Moduli Spaces, Derived Categories, and Motives

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

This project concerns several problems at the interface of algebraic geometry and number theory. There is a rich interplay between these two fields which are at the center of many of the most important past and current developments in mathematics. A number of important problems in number theory, such as questions about the number of solutions of Diophantine equations, can be reinterpreted in geometric terms allowing for the use of powerful tools from algebraic geometry. Conversely, many important geometric questions are motivated by arithmetic applications; for example, questions about the geometry of so-called moduli spaces. The proposed work has connections to other parts of mathematics, such as representation theory, combinatorics, and mathematical physics. More specifically, the project explores four principal areas related to this interplay between arithmetic and geometry. In collaboration with other experts form this field, the PI will investigate the arithmetic of algebraic varieties and its relationship with derived categories of coherent sheaves, building on earlier work on K3 surfaces. The PI will study the main components of moduli spaces, and various generalizations of log geometry. A third part of the project is aimed at understanding the motivic nature of sheaves on algebraic varieties over finite fields, and various invariants thereof. In the fourth part, which is more foundational in nature, the PI will work with collaborators on a new approach to the development of a six operations formalism for l-adic sheaves on stacks.