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CAREER: Statistical mechanics and knot theory in algebraic combinatorics

职业：代数组合中的统计力学和纽结理论

基本信息

批准号：
2046915
负责人：
Pavel Galashin
金额：
$ 39.93万
依托单位：
University of California-Los Angeles
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-07-01 至 2026-06-30
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2046915&HistoricalAwards=false
关键词：
CAREER Statistical mechanics knot theory

项目摘要

A mathematical knot is obtained by taking a piece of rope, tangling it in some way, and then joining the ends. A classical question in knot theory asks whether two knots can be obtained from each other by continuously transforming the rope. One way to distinguish two knots is to compute their knot invariants. Some of the most powerful knot invariants include the HOMFLY polynomial and its recent generalization known as Khovanov–Rozansky homology. For instance, the HOMFLY polynomial is used in molecular biology to study how DNA molecules are folded in space. In this project, we relate these knot invariants to objects arising naturally in algebraic combinatorics, a field which applies algebraic methods to study discrete objects such as binomial coefficients or triangulations of a polygon. The number of possible triangulations of a polygon is counted by the famous Catalan number sequence. One of the main results of the project gives a natural geometric interpretation of Catalan numbers, by means of relating them to Khovanov–Rozansky knot homology and the HOMFLY polynomial. The objects that appear along the way are interpreted from the point of view of statistical mechanics, which deals with macroscopic observations of a physical system consisting of a large number of particles. For example, the geometric spaces in question are directly linked to the Ising model at critical temperature, which describes ferromagnetic properties of a flat metal plate at the Curie point. The award also provides funding for the involvement of undergraduate students, graduate students and postdocs in the PI's research.The Grassmannian is stratified by spaces known as positroid varieties. In a joint project with Thomas Lam, the Principal Investigator (PI) studies the mixed Hodge structure on the cohomology of positroid varieties. The main result states that the bigraded Poincaré polynomial of the top-dimensional positroid variety is given by the (rational) q,t-Catalan number, introduced in the works of Garsia–Haiman and Loehr–Warrington. The proof proceeds by associating a link to each positroid variety, and relating its cohomology to the Khovanov–Rozansky homology of the associated link. The point count of the positroid variety is therefore given by a coefficient of the HOMFLY polynomial of the link. The PI has recently shown that the point count is given by certain observables in the stochastic six-vertex model. Separately, positroid varieties were connected to the Ising model in the joint work of the PI with Pavlo Pylyavskyy. In this project, the PI uses this relation to give a direct formula for boundary correlations of Baxter's critical Z-invariant Ising model. This formula is applied to questions of universality and conformal invariance of the model, studied by Smirnov et al.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

数学结是通过取一根绳子，以某种方式将其缠结，然后将两端连接起来而获得的。结理论中的一个经典问题是，是否可以通过不断变换绳子来获得两个结。区分两个结的一种方法是计算它们的结不变量。一些最强大的结不变量包括 HOMFLY 多项式及其最近的推广，称为 Khovanov-Rozansky 同调。例如，HOMFLY 多项式在分子生物学中用于研究 DNA 分子在空间中如何折叠。在这个项目中，我们将这些结不变量与代数组合学中自然产生的对象联系起来，代数组合学是应用代数方法来研究离散对象（例如二项式系数或多边形的三角剖分）的领域。多边形可能的三角剖分的数量是通过著名的加泰罗尼亚数列来计算的。该项目的主要成果之一是通过将加泰罗尼亚数与 Khovanov-Rozansky 结同调性和 HOMFLY 多项式联系起来，对加泰罗尼亚数进行自然的几何解释。沿途出现的物体是从统计力学的角度来解释的，统计力学涉及对由大量粒子组成的物理系统的宏观观察。例如，所讨论的几何空间与临界温度下的伊辛模型直接相关，该模型描述了金属平板在居里点的铁磁特性。该奖项还为本科生、研究生和博士后参与 PI 的研究提供资金。格拉斯曼系按称为正类簇的空间进行分层。在与 Thomas Lam 的联合项目中，首席研究员 (PI) 研究了正类簇上同调的混合 Hodge 结构。主要结果表明，顶维正类簇的二阶庞加莱多项式由（有理）q,t-Catalan 数给出，该数在 Garsia–Haiman 和 Loehr–Warrington 的著作中引入。通过将一个链接与每个正类变体相关联，并将其上同调与相关链接的 Khovanov-Rozansky 同调联系起来，进行证明。因此，正类变体的点数由链接的 HOMFLY 多项式的系数给出。 PI 最近表明，点数是由随机六顶点模型中的某些可观测值给出的。另外，在 PI 与 Pavlo Pylyavskyy 的联合工作中，正类变体被连接到伊辛模型。在这个项目中，PI 使用这种关系给出了 Baxter 临界 Z 不变伊辛模型边界相关性的直接公式。该公式应用于 Smirnov 等人研究的模型的普遍性和保角不变性问题。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。