Combinatorics in Algebra, Geometry, and Physics

Massachusetts Institute Of Technology, Cambridge MA

Investigators

Abstract

This project is dedicated to several questions in algebraic and geometric combinatorics. A key topic of the research is a geometric object, called the Grassmannian, that appears in the study of a wide variety of other mathematical topics, but also in physics, where it is related to certain quantum field theories. Despite its technical nature, some questions explored in the project are suitable for undergraduate and high school students, and the investigator will work with these students through the various programs, such as the Program for Research in Mathematics, Engineering, and Science for High School Students (PRIMES) and the Undergraduate Research Opportunities Program (UROP), available at his institution. The project addresses several questions about the positive Grassmannian, a beautiful geometrical object with a rich combinatorial structure. It is closely related to matroid theory, Schubert calculus, symmetric functions, and cluster algebras, as well as the study of scattering amplitudes of elementary particles. Several sub-projects are concerned with an extension of weakly separated collections and purity phenomenon to oriented matroids. This area of research is related to the study of combinatorics of zonotopal tilings and cluster algebras. The project explores questions about the chip-firing game for root systems. There are many fascinating conjectures and open problems in this area. Several questions concern generalized permutohedra, which are certain convex polytopes whose volumes and numbers of integer lattice points produce many classical combinatorial sequences. The research on generalized permutohedra involves a new analogue of the Tutte polynomial for hypergraphs and a remarkable duality, with links to the theory of knots and tropical geometry. The project also includes questions regarding the algebra of Chern forms and related power ideals. These are objects from commutative algebra involving combinatorics of graphs, forests, hyperplane arrangements, parking functions, and matroids. Many constructions that will be investigated in this work are related to convex polytopes and other polytope-like structures, such as permutohedra, amplituhedra, cosmological polytopes, and root polytopes. 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.