Seminar in Analysis and Quantum by Aron Wennman

Speaker:

Aron Wennman, KU Leuven, Belgium

Title:

An equivariant Weierstrass theorem for entire functions

Abstract:

A classical theorem of Weierstrass states that for any discrete set of points in the complex plane, there is an entire function which vanishes precisely on that set. In other words, the "zero set map" Z maps the space E of entire functions surjectively onto the space D of discrete sets, so it has a right inverse W.

The complex plane acts naturally by translation on both the spaces E and D, and the zero set map is equivariant, i.e., it commutes with these translations. A natural question is: Can we require that the "Weierstrass map” W commutes with translations as well? (The standard product constructions of entire functions with given zeros do not give this property).

We study this question in ongoing joint work with Konstantin Slutsky and Mikhail Sodin. In particular, excluding periodic zero sets, it’s possible to construct a Borel equivariant right inverse W, but not a continuous one. I will describe our results, and explain how we came to this question starting from natural questions in statistical physics.