is analyzedusing the special flow construction (a flow built over the mapping).Depending on the type of the passage time singularity, the asymptoticdispersion law can correspond to either subdiffusion or superdiffusion.The analytical predictions match the results of numerical simulations.The second example is the diffusion-advection problem in spatiallyperiodic, steady two-dimensional flows that contain closed cells, possiblyseparated by jets. The anomalous dispersion is predicted and foundnumerically on an intermediate time interval. On the large time scale, anormal diffusion (enhanced by the flow) takes place. The dispersiondisplays peculiar aging properties.

Joint work with M.A. Zaks, P. Poeschke and I.M. Sokolov, Humboldt University of Berlin, Germany

*Abstract:*

A central conjecture in inverse Galois theory, proposed by Dèbes and Deschamps, asserts that every finite split embedding problem over an arbitrary field has a geometric solution. This conjecture contains, in particular, an affirmative answer to the inverse Galois problem over the rationals and the Shafarevich conjecture that the absolute Galois group of the maximal cyclotomic extension of the rationals is profinite free. We shall prove a consequence of the Dèbes--Deschamps conjecture, namely that every finite embedding problem over an arbitrary Hilbertian field has a geometric solution after a finite (Galois) base change. Time permitting, we shall also present similar results in the situation of lifting solutions. This is joint work in progress with Arno Fehm.

*Abstract:*

In this talk I will discuss obstructions to having a Riemannian metric with non-positive sectional curvature on a locally CAT(0) manifold. I will focus on the obstruction in dimension = 4 given by Davis-Januszkiewicz-Lafont and show how their method can be extended to construct new examples of locally CAT(0) 4-manifolds M that do not have a Riemannian smoothing. The universal covers of these manifolds satisfy the isolated flats condition and contain a collection of 2-dimensional flats with the property that their boundaries at infinity form non-trivial links in the boundary 3-sphere.

*Abstract:*

The nodal set of a nice function defined on a smooth manifold or the Euclidean space is its zero set. The study of nodal sets of Gaussian random fields has positioned itself in the heart of several disciplines, including probability theory and spectral geometry, and, more recently, it has exhibited connections to number theory. We are interested in the asymptotic topology and geometry of the nodal lines in the high energy limit.

In the first part of the talk I will give an overview of the classical results in this field, and the related methods. In the second part of the talk I will describe the more recent progress,related to percolation properties of the nodal lines, borrowing from percolation theory, inspired by the beautiful percolation model due to Bogomolny-Schmit. Finally, I will describethe recent results obtained in a joint work with D. Beliaev and S. Muirhead on the relation between the percolation properties of the nodal sets and their connectivity measures, that were defined and whose existence was established in a joint work with P. Sarnak.

*Abstract:*

**Advisor: **Ron Rosenthal

**Abstract**: A random matrix is said to be sampled from the Ginibre ensemble if all of its entries are i.i.d., complex normal random variables with mean zero. In this work, we study the asymptotic behavior of the angles between pairs of eigenvectors of such matrices. In particular, we compute the limiting distribution of the angle for fixed pairs of eigenvalues, obtain precise bounds on the typical behavior of the maximal angle with high probability and find the limiting distribution for the location of the eigenvalues which attain the maximal angle with high probability. The talk will present the main results and demonstrate some of the techniques used to reach them.

*Abstract:*

T.B.A.

*Abstract:*

TBA

*Announcement:*

'What Is' the best way to celebrate the new year?

That's right! Seminar talk by Stav Berman on:

What Is CAT(0) Cube Complex?

There will be pizza, there will be beer but mainly there will be grad students talking about math! All talks will be given by grad students and will be at an introductory level.

Happy new year!

Lior

*Abstract:*

T.B.A.

*Announcement:*

You are invited to a:

** Distinguished Lecture Series by**

**Prof. Sergei Tabachnikov**

**(Pennsylvania State University)**

**Title: **Frieze patterns. Cross-ratio dynamics on ideal polygons.

**Abstract: **

In the first lecture I shall describe basic properties of frieze patterns. These are are beautiful combinatorial objects, introduced by Coxeter in the early 1970s. He was about 30 years ahead of time: only in this century, frieze patterns have become a popular object of study, in particular, due to their relation with the emerging theory of cluster algebras and to the theory of completely integrable systems. I shall prove the theorem of Conway and Coxeter that relates arithmetical frieze patterns with triangulations of polygons. There is an intimate, and somewhat unexpected, relation between three objects: frieze patterns, 2nd order linear difference equations, and polygons in the projective line (or ideal polygons in the hyperbolic plane).

**1st lecture**: Monday, January 6, 2020 at 15:30

**2nd lecture**: Wednesday, January 8, 2020 at 15:30

**3rd lecture**: Thursday, January 9, 2020 at 15:30

All lectures will be in Amado 232.

Light refreshments will be given in the faculty lounge on the 8th floor.

*Abstract:*

T.B.A.

*Abstract:*

T.B.A.

*Announcement:*

You are invited to a:

** Distinguished Lecture Series by**

**Prof. Sergei Tabachnikov**

**Pennsylvania State University **

**Title: ***Frieze patterns. Cross-ratio dynamics on ideal polygons.*

**Abstract: **In the next lectures I shall outline some recent work on frieze patterns, including their relation with the Virasoro algebra. Then I shall present cross-ratio dynamics on ideal polygons. Define a relation between labeled ideal polygons in the hyperbolic 3-space by requiring that the complex distances (a combination of the distance and the angle) between their respective sides equal a constant; the constant is a parameter of the relation. This gives a 1-parameter family of maps on the moduli space of ideal polygons in the hyperbolic space (or, in its real version, in the hyperbolic plane). I shall discuss complete integrability of this family of maps and related topics, including a continuous version of this relation that is intimately related with the Korteweg-de Vries equation.

**1st lecture**: Monday, January 6, 2020 at 15:30

**2nd lecture**: Wednesday, January 8, 2020 at 15:30

**3rd lecture**: Thursday, January 9, 2020 at 15:30

All lectures will be in Amado 232.

Light refreshments will be given in the faculty lounge on the 8th floor.

*Abstract:*

There are many formulas that express interesting properties of a finite group G in terms of sums over its characters. For estimating these sums, one of the most salient quantities to understand is the **character ratio**:

Trace(ρ(g)) / dim(ρ),

for an irreducible representation ρ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of the mentioned type for analyzing certain random walks on G.

Recently (https://www.youtube.com/watch?v=EfVCWWWNxvg&feature=youtu.be), we discovered that for classical groups G over finite fields there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant **rank**.

Rank suggests a new organization of representations based on the very few “Small” ones. This stands in contrast to Harish-Chandra’s “philosophy of cusp forms” (P-of-CF), which is (since the 60s) the main organization principle, and is based on the (huge collection) of “Large” representations.

This talk will discuss the notion of rank for the group GL(n) over finite fields, demonstrate how it controls the character ratio, and explain how one can apply the results to verify mixing time and rate for random walks.

This is joint work with Roger Howe (Yale and Texas A&M). The numerics for this work was carried by Steve Goldstein (Madison).

*Announcement:*

You are invited to a:

** Distinguished Lecture Series by**

**Prof. Sergei Tabachnikov**

**Pennsylvania State University **

**Title: ***Frieze patterns. Cross-ratio dynamics on ideal polygons.*

**Abstract: **In the next lectures I shall outline some recent work on frieze patterns, including their relation with the Virasoro algebra. Then I shall present cross-ratio dynamics on ideal polygons. Define a relation between labeled ideal polygons in the hyperbolic 3-space by requiring that the complex distances (a combination of the distance and the angle) between their respective sides equal a constant; the constant is a parameter of the relation. This gives a 1-parameter family of maps on the moduli space of ideal polygons in the hyperbolic space (or, in its real version, in the hyperbolic plane). I shall discuss complete integrability of this family of maps and related topics, including a continuous version of this relation that is intimately related with the Korteweg-de Vries equation.

**1st lecture**: Monday, January 6, 2020 at 15:30

**2nd lecture**: Wednesday, January 8, 2020 at 15:30

**3rd lecture**: Thursday, January 9, 2020 at 15:30

All lectures will be in Amado 232.

Light refreshments will be given in the faculty lounge on the 8th floor.

*Abstract:*

T.B.A.

*Announcement:*

You are invited to a:

** Distinguished Lecture Series by**

**Prof. David Jerison (MIT)**

Abstract:

In Lecture 1 we will begin by describing the Hot Spots Conjecture of J. Rauch.This question is an essential test of our understanding of the shapes of level sets of thesimplest eigenfunctions. We will then relate our question to a variety of others about levelsets and other kinds of interfaces - free boundaries, minimal surfaces, isoperimetric surfaces - as well as the KLS Hyperplane Conjecture in high dimensional convex analysis.

**1st lecture**: Monday, January 20, 2020 at 15:30

**2nd lecture**: Wednesday, January 22, 2020 at 15:30

**3rd lecture**: Thursday, January 23, 2020 at 15:30

All lectures will be in Amado 232.

Light refreshments will be given in the faculty lounge on the 8th floor.

*Abstract:*

T.B.A.

*Announcement:*

You are invited to a:

** Distinguished Lecture Series by**

**Prof. David Jerison (MIT)**

**Abstracts**:

In Lecture 2 we will discuss how methods from geometric measure theory and elliptic regularity theory (developed for minimal surfaces) apply to level sets. We will focus on a version of the Harnack inequality that tells us how level surfaces influence each other. This gets us part way towards understanding hot spots.

**THE LECTURE WILL BE AT 16:30 AND NOT AT 15:30 AS INDICATED IN THE POSTER**

All lectures will be in Amado 232.

Light refreshments will be given in the faculty lounge on the 8th floor.

*Announcement:*

TBA...

*Abstract:*

Let M be an orientable hyperbolic surface without boundary and let c be a closed geodesic in M. We prove that any side of any triangle formed by distinct lifts of c in the hyperbolic plane is shorter than c.

The talk will be presented for advanced undergraduate and beginning graduate students.

*Announcement:*

You are invited to a:

** Distinguished Lecture Series by**

**Prof. David Jerison (MIT)**

**Abstracts**:

In Lecture 3 we will explain how complex analysis and differential geometry, in particularas developed for minimal surface theory, can be used to characterize global solutions andprove rigidity and regularity results for free boundaries. This gives further insights into themissing ingredients that will be needed to understand level sets of eigenfunctions.

All lectures will be in Amado 232.

Light refreshments will be given in the faculty lounge on the 8th floor.

*Announcement:*

You are cordially invited to:

** the 33 ^{rd} **

**Elisha Netanyahu Memorial Lecture**

**Professor ****Claire Voisin**

**College de France, France**

**Title**: *Hodge structures in algebraic geometry*

**Abstract**:

Hodge structures of weight 1 appear implicitly in the study of Riemann surfaces and their Jacobians, which are associated complex tori. In higher weight (degree, dimension), the theory of Hodge structures has been developed by Griffiths starting from the Hodge decomposition theorem.

I will explain what a Hodge structure is and how it can be used to study the geometry of algebraic varieties (or Kähler manifolds).

**Reception will be held at 16:30 in the Faculty Lounge Amado Mathematics Building, 8th Floor.**

*Abstract:*

After summarizing 1D periodic Jacobi matrices, I will define periodic Jacobi matrices on infinite trees. I'll discuss the few known results and some interesting examples and then discuss lots and lots of interesting conjectures. This is joint work mainly with Nir Avni and Jonathan Breuer but also with Jacob Christensen, Gil Kilai and Maxim Zinchenko.

It is on the spectral theory of a class of operators on trees, for which there has been literature on the random case even in the theoretical physics literature but I am not aware of any application to anything close to real physics so this is probably better as a math talk but I leave it to you to sort it out if you are interested. I don’t care at all which it is called or even if it is jointly sponsored.