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

当前项目的总体主题是应用新发展的衍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。提出了两个主要议题。第一个是从拓扑共形场论的角度研究Spencer Bloch在1972年引入的半正则映射。主要的直觉是，半正则映射应该是在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最近与Dima Arinkin关于子流形的自交上的纤维结构存在性的结果与1988年Deligne和Illusie对代数Hodge定理的证明之间的关系。19世纪和20世纪见证了李理论和霍奇理论的发展，这是现代数学中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学和相关领域的理解。推导代数几何是一个新兴的、令人兴奋的数学领域，它处于代数几何和代数拓扑的交叉领域。本项目的工作将通过研究霍奇理论和代数几何经典问题的应用，加强我们对派生代数几何新发展思想的理解。预计将应用于其他领域，包括一些包含李理论问题应用的项目。

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.