Logic Seminar: Borel equivariant inverse to Weierstrass factorization theorem

(This talk reports on an ongoing joint work with Mikhail Sodin, Aron Wennman and Oren Yakir.)

Weierstrass factorization theorem—a classical result in complex analysis—represents any entire function as a (typically infinite) product based on the zeroes of the function.  Furthermore, it asserts that any discrete multiset in a complex plane serves as a set of zeroes for some entire function.

The space of entire functions naturally bears the structure of a standard Borel space, and so does the space of discrete multisets (aka, the space of non-negative divisors).  Each of these spaces comes with a Borel action of the complex plane via the argument shift, and the map assigning to an entire function its multiset of zeroes is easily seen to be equivariant.  The aforementioned theorem of Weierstrass asserts that this map has an inverse, and we investigate the question of whether it admits a Borel equivariant inverse.

There turns out to be two obstructions for such an inverse—one for the doubly periodic divisors and a different one for the 1-periodic divisors. In this talk, we will discuss these obstructions in some detail as well as outline the argument for the existence of a Borel equivariant inverse on the free part of the actions.