Activities This Week

Given two permutations A and B which “almost” commute, are they “close” to permutations A’ and B’ which really commute? Arzhantseva and Paunescu (2015) formalized this question and answered affirmatively. This can be viewed as a property of the equation $XY=YX$, and turns out to be equivalent to the following property of the group $Z^2 = \langle X,Y \vert XY=YX \rangle$ Every “almost-everywhere locally-defined” action of $X^2$ is close to a genuine action of $Z^2$. This leads to the notion of locally testable groups (aka “stable groups”).

We will take the opportunity to say something about “local testability” in general, which is an important subject in computer science. We will then describe some results and methods to decide whether various groups are locally testable. This will bring in some important notions in group theory, such as amenability, Kazhdan’s Property (T) and sofic groups.

Based on joint work with Alex Lubotzky.

By a result of Glimm, we know that classifying representations of non-type-I $C^*$-algebras up to unitary equivalence is essentially impossible (at least with countable structures). Instead of this, one either restricts to a tractable subclass or weakens the invariant.

In the theory of free semigroup algebras, the latter is done for Toeplitz-Cuntz algebras, and is achieved via two key results in the theory: the first is a theorem of Davidson, Katsoulis and Pitts on the $2\times 2$ structure of free semigroup algebras, and the second, a Lebesuge-von Neumann-Wold decomposition theorem of Kennedy.

This talk is about joint work with Ken Davidson and Boyu Li, where we generalize this theory to representations of Toeplitz-Cuntz-Krieger algebras associated to a directed graph $G.$ We prove a structure theorem akin to that of Davidson, Katsoulis and Pitts, and provide a Lebesuge-von Neumann Wold decomposition using Kennedy’s theorem. We discuss some of the difficulties and similarities when passing to the more general context of operator algebras associated to directed graphs.

תורת ההצגות היא תורה אשר חוקרת ייצוגים של חבורות כאוסף “סימטריות” של אובייקט כלשהו. למשל, ניתן להציג חבורה מסוימת כאוסף מטריצות (כלומר, אוסף טרנספורמציות לינאריות על מרחב ווקטורי נתון, אשר סגור תחת כפל והופכי). ייצוג כזה נקרא הצגה של חבורה. נדבר קצת על מבנה ההצגות של חבורות סופיות ולא סופיות, ונראה כיצד מופיעים בתורה זו הילוכים על גרפים ומספרי קטלן.

In this talk we discuss the fine scale $L^2$-mass distribution of toral Laplace eigenfunctions with respect to random position. For the 2-dimensional torus, under certain flatness assumptions on the Fourier coefficients of the eigenfunctions and generic restrictions on energy levels, both the asymptotic shape of the variance and the limiting Gaussian law are established, in the optimal Planck-scale regime. We also discuss the 3-dimensional case, where the asymptotic behaviour of the variance is analysed in a more restrictive scenario. This is joint work with Igor Wigman.

Zilber’s trichotomy conjecture, in modern formulation, distinguishes three flavours of geometries of strongly minimal sets — disintegrated/trivial, modular, and the geometry of an ACF. Each of these three flavours has a classic ``template’’ — a set with no structure, a projective space over a prime field, and an algebraically closed field, respectively. The class of ab initio constructions with which Hrushovski refuted the conjecture features a new flavour of geometries — non-modular, yet prohibiting any algebraic structure.

In this talk we take a step towards defining ``template’’ structures for the class of (CM-trivial) ab initio Hrushovski constructions. After presenting intuitively the standard ab initio Hrushovski construction, we generalize Hrushovski’s predimension function, showing that the geometries associated to certain Hrushovski constructions are, essentially, ab initio constructions themselves. If time permits, we elaborate on how these \emph{geometric} structures may generate the class of geometries of ab initio constructions under the Hrushovski fusion operation.

I shall review the framework of algebraic families of Harish-Chandra modules, introduced recently, by Bernstein, Higson, and the speaker. Then, I shall describe three of their applications. The first is contraction of representations of Lie groups. Contractions are certain deformations of representations with applications in mathematical physics. The second is the Mackey bijection, this is a (partially conjectural) bijection between the admissible dual of a real reductive group and the admissible dual of its Cartan motion group. The third is the hidden symmetry of the hydrogen atom as an algebraic family of Harish-Chandra modules.