Moduli Spaces, Derived Categories, and Motives

模空间、派生范畴和动机

• 批准号: 1601940
• 负责人: Martin Olsson
• 金额: $ 16.96万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-09-01 至 2019-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1601940&HistoricalAwards=false
• 关键词: Moduli; Spaces; Derived; Categories; Motives

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.

该项目涉及代数几何和数论接口的几个问题。 这两个领域之间存在着丰富的相互作用，它们是数学过去和当前许多最重要发展的中心。数论中的许多重要问题，例如有关丢番图方程的解数的问题，可以用几何术语重新解释，从而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都是由算术应用引发的；例如，有关所谓模空间几何的问题。拟议的工作与数学的其他部分有联系，例如表示论、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的相互作用相关的四个主要领域。 PI 将与该领域的其他专家合作，在 K3 曲面上的早期工作的基础上，研究代数簇的算术及其与相干滑轮的派生类别的关系。 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理解有限域上代数簇上滑轮的动机性质及其各种不变量。 在本质上更为基础的第四部分中，PI 将与合作者合作开发一种新方法来开发堆栈上的 l-adic 滑轮的六操作形式主义。