Combinatorics and its Applications

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Combinatorics is the area of mathematics that studies discrete structures. This area has close links with many other fields of pure and applied mathematics and with other sciences such as physics and biology. This project is dedicated to several problems in combinatorics and their applications to algebra, geometry, and theoretical physics. The project will lead to a better understanding of fundamental mathematical concepts and constructions and will have an impact in many other areas of research. The results will be disseminated among specialists in the field as well as to broader audiences. Special cases of some problems will be used for undergraduate and high school research projects. Graduate students will be involved in the research. This project will promote the general public knowledge and appreciation of mathematics.Many problems in this project revolve around special classes of polytopes and other polytope-like structures, such as permutohedra, root polytopes, positroids, and the positive Grassmannian. Several problems are concerned with generalized permutohedra. These problems involve an extension of the Tutte polynomial to hypergraphs and polymatroids, as well as mirror duality for generalized permutohedra. These problems have links with low-dimensional topology, specifically knot invariants, with tropical geometry, and with toric geometry. Several problems are related to the positive Grassmannian, which is a beautiful geometrical object with rich combinatorial structure. The combinatorial constructions and techniques developed in the study of the positive Grassmannian have surfaced in many other areas: inverse boundary problems, matroid theory, convex geometry, toric geometry, statistical mechanics, the theory of solitons, Fomin-Zelevinsky's cluster algebras, symmetric functions, affine Schubert calculus, Lusztig's canonical bases, matrix completion problems, and Schur positivity problems, as well as the study of scattering amplitudes of elementary particles. This project will study new links between the positive Grassmannian and geometry of polyhedral subdivisions. The project also includes problems on the purity phenomenon for oriented matroids, chip-firing games for root systems, algebras of Chern forms, and power ideals.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.