Combinatorial State Sums and Interval Flag Varieties

组合状态和和区间标志变量

• 批准号: 1700372
• 负责人: Allen Knutson
• 金额: $ 27万
• 依托单位: Cornell University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-06-01 至 2020-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1700372&HistoricalAwards=false
• 关键词: Combinatorial; State; Sums; Interval; Flag

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.

该研究项目解决了代数组合和代数几何形状，与对称性描述有关的数学领域以及多元多项式方程的解决方案。 该项目涉及对“状态总和”的研究，这是统计力学的一个术语，以描述某些产品总和（通常是多项式）的研究 - 以及对物理学的拓扑量子场理论的研究，并已与现代数学的几个领域建立了联系。代数组合中的许多重要数量可以作为状态总和计算。这些通常是在带有兼容瓷砖的四边形表面的所有标记上的总和，例如组装拼图拼图。这个了不起的事实是，这些总数经常独立于四边形，这表明它们可能具有更基本的定义。该项目研究了拓扑量子场理论的来源，该理论可能会统一这些状态总和，同时提出了更深入的探测方法。具体而言，改变表面上的四边形与通过空间进行发展有关，这表明探索更高和较低的类似物。该计划有许多适用于数学和统计力学研究生的次项目，他们将参与研究。该研究计划包括两个项目。第一个项目在一项努力中使用量子场理论和几何表示理论的灵感来阐明代数组合中产生的多项式公式。 有趣的多项式的公式通常采用线性多项式产物总和的形式。例如，与置换相关的“舒伯特多项式”可以将其写入置换的某些二维图上的总和。该项目将通过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）维量子场理论中的时间进化运算符内部的矩阵系数。由于长期进化是许多短期发展的综合，因此矩阵可以表示为简单矩阵的产物，准确地提供了产品的总和。第二步是将这些量子场理论的希尔伯特空间视为某些代数品种的同源组，之后是几何表示理论的方式。这自然意味着这些矩阵的某些换向特性，例如Yang-Baxter方程。 第二个项目旨在将Schubert演算的概括为线性子空间的链。