Title: The Weil conjectures
Speaker: Hui Langwen
Abstract: Given a system of polynomial equations, one may count their solutions over finite fields with q^n elements, which gives a finite number. Collecting these numbers into a generating function Z(t), one obtains an analogue of Riemann's zeta function for the corresponding variety over F_q. It turns out that this "zeta function" Z(t) is intimately related to the topology of the associated variety.
Weil observed this and proposed several conjectures regarding the property of Z(t), also indicating how these could be proved if one has a well-behaving cohomology for varieties. Assuming the existence of such a cohomology theory, a heuristic "proof" of some of these conjectures will be sketched in the talk. If time permits, the talk will also include the cohomology theory that eventually enabled the solution to the Weil conjectures, namely ℓ-adic cohomology introduced by Grothendieck.
