Interactions between Newton-Okounkov Bodies, Cluster Algebras, and Orbit Closures

Washington University, Saint Louis MO

Investigators

Abstract

The interplay between combinatorics, which studies with discrete structures, and algebraic geometry, which is concerned with solutions of polynomial equations, has immensely enriched both areas. Combinatorics provides discrete objects that can be used to encode information about an algebraic variety. Conversely, the rich structure of a combinatorial object is often better understood when seen as a discrete shadow of an algebro-geometric phenomenon. The research in this project develops the interplay between these areas utilizing tools which include degenerations of varieties and high-dimensional analogues of polygons. This project aims to understand various aspects of the interplay between combinatorics and algebraic geometry for Newton-Okounkov bodies, symmetric orbit closures, and subword complexes. Generalizing the relation between polytopes and toric varieties, Newton-Okounkov bodies provide tools from convex geometry to study algebraic varieties. The PI will develop the theory of Newton-Okounkov bodies by constructing combinatorial parameter spaces for these bodies together with a wall-crossing formula. The study of symmetric orbit closures is relevant to the theory of Kazhdan-Lusztig-Vogan polynomials and Harish-Chandra modules for a certain real Lie group, mirroring the connection of Schubert varieties to Kazhdan-Lusztig polynomials and representation theory. The PI will investigate the projections of symmetric orbit closures to Grassmannians. Knutson and Miller introduced subword complexes to study the cohomology of Schubert varieties. The PI will relate these complexes with toric degenerations of Schubert varieties and their desingularizations. 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.