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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.

当前项目的主要主题是应用新发展的派生代数几何领域的技术和直觉来解决或重新表述代数几何和复杂几何中的经典问题。提出了两个主要的研究课题。第一种是从拓扑共形场理论的角度研究Spencer Bloch在1972年引入的半正则映射。主要的直觉是半正则映射应该是开闭拓扑保形场理论研究中出现的所谓开闭映射的一部分。第二个主题是研究PI最近与Dima Arinkin关于子流形的导出自交上存在纤维结构的结果与1988年Deligne和Illusie的代数Hodge定理的证明之间的关系。19和20世纪见证了李理论和霍奇理论的发展，这两个理论是现代数学中最有影响力的两个领域。这些理论直接影响了我们对量子物理及相关领域的理解。导代几何是一个新的、令人兴奋的数学领域，它位于代数几何和代数拓扑的交界处。这个项目的工作将通过研究在Hodge理论和代数几何中的经典问题的应用来加深我们对新发展的派生代数几何思想的理解。应用到其他领域是预期的，一些包含应用到李氏理论问题的项目也被包括在内。