Polytopes in Combinatorics and Algebra

多面体是具有平面的几何对象：二维的多边形、三维的多面体和更高维的泛化。多面体在数学中普遍存在，在整个科学和工程中扮演着重要的角色。最简单的n维多面体只有n+1个顶点，称为简单：三角形、四面体及其高维推广。虽然在任何维度上确定单纯形的体积是很简单的，但计算高维的一般多面体的体积可能是一件具有挑战性的事情。计算体积的一种方法是将一个多面体分解成几个简单的。这项研究项目研究了称为流多面体和根多面体的特殊多面体家族的解剖、体积和整点。流多面体和根多面体自然会出现在数学的几个领域中，例如表示论和代数几何。这个项目的结果将对这些数学领域以及科学和工程应用产生影响。代数组合数学中的一些重要猜想和问题都是以流多面体和根多面体为核心的。这些特殊的多面体可以被系统地解剖成简单的形状。该项目将调查Haglund关于对角调和空间的重希尔伯特级数的猜想，该猜想可以用流多面体的整点上的某个权函数的和来表示。本研究还将研究Knutson和Miller引入的子词复合体，这些子词复合体旨在通过将舒伯特多项式和行列式理想与根多面体联系起来来说明它们的组合。

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.