CAREER: Statistical mechanics and knot theory in algebraic combinatorics

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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.