# Convolutions of polynomial maps: a crossroads of algebra, geometry and analysis

Speaker
Yotam Hendel, Universite de Lille
Date
30/10/2022 - 13:00 - 12:00Add to Calendar 2022-10-30 12:00:00 2022-10-30 13:00:00 Convolutions of polynomial maps: a crossroads of algebra, geometry and analysis Let f=(f_1,...,f_m) be a tuple of m polynomials with integer coefficients in n variables.  Then f can naturally be considered as a complex map f:C^n->C^m, and one may approach studying f from several different perspectives: 1) Arithmetic, by counting the number of solutions to the system of equations {f_i = c_i mod r}, for different choices of integers c_i and r. 2) Geometric, by studying the geometry and singularities of the fibers (level-sets) of the complex map f:C^n->C^m. 3) Analytic, by taking a smooth measure \mu with compact support in C^n, and studying analytic  (e.g. integrability) properties of the resulting pushforward measure f_*(\mu) on C^m. In this talk I will explain how these different approaches relate to one another. I will then discuss a natural algebraic convolution operation  between maps, which allows one to study the behavior of certain algebraic families of random walks using these connections.  This is based on a series of works from recent years joint with I. Glazer. Seminar Room, Math Building 216 אוניברסיטת בר-אילן - המחלקה למתמטיקה mathoffice@math.biu.ac.il Asia/Jerusalem public
Place
Seminar Room, Math Building 216
Abstract

Let f=(f_1,...,f_m) be a tuple of m polynomials with integer coefficients in n variables.

Then f can naturally be considered as a complex map f:C^n->C^m, and one may approach studying f from several different perspectives:
1) Arithmetic, by counting the number of solutions to the system of equations {f_i = c_i mod r}, for different choices of integers c_i and r.
2) Geometric, by studying the geometry and singularities of the fibers (level-sets) of the complex map f:C^n->C^m.
3) Analytic, by taking a smooth measure \mu with compact support in C^n, and studying analytic

(e.g. integrability) properties of the resulting pushforward measure f_*(\mu) on C^m.

In this talk I will explain how these different approaches relate to one another. I will then discuss a natural algebraic convolution operation

between maps, which allows one to study the behavior of certain algebraic families of random walks using these connections.

This is based on a series of works from recent years joint with I. Glazer.

תאריך עדכון אחרון : 27/10/2022