פעילויות השבוע

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.

This is about joint work with Shuji Saito. We will begin the talk with a quick introduction to Voevodsky‘s theory of motives, and how Kahn-Saito-Yamazaki generalise this theory to allow one to treat cohomology theories that are not necessarily A^1-invariant, and have a notion of ramification. We finish by discussing a blowup formula in Kahn-Saito-Yamazaki‘s category of motives with modulus, and how this produces a new proof of this blowup formula for cyclic homology.