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.