CAREER: Statistical mechanics and knot theory in algebraic combinatorics

数学上的结是这样的：取一根绳子，以某种方式缠绕，然后将两端连接起来。结理论中的一个经典问题是，是否可以通过不断变换绳子而得到两个结。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多项式和它最近的推广称为Khovanov-Rozansky同调。例如，HOMFLY多项式在分子生物学中用于研究DNA分子如何在空间中折叠。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，代数组合学是一个应用代数方法研究离散对象的领域，如二项式系数或多边形的三角剖分。一个多边形可能的三角剖分数是用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结同调和HOMFLY多项式联系起来，给出了加泰罗尼亚数的自然几何解释。沿途出现的物体是从统计力学的角度来解释的，统计力学处理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Ising模型直接相关，该模型描述了居里点处平坦金属板的铁磁特性。该奖项还为参与PI研究的本科生、研究生和博士后提供资金。格拉斯曼系是由称为类正变异的空间分层的。在与Thomas Lam的合作项目中，首席研究员（PI）研究了正电子品种上同源性的混合Hodge结构。主要结果表明，在Garsia-Haiman和Loehr-Warrington的著作中引入的（有理数）q，t-Catalan数给出了顶维正极变量的变阶poincar<s:1>多项式。证明通过将一个链与每个正电子变体相关联，并将其上同调与相关链的Khovanov-Rozansky同调相关联来进行。因此，正态变量的点计数由连杆的HOMFLY多项式的系数给出。PI最近表明，点计数是由随机六顶点模型中的某些观测值给出的。另外，在PI与Pavlo pylyavsky的联合工作中，正电子品种与Ising模型相关联。在这个项目中，PI使用这个关系给出了Baxter临界z不变Ising模型的边界相关性的直接公式。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.