CAREER: Algebraic Combinatorics and its Applications

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This CAREER proposal includes problems related to algebraic and enumerativecombinatorics and its applications to geometry of polytopes, representationtheory, inverse boundary problems, and algebraic geometry. The proposaldiscusses permutohedra and their generalizations, which are certain convexpolytopes related to Coxeter arrangements. The PI suggests three differentapproaches to calculation of volumes and Ehrhart polynomials of such polytopes.He introduces new mixed Eulerian numbers with remarkable combinatorialproperties. Various generalizations of permutohedra include Stasheff'sassociahedra, Pitman-Stanley polytopes, polytopes related to wonderfulcompactifications of De Concini-Procesi, etc. There are intriguing parallelsbetween these polytopes and generalized associahedra related toFomin-Zelevinsky's theory of cluster algebras. The next part of the proposalis related to Schur positivity. The PI with collaborators have recentlyresolved several open Schur positivity problems that attracted a lot ofattention, including Fomin-Fulton-Li-Poon's conjecture and Okounkov'sconjecture. They plan to apply their techniques to prove several otherprominent conjectures. The proposal mentions a general Schur positivityconjecture involving a mysterious polytope, which might have relations with theKlyachko cone and lead to new interesting combinatorics. The proposaldiscusses the inverse boundary problem for networks and its relations withtotal positivity. Networks parametrize the totally positive cells on theGrassmannian. This construction has links with Fomin-Zelevinsky's results ondouble Bruhat cells and with their theory of cluster algebras.This CAREER proposal describes new initiatives in research and education in thearea of combinatorics. Combinatorial techniques play an increasingly importantrole in other fields such as algebra, geometry, computer science, probabilitytheory, physics, biology, cryptography, etc. Both the research and theeducational components of the proposal aim on applications of combinatorics.The proposal discusses several important combinatorial problems on thefrontiers of modern mathematical research. These problems involve counting orenumerating various discrete mathematical objects, say, vertices of a polytopeor pieces in a decomposition of a complicated geometrical object into simplerobjects. These problems would help to understand, clarify, and simplifynontrivial mathematical concepts and constructions. The proposed researchproject would have impact in several fields, including algebraic geometry(which studies geometrical objects using algebra), representation theory (whichstudies symmetries), theory of convex polytopes, and theoretical physics. Theeducational part of the proposal includes plans for a new course on modernapplications of combinatorics. The PI is planning to encourage graduate andundergraduate research. Combinatorial problems are appealing for undergraduatestudents because these problems are intuitive and easy to formulate but yetquite challenging. The proposal describes plans for using interactive visualtools and demos, applets, and animations in teaching future combinatoricscourses.