Questions in Algebraic and Geometric Combinatorics

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Combinatorics arises naturally in many other fields of mathematics. Polytopes, one of the central subjects of geometric combinatorics, have numerous applications not only in branches in pure math such as algebra, algebraic geometry and number theory, but also in other fields like statistics, economics, and optimization. The simplest way to understand polytopes is that they are high-dimensional generalizations of polygons, and they can be constructed by taking the intersection of half spaces. Familiar three-dimensional polytopes include tetrahedra, cubes, octahedra, and dodecahedra. There are many aspects of polytopes one can study. Counting integer points is a fundamental enumerative problem, which has real-life applications in counting the number of integer solutions of a set of linear constraints in multiple variables. This is related to one of the two major research directions in this project. The other major direction is on construction of polytopes satisfying certain conditions. In general, many of the problems addressed in this project have a combinatorial nature, which makes them sufficiently accessible that they may be integrated into course material and student research projects. In particular, research described in two parts of the project involves simple combinatorial objects, and the PI plans to build one topic into an undergraduate research project.In the 1960s, Ehrhart discovered that the number of lattice points in dilations of polytopes is counted by a polynomial, called Ehrhart polynomial. The first part of the project is focused on the study of Ehrhart positivity, valuations of polytopes, and related questions. Topics include (1) studying Ehrhart positivity problems on Tesler polytopes and Birkhoff polytopes; (2) investigating the uniqueness of Berline-Vergne's valuation; (3) studying Fischer-Pommersheim's alpha-construction for McMullen's formula; (4) exploring the connection between Ehrhart positivity and properties of h^*-polynomials. In the second part, the PI will focus on problems related to constructions of polytopes. Instead of defining a polytope directly, one can start with a fan (or a poset), and ask whether there exists a polytope whose normal fan (or whose face lattice) is the given one. The case of posets is called the realization problem, which builds a nice bridge between the study of combinatorial objects (posets) and geometric objects (polytopes). Based on her recent work on nested generalized permutohedra and their connection with permuto-asscociahedra, the PI will study the realization problem on "hybrid-posets".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.