Algebra Seminar - Padmavathi Srinivasan, ICERM

Title: Weil conjectures, enriched zeta functions, and the topology of real points

Abstract:

For a variety defined over the rationals, the Weil conjectures relate the Betti numbers of the set of complex solutions to the number of solutions modulo prime powers, packaged into the zeta function. Can the zeta function also tell us about the Betti numbers of the set of real solutions? In this talk, we will present a quadratic enrichment of the zeta function and show that it naturally encodes the Betti numbers of the real points in some situations. We will also present a rationality result for the enriched zeta function for cellular varieties. This is joint work with Margaret Bilu, Wei Ho, Isabel Vogt, and Kirsten Wickelgren.