Polytopes in Combinatorics and Algebra

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Polytopes are geometric objects with flat sides: polygons in two dimensions, polyhedra in three dimensions, and higher-dimensional generalizations. Polytopes are ubiquitous in mathematics and play important roles throughout science and engineering. The simplest n-dimensional polytopes have only n+1 vertices and are called simplices: triangles, tetrahedra, and their higher-dimensional generalizations. While it is straightforward to determine the volume of a simplex in any dimension, the volume of a general polyhedron in high dimension can be challenging to compute. One way to calculate volume is to dissect a polytope into simplices. This research project studies dissections, volumes, and integer points of special families of polytopes known as flow and root polytopes. Flow and root polytopes naturally appear in problems in several areas of mathematics, such as representation theory and algebraic geometry. Results of this project will have impact in these areas of mathematics as well as in scientific and engineering applications.A number of important conjectures and questions in algebraic combinatorics have flow polytopes and root polytopes at their core. These special polytopes can be systematically dissected into simplices. The project will investigate a conjecture of Haglund about the bigraded Hilbert series of the space of diagonal harmonics that can be stated in terms of a sum of a certain weight function over the integer points of a flow polytope. The research will also study the subword complexes introduced by Knutson and Miller, conceived to illustrate the combinatorics of Schubert polynomials and determinantal ideals, by relating them to root polytopes.