Algebraic Combinatorics and Its Applications

Combinatorics is the area of mathematics dedicated to the study of discrete structures, such as diagrams, graphs, various combinations, polytopes, etc. A typical combinatorial question is ``How to count these structures?'' Combinatorial problems appear in various areas of pure and applied mathematics, physics, and other sciences. Several problems in this research project are related to the latest advances in the fundamental physics. This study will lead to a better understanding of the fundamental processes of the nature: interactions between elementary particles. A unique feature of combinatorics is that combinatorial objects are quite visual and easy to understand. Many combinatorial problems can be explained to a high-school student. The PI and RAs will disseminate research results among various audiences. The PI will encourage graduate, undergraduate, and high-school students to work on the combinatorial problems. He will support MIT PRIMES "Program for Research in Mathematics, Engineering and Science for High School Students". This project will promote public knowledge of combinatorics and mathematics in general.

This research project is devoted to the study of several problems in algebraic combinatorics and their applications. Many of the problems are related to the positive Grassmannian, a beautiful geometrical object with a rich combinatorial structure. The combinatorial structures and techniques that were developed in the study of the positive Grassmannian also surface in other areas: inverse boundary problems, matroid theory, convex geometry, toric geometry, statistical mechanics, theory of solitons, Fomin-Zelevinsky's cluster algebras, symmetric functions, affine Schubert calculus, Lusztig's canonical bases, matrix completion problems, Schur positivity problems, as well as the study of scattering amplitudes of elementary particles - one of the most fundamental field in physics. Several problems are related to power ideals, which are objects from commutative algebra linked with graphs, hyperplane arrangements, parking functions, matroids, and other combinatorial structures. The project involves problems on generalized permutohedra, which are certain convex polytopes, whose volumes and numbers of integer lattice points express many classical combinatorial sequences: the Eulerian numbers, the Catalan numbers, the numbers of trees and forests, and their generalizations. They have links with low-dimensional topology and tropical oriented matroids, and concern a interesting new analogue of the Tutte polynomial. The project also contains problems on several classes of convex polytopes: root polytopes, alcoved polytopes, and flow polytopes.