Combinatorial State Sums and Interval Flag Varieties

本研究项目涉及代数组合学和代数几何学的问题，这些数学领域与对称性的描述和多元多项式方程的求解有关。 该项目涉及“状态和”的研究-统计力学中的一个总括性术语，用来描述某些乘积的和（通常是多项式）-以及拓扑量子场论的研究，它产生于物理学，后来被证明与现代数学的几个领域有联系。代数组合学中的许多重要量可以计算为状态和;这些通常是在具有兼容瓦片的四边形曲面的所有标记上的和，就像组装拼图一样。值得注意的事实是，这些总数往往是独立的四边形，这表明他们可能有一些更基本的定义。这个项目研究了拓扑量子场论的一个来源，它可能统一这些状态和结果，同时提出了更深入地探索它们的方法。具体来说，改变一个表面上的四边形与它在空间中的演化有关，这意味着探索更高和更低维的类似物。本研究计画有许多子计画，适合数学与统计力学研究生参与研究。第一个项目利用量子场论和几何表示论的灵感，奋进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，与置换相关联的“舒伯特多项式”可以被写为置换的某些二维图上的和。该项目将分两步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。由于长时间演化是许多短时间演化的合成，矩阵可以表示为简单矩阵的乘积，精确地给出乘积之和。第二步是按照几何表示论的方式，把这些量子场论的希尔伯特空间看作某些代数簇的同调群。这自然意味着这些矩阵的一些交换性质，例如杨-巴克斯特方程。 第二个项目的目的是发展一个推广的舒伯特演算链的线性子空间。

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.