CAREER: Moduli Spaces and Derived Categories

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Many phenomena in mathematics and mathematical physics are described by systems of polynomial equations. The set of solutions of such a system of equations can have a very complicated and interesting shape. Typically the equations involve numbers which are regarded as “parameters.” The problem of determining how the geometric properties of the set of solutions change as one varies these parameters is called a “moduli problem,” and moduli problems are important to many subjects in algebra and geometry. This project will introduce a new approach to studying moduli problems, as well as applications of this new approach to several open problems inspired by high energy theoretical physics. The PI will work towards the training of students in this research field, through research projects with undergraduate students, mentoring of graduate students, and lectures aimed to k-12 students. One of the PI's goals is to build connections with industry where machine learning is involved. Specifically, the Principal Investigator will use recent developments in the theory of algebraic stacks and derived algebraic geometry to develop a general approach to moduli problems in algebraic geometry. The PI’s recent work on extending the methods and results of geometric invariant theory to arbitrary moduli problems has led to a relatively complete framework for constructing moduli spaces and for breaking large moduli problems into pieces which are easier to study. This project investigates some extensions of the foundational aspects of this work, as well as applications to several problems in enumerative geometry. The project also proposes applications to studying derived categories of coherent sheaves.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.