Embedded Files
Title: Symmetric function generalizations of the chromatic polynomial
Abstract: I will discuss two different symmetric function generalizations of the chromatic polynomial. The famous classical one is Richard Stanley’s chromatic symmetric function, which has led to a great deal of activity in combinatorics and algebraic geometry. In particular, I will recall a refinement, introduced in my work with John Shareshian, of the Stanley–Stembridge e-positivity conjecture and its connection to Hessenberg varieties. The other generalization was introduced more recently in my work with Rafael González D’Léon on weighted bond lattices. An e-positivity result and conjecture for these non-Stanley chromatic symmetric functions will also be discussed.
Title: Solvable Lattice Models and Quantum Groups
Abstract: Solvable lattice models arose in statistical
mechanics, and they were a key motivating example
in the invention of quantum groups in the 1980's.
Beyond their origin in statistical mechanics,
they have applications in different areas. We
will look at examples where the partition functions
are polynomials of interest in algebraic combinatorics.
Then we will look at examples of parametrized
Yang-Baxter equations and their relationship to
quantum group theory, where we will encounter
some interesting phenomena, even in the simplest
case of the six-vertex model.
Title: Dimer face polynomials in knot theory and cluster algebras
Abstract: The set of dimers (aka perfect matchings) of a connected bipartite plane graph G has the structure of a distributive lattice, as shown by Propp. The order relation on this lattice is induced by the height of a dimer. In this talk, I'll focus on the dimer face polynomial of G, which is the height generating function of all dimers of G. This polynomial has close connections to knot invariants on the one hand, and cluster algebras on the other. I'll discuss joint work with Mészáros, Musiker and Vidinas in which we explore these connections. No knowledge about knots or cluster algebras will be assumed.
Title: The quasisymmetric flag variety and equivariant forest polynomials
Abstract: Schubert polynomials are a family of combinatorial polynomials that make it possible to study of the (equivariant) cohomology ring of the flag variety GLn/B using the combinatorics of permutations. Symmetric polynomials, a favorite family in algebraic combinatorics, play a central role in this framework. In this talk, I will recount the story of Schubert and symmetric polynomials to highlight parallels with a new framework developed in collaboration with Bergeron, Nadeau, Spink, and Tewari. Our work closely mirrors this classical narrative, with some key replacements: permutation combinatorics gives way to Catalan combinatorics in various forms, and symmetric polynomials are replaced by quasisymmetric polynomials. I will present key background concepts at an accessible, conceptual level, assuming no prior familiarity with flag varieties, Schubert calculus, or quasisymmetric polynomials.
Page updated
Report abuse