Moduli Spaces, Derived Categories, and Motives

这个项目涉及代数几何和数论的接口的几个问题。 这两个领域之间有着丰富的相互作用，它们是过去和现在数学中许多最重要的发展的中心。数论中的一些重要问题，如丢番图方程的解的个数问题，可以用几何术语重新解释，允许使用代数几何中的强大工具。相反，许多重要的几何问题是由算术应用激发的;例如，关于所谓模空间的几何问题。这项工作与数学的其他部分有联系，如表示论、组合学和数学物理。更具体地说，该项目探索了与算术和几何之间相互作用有关的四个主要领域。 在与其他专家合作形成这一领域，PI将研究代数簇的算法及其与相干层的派生类别的关系，建立在早期的工作K3表面。 PI将研究模空间的主要组成部分，以及对数几何的各种推广。 该项目的第三部分旨在了解有限域上代数簇上层的动机性质及其各种不变量。 在第四部分，这是更基础的性质，PI将与合作者在一个新的方法来开发一个六个操作形式主义的l-adic层堆栈上。

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.