The Weil Conjectures and A1-homotopy Theory

Explore the fascinating intersection of algebraic topology and number theory in this colloquium talk by Kirsten Wickelgren from Duke University. Delve into the Weil conjectures, a groundbreaking set of propositions from 1948 that establish a profound connection between algebraic topology and solutions to equations over finite fields. Discover how the zeta function of a variety over a finite field serves as both a generating function for solution counts and a product of characteristic polynomials of cohomology group endomorphisms. Learn about A1-homotopy theory and its applications in this context. Examine an enriched version of the zeta function with coefficients in a group of bilinear forms, revealing new insights into the relationship between finite field solutions and the topology of associated real manifolds. Gain exposure to cutting-edge research in this field, presented in collaboration with Tom Bachmann, Margaret Bilu, Wei Ho, Padma Srinivasan, and Isabel Vogt.