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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.

A group is said to have a fixed-point 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 fixed-point properties with respect to (classes of) Banach spaces. I will survey some results regarding groups with and without these fixed-point properties and then present a recent result of mine regarding fix-point properties for random groups with respect to l^p spaces.

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 simplied 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 self-ignition 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.

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 C-algebra 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, C-algebras, 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.

Using the Bruhat decomposition, a general linear group over a p-adic 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 p-adic groups. I will show how the class of RSK representations (developed with Erez Lapid) serves as a bridge between the Langlands classification for the p-adic group and the classical Specht construction of the finite domain.

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 “low-degree” 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 first-degree case. We shall present several results on first-degree functions on the discrete cube, including the recent proof of Tomaszewski’s conjecture (1986) which asserts that any first-degree 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 low-degree function is bounded, or is two-valued, tell us about its structure.

Based on joint work with Ohad Klein

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 Four-Color-Theorem 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 Ramsey-Theory.

Joint work with James Davis, Chaya Keller, Linda Kleist and Bartosz Walczak

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.

Celebrating Michael Lin’s 80th birthday. Place: Ben-Gurion university of the Negev. Room -103 in building 14 Mendel building (across from Aroma branch on the west side of the campus). Time: Tuesday 31/5/2022 between 10:30-16:30.

Schedule:

10:00 - 10:30 Coffee 10:30 - 11:20 Omri Sarig (Weizmann institute) 11:40 - 13:00 Jon Aaronson (Tel-Aviv University) 13:00 - 14:30 Lunch (the lunch will take place in room -101 of the math building) 14:30 - 15:20 Eli Glasner (Tel-Aviv University, joint with Colloquium) 15:30 - 16:20 Ariel Yadin (Ben Gurion University)

Organizers: Yair Glasner, Tom Meyerovitch, and Guy Cohen Zoom broadcasting LINK Meeting ID: 820 8565 9434 The conference website: https://sites.google.com/view/michael-lins-80th-birthday/home

The dimension of the space of multilinear products of higher commutators is equal to the number of derangements, $[e^{-1}n!]$. Our search for a combinatorial explanation for this fact led us to study representations of left regular bands, whose resolution is obtained through analysis of cubical partial representations. There are applications in combinatorics, probability, and nonassociative algebra.

Asymptotic representation theory is an important and quickly developing area of mathematics rich in connections to other fields, such as, e.g., probability, algebraic combinatorics, and mathematical physics. I will survey the basic ideas and results of asymptotic representation theory, mostly of symmetric groups, and then focus on some recent contributions.

By work of Ratner, Margulis, Dani and many others, unipotent flows on homogeneous spaces have strong measure theoretic and topological rigidity properties. By work of Eskin-Mirzakhani and Eskin-Mirzakhani-Mohommadi, the action of SL(2,R) and the upper triangular subgroup of SL(2,R) on strata of translation surfaces have similar rigidity properties. We will describe how some of these results fail for the horocycle flow on strata of translation surfaces. In particular, 1) There exist horocycle orbit closures with fractional Hausdorff dimension. 2) There exist points which do not equidistribute under the horocycle flow with respect to any measure. 3) There exist points which equidistribute distribute under the horocycle flow to a measure, but they are not in the topological support of that measure. This is joint work with Jon Chaika and John Smillie. The talk will be elementary and will require no prior background in dynamics.