CAREER: Equivariant and Infinite-Dimensional Combinatorial Algebraic Geometry

Ohio State University, The, Columbus OH

Investigators

Abstract

Symmetry plays a fundamental role in our mathematical understanding of the world. There are discrete symmetries, like the reflection of a face in a mirror or the hexagonal pattern of floor tiles; and there are continuous groups of symmetries, like the rotations of a sphere about various axes. Among continuous groups, there are those that involve finitely many parameters, as well as infinite-dimensional ones. Objects with finite-dimensional groups of symmetries often admit explicit "combinatorial" descriptions. Infinite-dimensional groups are naturally more complicated. A key goal of this project is to develop a more concrete understanding of infinite-dimensional groups of symmetries and the spaces on which they act. As part of the educational component of the project, the PI will run workshops related to the research and will develop an undergraduate course linking combinatorial mathematics to applications in the community. The research program involves three main topics: Schubert polynomials, which give formulas for the degrees of algebraic varieties locally defined by rank conditions on matrices; Newton-Okounkov bodies, which are convex bodies encoding the geometry of line bundles on projective varieties; and quantum K-theory, which contains refined information about curves on varieties. The main objectives are (1) to complete a package of Schubert polynomials, tying them to the geometry of infinite-dimensional Grassmannians and flag varieties; (2) to numerically compute Newton-Okounkov bodies and use them to construct mirror-symmetric duals of special varieties; and (3) to establish foundational results about the quantum K-theory of homogeneous varieties. A further aim is to apply degeneracy locus techniques to questions ranging from nonlinear optimization to the geometry of linear systems on algebraic curves. 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.