Activities This Week

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.

A family of lines through the origin in Euclidean space is called equiangular if any pair of lines defines the same angle. The problem of estimating the maximum cardinality of such a family in $R^n$ was extensively studied for the last 70 years. Answering a question of Lemmens and Seidel from 1973, in this talk we show that for every fixed angle$\theta$ and sufficiently large $n$ there are at most $2n-2$ lines in$R^n$ with common angle $\theta$. Moreover, this is achievable only when $\theta =\arccos \frac{1}{3}$. Various extensions of this result to the more general settings of lines with $k$ fixed angles and of spherical codes will be discussed as well. Joint work with I. Balla, F. Drexler and P. Keevash.

A Gaussian stationary process is a random function f:R–>R or f:C–>C, whose distribution is invariant under real shifts, and whose evaluation at any finite number of points is a centered Gaussian random vector. The mathematical study of these random functions goes back at least 75 years, with pioneering works by Kac, Rice and Wiener. Nonethelss, many basic questions about them, such as the fluctuations of their number of zeroes, or the probability of having no zeroes in a large region, remained unanswered for many years.

In this talk, we will give an introduction to Gaussian stationary processes, and describe how a spectral perspective combined with tools from harmonic, real and complex analysis, yields new results about such long-lasting questions.