This is a project with three investigators in the area of mathematics called number theory. Professor Katz will investigate varieties defined over finite fields in particular their cohomology, the distribution of the zeros of their zeta and L functions and monodromy. Professor Yu will study the analogue of the Sato-Tate conjecture in terms of Drinfeld Modules. Professor Vakil will concentrate on Chow groups of certain moduli spaces including the moduli space of stable n-pointed genus g curves.
The main focus of this project can be described as the study of certain kinds of solutions to polynomial equations called modular solutions. Many of the applications computer scientists have for number theory involve these kinds of solutions. A variety over a finite field is a collection of all the modular solutions to a set of polynomial equations. Professor Katz will consider questions involving the number of such solutions, how this number changes as the polynomials change, and how solution sets of different polynomials are related. Many abstract versions of these mathematical objects can help shed light on these questions. Drinfeld modules can be used as a generalization of a variety, and moduli spaces can be used to study collections of varieties as a whole. In this project, three investigators will try to combine several different techniques to a basic problem in mathematics.
