3:00–4:00 pm Eckhart Hall, Room 202
The Weil Conjectures and A1-homotopy Theory
The celebrated Weil conjectures from 1948 propose a beautiful connection between algebraic topology and the number of solutions to equations over finite fields: the zeta function of a variety over a finite field is simultaneously a generating function for the number of solutions to its defining equations and a product of characteristic polynomials of endomorphisms of cohomology groups. The ranks of these cohomology groups are the Betti numbers of the associated complex manifold. This talk will describe the Weil conjectures, some A1-homotopy theory, and then enrich the zeta function to have coefficients in a group of bilinear forms. The enrichment provides a connection between the solutions over finite fields and the topology of the associated real manifold.
The new work in this talk is joint with Tom Bachmann, Margaret Bilu, Wei Ho, Padma Srinivasan and Isabel Vogt.
