This CAREER proposal includes problems related to algebraic and enumerative combinatorics and its applications to geometry of polytopes, representation theory, inverse boundary problems, and algebraic geometry. The proposal discusses permutohedra and their generalizations, which are certain convex polytopes related to Coxeter arrangements. The PI suggests three different approaches to calculation of volumes and Ehrhart polynomials of such polytopes. He introduces new mixed Eulerian numbers with remarkable combinatorial properties. Various generalizations of permutohedra include Stasheff's associahedra, Pitman-Stanley polytopes, polytopes related to wonderful compactifications of De Concini-Procesi, etc. There are intriguing parallels between these polytopes and generalized associahedra related to Fomin-Zelevinsky's theory of cluster algebras. The next part of the proposal is related to Schur positivity. The PI with collaborators have recently resolved several open Schur positivity problems that attracted a lot of attention, including Fomin-Fulton-Li-Poon's conjecture and Okounkov's conjecture. They plan to apply their techniques to prove several other prominent conjectures. The proposal mentions a general Schur positivity conjecture involving a mysterious polytope, which might have relations with the Klyachko cone and lead to new interesting combinatorics. The proposal discusses the inverse boundary problem for networks and its relations with total positivity. Networks parametrize the totally positive cells on the Grassmannian. This construction has links with Fomin-Zelevinsky's results on double Bruhat cells and with their theory of cluster algebras.
This CAREER proposal describes new initiatives in research and education in the area of combinatorics. Combinatorial techniques play an increasingly important role in other fields such as algebra, geometry, computer science, probability theory, physics, biology, cryptography, etc. Both the research and the educational components of the proposal aim on applications of combinatorics. The proposal discusses several important combinatorial problems on the frontiers of modern mathematical research. These problems involve counting or enumerating various discrete mathematical objects, say, vertices of a polytope or pieces in a decomposition of a complicated geometrical object into simpler objects. These problems would help to understand, clarify, and simplify nontrivial mathematical concepts and constructions. The proposed research project would have impact in several fields, including algebraic geometry (which studies geometrical objects using algebra), representation theory (which studies symmetries), theory of convex polytopes, and theoretical physics. The educational part of the proposal includes plans for a new course on modern applications of combinatorics. The PI is planning to encourage graduate and undergraduate research. Combinatorial problems are appealing for undergraduate students because these problems are intuitive and easy to formulate but yet quite challenging. The proposal describes plans for using interactive visual tools and demos, applets, and animations in teaching future combinatorics courses.
