Activities This Week

I shall give simplicial reformulations of several equivalent characterisations of stability of a first order theory. In particular, I shall reformulate without any word of logic the statement of the Sela’s theorem that the theory of a free group is stable.

I will make an effort to be self-contained and remind the definitions of my previous talk. In particular, I shall define a semantics of first order formulas with respect to any simplicial set, where a predicate is assigned a subset of simplicies rather than n-tuples. From this point of view, the type space functor defined last time, is the canonical model of a first order theory.

Delsarte’s method provides a framework for analysis of sphere packing problems in terms of positive definite functions. Constrained packing, where only a prespecified set of angles between points may occur, is, in turn, naturally governed by the interpolation theory for positive definite functions. In this talk, we discuss the general theory of positive definite functions on the sphere, Delsarte’s method and its geometric kernel embedding interpretation, related interpolation problems, and other topics.

Based on joining work with Sujit Sakharam Damase.

In this talk we will be concerned with actions of finite groups, and more specifically with elements fixing “few” points in such actions. One of the goals will be to explain why this is interesting, highlighting applications to polynomials, random generation and, time permitting, monodromy groups of curves.

We will discuss questions of invertibility of large random matrices, and estimating the smallest singular value of random matrices in the non-asymptotic setting, via methods of asymptotic geometric analysis. We will discuss a construction of an efficient net on the high-dimensional sphere, and its applications to the questions of invertibility of inhomogeneous and log-concave random matrices.

An old question of Poincaré concerns creating periodic points via perturbations of a diffeomorphism. While this question was initially studied in the 60s, various facets of it remain largely open. Recently, several advances were made in the context of Hamiltonian diffeomorphisms, which are diffeomorphisms arising from classical mechanics. I will discuss a variant of this problem in the presence of certain invariant sets, as well as an application to invariant measures. This is based on a joint work with Erman Cineli and Sobhan Seyfaddini.

כל יריעה חלקה ”נראית מקומית כמו $\mathbb{R}^N$“. במקרה זה גאומטריה, טופולוגיה, ואלגברה הן כולן מקומית טריויאליות. אבל לא הכל חלק בעולם שלנו, למשל העקומים הנתונים על-ידי המשוואות $xy=0$ או $y^2=x^d$.

כל נקודה סינגולרית כוללת בתוכה:

• חוגים, אידאלים, מודולים;
• חבורה יסודית, הומולוגיות הומוטופיות;
• מרחבי “מודולים”

וכו.

אני אציג מבוא נגיש לתחום זה.