Polytopes in Combinatorics and Algebra

多面体是具有平边的几何对象：二维多边形、三维多面体和高维泛化。多面体在数学中无处不在，在科学和工程中发挥着重要作用。最简单的n维多面体只有n+1个顶点，被称为简单体：三角形、四面体和它们的高维推广。虽然在任何维度上确定单纯形的体积是很简单的，但在高维上计算一般多面体的体积可能是一项挑战。计算体积的一种方法是将多面体分解成简单体。这个研究项目研究了被称为流多面体和根多面体的特殊多面体家族的解剖、体积和整数点。流动多面体和根多面体自然地出现在许多数学领域的问题中，例如表示理论和代数几何。该项目的研究结果将对这些数学领域以及科学和工程应用产生影响。在代数组合学中，许多重要的猜想和问题都以流动多面体和根多面体为核心。这些特殊的多面体可以系统地分解成简单体。该项目将研究Haglund关于对角谐波空间的梯度Hilbert级数的一个猜想，对角谐波空间可以用流多面体的整数点上的某个权重函数的和来表示。该研究还将研究由Knutson和Miller引入的子词复合体，这些子词复合体旨在通过将Schubert多项式和行列式理想与根多面体联系起来，来说明它们的组合学。

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.