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.

当前项目的总体主题是在新开发的代数几何形状领域应用技术和直觉，以在代数几何形状和复杂的几何形状中解决或重塑经典问题。提出了两个主要主题。第一个涉及研究Spencer Bloch于1972年从拓扑结构域理论的角度研究的半定期图。主要的直觉是，半规范性图应成为所谓的开放式封闭图的一部分，该图出现在开放式拓扑结构域理论研究中。第二个主题涉及研究PI的最新结果与Dima Arinkin对纤维化结构的存在，这是在Submanifold的衍生自我交流中与代数Hodge定理的Deligne和Illusie证明。 19世纪和20世纪看到了谎言理论和霍奇理论的发展，这是现代数学最具影响力的两个领域。这些理论直接影响了我们对量子物理和相关领域的理解。派生的代数几何形状是一个新的令人兴奋的数学领域，位于代数几何和代数拓扑结构的界面上。该项目的工作将通过研究霍奇理论和代数几何学中的经典问题的应用来增强我们对新发展的代数几何思想的理解。预计将在其他领域的应用程序应用，其中许多项目包含谎言理论中的问题。