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.

通过取绳，以某种方式缠结，然后连接末端来获得数学结。结理论中的一个经典问题询问是否可以通过不断转换绳索来相互获得两个结。区分两个结的一种方法是计算他们的结不变。一些最强大的结不变剂包括Homfly多项式及其最近的概括，称为Khovanov -Rozansky同源性。例如，Homfly多项式用于分子生物学，以研究如何在空间中折叠DNA分子。在该项目中，我们将这些结与代数组合物自然产生的对象联系起来，该领域采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形。多边形的可能的三角数量由著名的加泰罗尼亚数字序列计数。该项目的主要结果之一通过将它们与Khovanov -Rozansky结的自然几何解释对加泰罗尼亚数字进行了解释。从沿途出现的对象是从统计力学的角度来解释的，该物体涉及由大量粒子组成的物理系统的宏观观察。例如，所讨论的几何空间直接与临界温度下的ising模型相关联，该模型描述了库里点处平坦金属板的铁磁特性。该奖项还为本科生，研究生和博士后参与PI研究提供了资金。格拉曼尼亚人被称为正品种的空间分层。在与托马斯·林（Thomas Lam）的联合项目中，首席研究员（PI）研究了混合霍奇结构，这些结构是关于阳性品种的共同体。主要的结果指出，高级果实的poincaré多项式由（理性的）Q，T-Catalan编号给出，该数字是在Garsia-Haiman和Loehr-Warrington的作品中引入的。证明是通过将链接与每个积极品种联系起来的，并将其同谋与相关链接的Khovanov -Rozansky同源性联系起来。因此，贴子素品种的点计数是由链接的Homfly多项式批评给出的。 PI最近表明，在随机六个vertex模型中某些可观察结果给出了点计数。另外，在PI与Pavlo Pylavskyy的PI联合工作中，阳性品种连接到Ising模型。在这个项目中，PI使用此关系为Baxter关键Z不变ISING模型的边界相关性提供了直接公式。该公式适用于该模型的普遍性和保形不变性的问题，Smirnov等人的Studiod。该奖项反映了NSF的法定任务，并使用基金会的知识分子和更广泛的影响评估标准，通过评估来诚实地通过评估来诚实地支持。