Mar 22

Symplectic maps: algebra, geometry, dynamics

Leonid Polterovich (Tel Aviv University)

Symplectic maps can be considered as symmetries of a geometric structure
(a symplectic form) on a manifold, and as a mathematical model of admissible
motions of classical mechanics. I discuss a number of rigidity phenomena
of algebraic, geometric, and dynamical nature exhibited by these maps,
focusing on a recent work with Egor Shelukhin.

Mar 29

Fixedpoint properties for random groups

Izhar Oppenheim (BGU)

A group is said to have a fixedpoint property with respect to some class of metric spaces if any isometric action of the group on any space in the class admits a fixed point.
In this talk, I will focus on fixedpoint properties with respect to (classes of) Banach spaces. I will survey some results regarding groups with and without these fixedpoint properties and then present a recent result of mine regarding fixpoint properties for random groups with respect to l^p spaces.

Apr 5

About the Dynamics of Polydispersed Fuel

Shlomo Hareli (BGU)

A poly disperse fuel spray consist of thousands of droplets in various volumes
and shapes. The Combustion of the poly disperse is a chemical process which
releases useful thermal energy. The poly disperse fuel droplets are described by
a discrete function  the particle (droplet) size distribution (PSD).
Models of the combustion process which accounts for each droplet are im
practicable as they require a considerable amount of computations. As a result,
approximations are used to describe the combustion process. The approxima
tions fail to describe the particle PSD adequately.
We propose a simplied theoretical model which allow us to use continuous
distribution functions to approximate any PSD (experimental or theoretical)
during the combustion process much more accurately then previous ap
proximations. The time depended distribution functions allow us to in
vestigate the dynamics of the poly disperse fuel elegantly and even permit an
analytical study. The model provided some new theoretical insights.
Our main results show that during the selfignition process, the radii of the
droplets decreased as expected, and the number of smaller droplets increased
in inverse proportion to the radius. An important novel result (visualized by
graphs) demonstrates that the mean radius of the droplets initially increases for
a relatively short period of time, which is followed by the expected decrease.

Apr 12

Relations between dynamics and C*algebras: Mean dimension and radius of comparison

Chris Phillips (University of Oregon)

This is joint work with Ilan Hirshberg.
For an action of an amenable group G on a compact metric space X, the mean dimension mdim (G, X) was introduced by Lindenstrauss and Weiss. It is designed so that the mean dimension of the shift on ([0, 1]^d)^G is d. Its motivation was unrelated to C*algebras.
The radius of comparison rc (A) of a C*algebra A was introduced by Toms to distinguish counterexamples in the Elliott classification program. The algebras he used have nothing to do with dynamics.
A construction called the crossed product C^* (G, X) associates a Calgebra to a dynamical system. Despite the apparent lack of connection between these concepts, there is significant evidence for the conjecture that rc ( C^ (G, X) ) = (1/2) mdim (G, X) when the action is free and minimal. We will explain the concepts above; no previous knowledge of mean dimension, Calgebras, or radius of comparison will be assumed. Then we describe some of the evidence. In particular, we give the first general partial results towards the direction rc ( C^ (G, X) ) \geq (1/2) mdim (G, X). We don’t get the exact conjectured bound, but we get nontrivial results for many of the known examples of free minimal systems with mdim (G, X) > 0.

Apr 26

In between finite and padic groups  the case of permutations

Maxim Gurevich (Technion)

Using the Bruhat decomposition, a general linear group over a padic field may be thought of as a “quantum affine” version of a finite group of permutations. I would like to discuss some analogies and explore the implications of this point view on the spectral properties of the two groups.
For one, restriction of an irreducible smooth representation to its finite counterpart gives the correct notion of the wavefront set  an invariant of arithmetic
significance which is often approached using microlocal analysis.
From another perspective, the class of cyclotomic Hecke algebras is a natural interpolation between the finite and padic groups. I will show how the class of RSK representations (developed with Erez Lapid) serves as a bridge between the Langlands classification for the padic group and the classical Specht construction of the finite domain.

May 3

TBA

Faculty meeting


May 10

Can you hear the shape of a lowdegree Boolean function?

Nathan Keller (BIU)

Analysis of Boolean functions aims at “hearing the shape” of functions on the discrete cube {1,1}^n – namely, at understanding what the structure of the (discrete) Fourier transform tells us about the function.
In this talk, we focus on the structure of “lowdegree” functions on the discrete cube, namely, on functions whose Fourier coefficients are concentrated on “low” frequencies. While such functions look very simple, we are surprisingly far from understanding them well, even in the most basic firstdegree case.
We shall present several results on firstdegree functions on the discrete cube, including the recent proof of Tomaszewski’s conjecture (1986) which asserts that any firstdegree function (viewed as a random variable) lies within one standard deviation from its mean with probability at least 1/2. Then we shall discuss several core open questions, which boil down to understanding, what does the knowledge that a lowdegree function is bounded, or is twovalued, tell us about its structure.
Based on joint work with Ohad Klein

May 17

A Solution to Ringel’s Circle Problem (1959)

Shakhar Smorodinsky (BGU)

In 1959 Gerhard Ringel posed the following problem which remained open for over 60 years.
Suppose we are given a finite family $\C$ of circles in the plane no three of which are pairwise tangent at the same point.
Is it possible to always color the circles with five colors so that tangent circles get distinct colors.
When the circles are not allowed to overlap (i.e., the discs bounded by the circles are pairwise interiorly disjoint) then the number of colors that always suffice
is four and this fact is equivalent to the FourColorTheorem for planar graphs.
We construct families of circles in the plane such that their tangency graphs have arbitrarily large
girth and chromatic number. Moreover, no two circles are internally tangent and no two circles are concentric.
This provides a strong negative answer to Ringel’s 1959 open problem.
The proof relies on a (multidimensional) version of Gallaiӳ theorem with polynomial constraints,
which we derive using tools from RamseyTheory.
Joint work with James Davis, Chaya Keller, Linda Kleist and Bartosz Walczak

May 24

Approximated and stable groups

Arie Levit (Tel Aviv University)

In the study of infinite discrete groups it is useful to consider imperfect approximations by finitary models (either permutations or matrices). I will talk about the stability of such approximations, i.e. can it always be corrected to a perfect approximation, focusing mostly on amenable groups. The involved techniques include ergodic theory and dynamics as well as character theory of infinite groups. Some directions and open problems will be presented.
