Collaborative Research: Calculus beyond Schubert

University Of North Carolina At Chapel Hill, Chapel Hill NC

Investigators

Abstract

This research project aims to resolve outstanding questions in enumerative algebraic geometry. Broadly speaking, methods for finding simultaneous solutions to multiple equations have significant implications for progress in physics, computer science, and engineering. These solutions may be expressed in terms of the intersection of certain geometric spaces. The motivating question in enumerative geometry is to predict the number of geometric figures with specified properties that satisfy a list of conditions. For example, we may seek the number of curves that contain a set of points and are tangent to a list of lines. Surprisingly, while it can be very difficult to determine the precise list of figures that satisfy the conditions, it is often possible to predict the number of such figures. The search for exact formulas for the number of solutions in enumerative geometry is an active area of research with relations to numerous fields, including geometry, combinatorics, representation theory, complexity theory in computer science, and mirror symmetry in theoretical physics. This grant will support continued work in these areas by the investigators and their graduate students. An effective approach to solving enumerative geometric problems is to understand the intersection theory of moduli spaces of geometric figures. The field of Schubert calculus loosely applies to these investigations among large classes of varieties in homogeneous spaces. Various cohomology theories can be used to extract enumerative information. For example, singular cohomology is useful for counting the number of points in intersections of geometric figures, quantum cohomology is designed for counting curves meeting other figures, and equivariant cohomology produces more general geometric invariants that depend on a group action. Cohomology theories of spaces with group actions often have naturally defined bases, and geometric invariants related to such bases tend to possess intriguing positivity properties, often related to beautiful combinatorial structures that capture the essential aspects of the geometry. The investigators will study these phenomena in the context of several geometric spaces, including flag varieties and their cotangent bundles, bow varieties, and Hessenberg varieties. Techniques include intersection theory, equivariant localization, symmetric functions, Hecke algebra actions, and geometric representation theory. 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.