Motivic, Stringy, and Gromov-Witten Invariants

Algebraic geometry is the study of varieties, which are geometric objects defined by polynomial equations. Examples of varieties include the surface of a sphere and the surface of a donut. These two examples have something important in common: they are both smooth, i.e., if one zooms in close to these objects, they eventually look flat. Smooth varieties naturally arise throughout the sciences. For example for many physical systems, one expects that small changes to the starting configuration will only lead to small changes in the short-term behavior of that system. Considering the underlying geometric object that determines the behavior of that system, such an expectation is essentially the assumption that this geometric object is smooth. Therefore improving our understanding of smooth varieties, in addition to being a central goal in algebraic geometry, is of broad significance beyond mathematics. Many of our most powerful tools for studying smooth varieties rely crucially on varieties that are not smooth, i.e., those that have "sharp" or "pointy" pieces called singularities. Obtaining a better understanding of these singularities is also a central goal in algebraic geometry. The PI will develop new techniques in the study of singularities with an emphasis on certain invariants that arise in theoretical physics. The PI will also conduct activities in outreach, mentoring, and conference and seminar organizing. More specifically, the PI will conduct research on stringy invariants and Gromov-Witten invariants and will pursue interactions between these invariants and combinatorial algebraic geometry, including connections to toric varieties, hyperplane arrangements, and matroids. The PI will use motivic integration for smooth Artin stacks to study stringy invariants of singular varieties. With a view toward finding a cohomological interpretation for stringy Hodge numbers, the PI will pursue a strategy for describing stringy Hodge numbers in terms of crepant resolutions via Artin stacks. Furthermore, The PI will use related techniques to develop a McKay correspondence for reductive groups. The PI will also study new matroid invariants related to the Gromov-Witten theory of wonderful models of hyperplane arrangements. In particular, the PI will pursue a strategy for defining quantum cohomology rings for arbitrary matroids. This strategy involves an interplay between Gromov-Witten theory, tropical curves, and polyhedral geometry. 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.