Oct 27

The Dimer Model in 3 dimensions

Nishant Chandgotia (Tata Institute of Fundamental Research  Centre for Applicable Mathematics)

The dimer model, also referred to as domino tilings or perfect matching, are tilings of the $Z^d$ lattice by boxes exactly one of whose sides has length 2 and the rest have length 1. This is a very wellstudied statistical physics model in two dimensions with many tools like height functions and Kasteleyn determinant representation coming to its aid. The higher dimensional picture is a little daunting because most of these tools are limited to two dimensions. In this talk I will describe what techniques can be extended to higher dimensions and give a brief account of a large deviations principle for dimer tilings in three dimensions that we prove analogous to the results by Cohn, Kenyon and Propp (2000).
This is joint work with Scott Sheffield and Catherine Wolfram.

Nov 3, In Room 303 in building 28 (or via zoom)

RecurrenceOnline

Tomas Persson (Lund University)

Recurrence is a classical topic in ergodic theory and
dynamical systems, which goes back to Poincaré’s recurrence theorem. I
will talk about old, less old, and new results on recurrence. In
particular, I will talk about how to obtain asymptotic results on the
number of times a typical point returns to a shrinking neighbourhood
around itself.

Nov 10

Sampling a random field along a stationary process, related questions in ergodic theory

JeanPierre Conze (French National Centre for Scientific Research)


Nov 17

Generalizations of Furstenberg’s ×2 × 3 Theorem

Michel Abramoff (BenGurion University)


Nov 24

Stable processes indexed by amenable groups: from probability to nonsingular ergodic theory

Nachi Avraham (The Hebrew University)

Stable processes is an important class of stochastic processes, including Gaussian processes, Cauchy processes and Levy processes. In an analogy to that the ergodicity of a Gaussian process is determined by the spectral measure, it was shown by Rosinski and Samorodnitsky that the ergodicity of a stationary symmetric stable process is characterized by its spectral representation. While this result was known when the process is indexed by $\mathbb{Z}^d$ or $\mathbb{R}^d$, the classical techniques fail when it comes to nonAbelian groups and it was an open question whether the ergodicity of such processes admits a similar characterization.
In this talk I will introduce the fundamentals of stable processes, the ergodic theory behind their spectral representation, and the key ideas of the characterization of the ergodicity for processes indexed by amenable groups. If time permits, I will mention recent results in nonsingular ergodic theory that allow the constructions of weaklymixing but not stronglymixing stable processes indexed by many groups (Abelian groups, Heisenberg group).

Dec 1

Remarkable symbolic dynamical systems associated with some multidimensional continued fraction algorithms

Mélodie Andrieu (BarIlan University)


Dec 8

Minkowski’s Conjecture in Function Fields

Noy Soffer Aranov (Technion)

A fascinating question in the geometry of numbers and diophantine approximation pertains to the maximal covering radius of a lattice with respect to a fixed function. An important covering radius is the multiplicative covering radius, since it is invariant under the diagonal group and relates to the Littlewood’s conjecture. Minkowski conjectured that the multiplicative covering radius of a unimodular lattice in $R^d$ is bounded by above by $1/2^d$ and that this upper bound is unique to the diagonal orbit of the standard lattice. Minkowski’s conjecture is known to be true for $d\leq 10$, yet there isn’t a general proof for higher dimensions.
In this talk, I will discuss the function field (positive characteristic) analogue of Minkowski’s conjecture, which we stated and proved for every dimension. The proofs and the results are surprisingly different from the real case and have implications in geometry of numbers and dynamics. This talk is based on joint work with Uri Shapira.

Dec 15

Structure theorem for the GowersHostKra seminorms

Or Shalom (Institute of advanced studies)

Szemeredi’s theorem asserts that in every subset of the natural numbers of positive density one can find an arithmetic progression of arbitrary length. In 2001, Gowers gave a quantitative proof for this theorem. A key definition in his work are the Gowers norms which measure the randomness of subsets of the natural numbers. Inspired by Furstenberg’s ergodic theoretical proof of Szemeredi’s theorem, Gowers proved the following dichotomy: Either the given set is close to a random set with respect to these norms, or it admits some algebraic structure. Gowers then proved that in each of these cases Szemeredi’s theorem holds. Later, Host and Kra studied the structure of certain ergodic systems associated with an infinitary version of the Gowers norms. Inspired by their work, Green, Tao and Ziegler improved Gowers’ structure theorem showing that a function (or a set) with large Gowers norm must correlate with a nilsequence. This result is known as the inverse theorem for the Gowers norms. Recently, Jamneshan and Tao proved (roughly speaking) that a generalization of the HostKra theorem for ergodic systems associated with actions of the largest countable abelian group $\mathbb{Z}^\omega$ will imply the most general version of the inverse theorem for the Gowers norms. In this talk I will survey the above in more detail and mention some recent developments about these structure theorems.

Dec 22

Probabilistic Laws on Groups

Guy Blachar (BarIlan University)

Suppose a finite group satisfies the following property: If you take two random elements, then with probability bigger than 5/8 they commute. Then this group is commutative.
Starting from this wellknown result, it is natural to ask: Do similar results hold for other laws (pgroups, nilpotent groups…)? Are there analogous results for infinite groups? Are there phenomena specific to the infinite setup?
We will survey known and new results in this area. New results are joint with Gideon Amir, Maria Gerasimova and Gady Kozma.

Dec 29

The seminar is cancelled

Action Now day in TelAviv


Jan 5

Classification and statistics of cutandproject sets

Yotam Smilansky (Rutgers University)

Cutandproject point sets are constructed by identifying a strip of a fixed ndimensional lattice (the “cut”), and projecting the lattice points in that strip to a ddimensional subspace (the “project”). Such sets have a rich history in the study of mathematical models of quasicrystals, and include wellknown examples such as the Fibonacci chain and vertex sets of Penrose tilings. Dynamical results concerning the translation action on the hull of a cutandproject set are known to shed light on certain properties of the point set itself, but what happens when instead of restricting to translations we consider all volume preserving linear actions?
A homogenous space of cutandproject sets is defined by fixing a cutandproject construction and varying the ndimensional lattice according to an SL(d,R) action. In the talk, which is based on joint work with René Rühr and Barak Weiss, I will discuss this construction and introduce the class of RatnerMarklofStrömbergsson measures, which are probability measures supported on cutandproject spaces that are invariant and ergodic for the group action. A classification of these measures is described in terms of data of algebraic groups, and is used to prove analogues of results about a Siegel summation formula and identities and bounds involving higher moments. These in turn imply results about asymptotics, with error estimates, of pointcounting and patchcounting statistics for typical cutandproject sets.

Jan 12

An Advertisement for Coarse Groups and Coarse Geometry

Arielle Leitner (Weizmann Institute and Afeka College of Engineering)

Coarse structures are used to study the large scale geometry of a space. For example, although the integers and the real line are different topologically, they look the same from “far away”, in the sense that any geometric configuration in the real line can be approximated by one in the integers, up to some uniformly bounded error. A coarse group is a group object in the category of coarse spaces, for example, this means the group operation is only “coarsely associative,” etc. In joint work with Federico Vigolo we study coarse groups. This talk will be an advertisement for our work, as we walk through examples that illustrate some of our main results, and connections to other subjects.

Jan 19

Dynamical questions arising from Dirichlet’s theorem on Diophantine approximation

Anurag Rao (Technion)

We study the notion of Dirichlet improvability in a variety of settings and make a comparison study between Dirichletimprovable numbers and badlyapproximable numbers as initiated by DavenportSchmidt. The question we try to answer, in each of the settings, is – whether the set of badlyapproximable numbers is contained in the set of Dirichletimprovable numbers. We show how this translates into a question about the possible limit points of bounded orbits in the space of twodimensional lattices under the diagonal flow. Our main result gives a construction of a full Hausdorff dimension set of lattices with bounded orbit and with a prescribed limit point. Joint work with Dmitry Kleinbock.

Feb 23

SEMINAR POSTPONED TO APRIL 20th: Mixing sequences for nonmixing locally compact Abelian groups actions

El Houcein El Abdalaoui (CNRSUniversité de Rouen)

Mixing is an important spectral property of dynamical systems and it can be described concretely. But, “In general a measure preserving transformation is” only “mixing” along a sequence of density one, by the RhoklinHalmos theorem. On the other hand, mixing on some sequences implies mixing. Formally, the mixing can be defined by demanding that the ergodic averages along any increasing sequence converge in mean, thanks to the BlumHanson theorem. In my talk, I will present my recent joint contribution with Terry Adams to Bergelson’s question asked online during the Lille conference 2021: Does mixing on the squares imply mixing?
We first obtain a characterization of a sequence for which mixing on it implies mixing. We further establish that there are nonmixing maps that are mixing on appropriate sequences. We extend also our results to the group action with the help of the HostParreau characterization of the set of continuity from Harmonic Analysis. We further extended our result to the Real line action. As a open question, we ask pour extension our our result to the case of noncommutative case and specially Heisenberg group action.
