Applications of derived algebraic geometry to problems in Hodge and Lie theory

派生代数几何在霍奇和李理论问题中的应用

基本信息

批准号：
1200721
负责人：
Andrei Caldararu
金额：
$ 21.11万
依托单位：
University of Wisconsin-Madison
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2012
资助国家：
美国
起止时间：
2012-08-01 至 2018-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1200721&HistoricalAwards=false
关键词：
Applications derived algebraic geometry problems

项目摘要

The overarching theme of the current project is applying techniques and intuitions from the newly developed field of derived algebraic geometry to solve or rephrase classical problems in algebraic geometry and complex geometry. Two main topics are proposed. The first one involves studying the semi-regularity map introduced by Spencer Bloch in 1972 from the point of view of topological conformal field theory. The main intuition is that the semi-regularity map should be a part of the so-called open-closed map that appears in the study of open-closed topological conformal field theories. The second topic involves studying the relationship between the PI's recent result with Dima Arinkin on the existence of a fibration structure on the derived self-intersection of a submanifold and the 1988 proof of Deligne and Illusie of the algebraic Hodge theorem. The 19th and 20th century saw the development of Lie theory and Hodge theory, two of the most influential areas of modern mathematics. These theories have had direct influence on our understanding of quantum physics and related fields. Derived algebraic geometry is a new and exciting field of mathematics, lying at the interface of algebraic geometry and algebraic topology. The work in this project will enhance our understanding of the newly developed ideas of derived algebraic geometry, by studying applications to classical problems in Hodge theory and algebraic geometry. Applications to other fields are expected, with a number of projects containing applications to problems in Lie theory being included.

当前项目的首要主题是应用派生代数几何新发展领域的技术和直觉来解决或改写代数几何和复杂几何中的经典问题。提出了两个主要主题。第一个是从拓扑共角场论的角度研究Spencer Bloch于1972年提出的半正则图。主要直觉是，半正则映射应该是开闭拓扑共形场论研究中出现的所谓开闭映射的一部分。第二个主题涉及研究 PI 与 Dima Arinkin 的最新结果（关于子流形导出自相交上纤维结构的存在性）与 1988 年 Deligne 和 Illusie 的代数 Hodge 定理证明之间的关系。十九世纪和二十世纪见证了李理论和霍奇理论的发展，这是现代数学最有影响力的两个领域。这些理论对我们对量子物理及相关领域的理解产生了直接影响。派生代数几何是一个令人兴奋的新数学领域，位于代数几何和代数拓扑的交汇处。该项目的工作将通过研究霍奇理论和代数几何中经典问题的应用，增强我们对派生代数几何新思想的理解。预计会应用于其他领域，其中包括许多包含李理论问题应用的项目。