Research Projects

The project aims to explore the application of derived algebraic geometry from a motivic viewpoint to study various cohomology theories of moduli stacks. The central goal is to develop tools for studying refined enumerative invariants. This project originated from the work of the PI and Khan, where we constructed categories of motivic sheaves on derived algebraic stacks. The cohomology with coefficients in these sheaves was shown to recover cohomology theories—such as Chow groups, algebraic K-theory, and cobordism—for derived algebraic stacks. Furthermore, Grothendieck’s six-functor formalism, established for these motivic categories, allowed us to build operations and functorialities for these cohomology theories. Using our approach, we have extended these cohomology theories to all derived algebraic stacks along with all their desirable basic properties. This categorical approach has also enabled us to work with these cohomology theories in a unified way. In subsequent projects, the PI, working with several coauthors, has used these techniques to develop localization methods for derived moduli stacks. Since moduli stacks are often complicated spaces, localization techniques à la Segal concentration, Atiyah–Bott torus localization, and cosection localization allow us to reduce computations on moduli stacks to simpler subspaces. These techniques, which were previously known under restrictive assumptions on moduli stacks, have already been successfully used in the works of Ellingsrud–Strømme, Kontsevich, Graber–Pandharipande, and others to compute enumerative invariants. Our results have vastly expanded the scope of the applicability of these techniques to much more general moduli stacks and several different cohomology theories. We expect these extensions to have broad applications in enumerative computations. The future goal of this project is to use these motivic techniques to further develop and study classical problems for moduli stacks in the context of refined cohomology theories, from the robustly developing perspective of derived algebraic stacks.