Activities This Week

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.

A group is said to be invariably generated (IG) by a set S, if any conjugation of elements of S still generates G, and topologically invariably generated (TIG) by S if every such conjugation generates G topologically.

I will give a short review on this notion, and present new results, from a joint work with Gennady Noskov.

In this talk, I will discuss specific cases of inverse systems of groupoids, and the dual directed systems of groupoid C*-algebras. This is based on a recent paper with Kyle Austin (with early contributions by Joav Orovitz).

I will start with a general discussion of inverse systems of groupoids for which limits can be shown to exist, followed by a particular construction of approximating a given sigma-compact groupoid equipped with a Haar system of measures by an inverse system of second countable groupoids. I will conclude by discussing connections to results about C*-algebras.

Kyle gave a talk a few weeks ago mentioning some of the results in our paper; overlap will be kept to a minimum, while still making the talk self-contained.