Combinatorial State Sums and Interval Flag Varieties

该研究项目涉及代数组合学和代数几何中的问题，这些数学领域涉及对称性的描述和多元多项式方程的解。该项目包括对“状态和”（state sum）的研究——统计力学中描述某些乘积和（典型的多项式）的统称——以及拓扑量子场论（topological quantum field theory）的研究，后者起源于物理学，并被证明与现代数学的几个领域有联系。代数组合中的许多重要量可以用状态和来计算；这些通常是四边形表面上所有标签的总和，就像组装拼图一样。值得注意的事实是，这些总数经常独立于四边形，这表明它们可能有一些更基本的定义。该项目研究了拓扑量子场论中可能统一这些状态和结果的来源，同时提出了更深入探索它们的方法。具体来说，改变表面上的四边形与在空间中进化有关，这意味着探索高维和低维的类似物。本项目有许多子课题，适合数学和统计力学研究生参与研究。这个研究计划包括两个项目。第一个项目利用量子场论和几何表示理论的灵感，努力阐明代数组合学中出现的多项式公式。有趣的多项式的公式通常采用线性多项式的乘积和的形式。例如，与排列相关的“舒伯特多项式”可以写成该排列的某些二维图的和。本项目将分两个步骤更深入地研究这些多项式公式。第一步是将多项式视为（1+1）维量子场论中时间演化算子内的矩阵系数。由于长时间进化是许多短时间进化的组合，因此矩阵可以表示为更简单矩阵的乘积，从而精确地给出乘积的和。第二步是按照几何表示理论的方式，将这些量子场论的希尔伯特空间视为某些代数变体的同调群。这自然意味着这些矩阵的一些交换性质，比如Yang-Baxter方程。第二个项目旨在将舒伯特演算推广到线性子空间链。

This research project addresses questions in algebraic combinatorics and algebraic geometry, areas of mathematics concerned with the description of symmetry and with the solution of multivariate polynomial equations. The project involves the study of "state sums" -- a catchall term from statistical mechanics to describe certain sums of products (typically of polynomials) -- and the study of topological quantum field theory, which arose in physics and has turned out to have connections to several areas of modern mathematics. Many important quantities in algebraic combinatorics can be computed as state sums; these are often sums over all the labelings of a quadrangulated surface with compatible tiles, like assembling a jigsaw puzzle. The remarkable fact is that these totals are frequently independent of the quadrangulation, suggesting that they may have some more fundamental definition. This project investigates a source from topological quantum field theory that might unify these state sum results, while suggesting ways to probe them more deeply. Specifically, changing a quadrangulation on a surface relates to evolving it through space, which suggests exploring both higher- and lower-dimensional analogues. This program has many subprojects suitable for graduate students in mathematics and statistical mechanics, who will be involved in the research.This research program comprises two projects. The first project uses inspiration from quantum field theory and geometric representation theory in an endeavor to elucidate polynomial formulae arising in algebraic combinatorics. Formulae for interesting polynomials often take the form of sums of products of linear polynomials. For example, the "Schubert polynomial" associated to a permutation can be written as a sum over certain 2-d diagrams for the permutation. The project will look more deeply into these polynomial formulae, in two steps. The first step is to see a polynomial as a matrix coefficient inside a time-evolution operator in a (1+1)-dimensional quantum field theory. Since long-time-evolution is a composite of many short-time evolutions, the matrix might be expressed as a product of simpler matrices, giving exactly the sum over products. The second step is to regard the Hilbert spaces of these quantum field theories as homology groups of certain algebraic varieties, after the manner of geometric representation theory. This would naturally imply some commutation properties of these matrices, such as the Yang-Baxter equation. The second project aims to develop a generalization of Schubert calculus to chains of linear subspaces.