Singularity properties of convolutions of algebraicmorphisms and probabilistic Waring type problems Abstract:

Abstract: Let G be a connected algebraic group. We define and study a convolution operation between algebraic morphisms intoG. We show that this operation yields morphisms with improved singularityproperties, and in particular, that under reasonable assumptions one can alwaysobtain a flat morphism with reduced fibers of rational singularities (termed anFRS morphism) after enough convolutions. The FRS property is of high importance since (FRS) morphisms can becharacterized by good asymptotic behaviour of the number of points of theirfibers over finite rings of the form Z/p^kZ. This further allows us to interpret the FRS property through probabilisticlenses. We discuss some of the above, motivated by the special case of word maps whichcan be viewed as a relative analogue in the settings of p-adic groups of Waring's problem from 1770 (seearXiv:1912.12556). Joint work with Itay Glazer.

algebraic geometryalgebraic topologycategory theoryrings and algebrasrepresentation theory

Audience: researchers in the topic

Seminar on Representation Theory and Algebraic Geometry