Abstract: 

The talk is about answering questions of the form “Given an interesting family of functions, what is the structure of their zero sets in the complex numbers?” For example, where are the zeros of orthogonal polynomials, `random’ real polynomials, graph polynomials, partition functions,…? The main focus of the talk is on the zeros of a family of modular forms, so-called Eisenstein series, defined as lattice sums. Contrary to their cousins, the Hecke eigenforms, whose zeros equidistribute in the complex upper half plane, the zeros of Eisenstein series converge (in the sense of Hausdorff distance) to a specific configuration of geodesics. Digging a bit deeper in the complex analysis, we can sometimes prove transcendence of the zeros and prove that the limiting measure of the counting measure of the zeros is a Haar measure. A supporting part is played by the largely forgotten Dutch mathematician Kluyver. (Joint work with Sebastian Carrillo and Berend Ringeling.)