Program

Lecture Series

Thomas Lam (University of Michigan)

Title: From positroid varieties to Catalan combinatorics

Abstract: Positroid varieties are subvarieties of the Grassmannian, defined as the intersection of cyclically rotated Schubert varieties.  These varieties enjoy a range of beautiful combinatorial and geometric properties and have appeared in diverse settings: Schubert calculus, cluster algebras, Poisson geometry, total positivity, mirror symmetry, scattering amplitudes, and Catalan combinatorics.

In this lecture series, we will start by discussing the fundamental combinatorics and geometry of positroid varieties developed by Postnikov and in my joint work with Knutson and Speyer.  We will then focus on the topology of open positroid varieties where connections with cluster algebras, link homology, and Catalan combinatorics appear.  We develop a parallel between the classical relation between Grassmannians and binomial coefficients, and a novel relation between positroids and Catalan numbers in my work with Galashin. 

Anton Mellit (University of Vienna)

Title: Introduction to the A_{q,t} algebra techniques

Abstract: Erik Carlsson and myself invented the so-called A_{q,t} algebra in order to prove the shuffle conjecture. On one side, the algebra helps to compute certain combinatorial sums over parking functions, and on the other side the operators from the theory of Macdonald polynomials. Subsequent proofs of the rational shuffle conjecture and the delta conjecture has been also based on this algebra. The definition of the algebra by generators and relations is a curious mix between the double affine Hecke algebra and the elliptic Hall algebra. I will give the algebraic definition and explain relationship to the LLT polynomials and the operators. Then, I will explain how the algebra admits a very natural geometric interpretation in terms of the parabolic Hilbert schemes. In particular, I will show how the recent proof of the Stanley-Stembridge conjecture due to Hikita can be deduced from this interpretation. These results are based on joint works with Carlsson, d'Adderio, Gorsky, Griffin, Romero, Weigl and Wen.

Mark Shimozono (Virginia Tech)

Title: Graded multiplicities of classical type (slides for lecture 1, 2, and slides for lecture 3, 4)

Abstract: We will discuss various generalizations of Kostka polynomials coming from graded representations of Lie groups of classical type. Among these are the Schur coefficients of Lapointe, Lascoux, Morse k-Schur functions and new combinatorial formulas for type C Lusztig q-analogues of weight multiplicity (found recently by Hyeonjae Choi, Donghyun Kim, and Seung Jin Lee) and common generalizations.

Talks

Syu Kato (Kyoto University)

Title: Geometry of Dyck Paths (slides)

Abstract: There are two major research trends in the theory of symmetric functions arising from Dyck paths. One is the theory of Catalan symmetric functions and its geometric realization proposed by Chen-Haiman, following the works of Broer and Shimozono-Weyman. The symmetric function part of this story was established by a series of works of Blasiak-Morse-Pun-Summers. Another is the study of chromatic symmetric functions of graphs, that utilize the fact that Dyck paths roughly correspond to unit interval graphs. We first exhibit a family of smooth projective algebraic varieties that realize Catalan symmetric functions, and explain how it implies the geometric predictions of Broer, Shimozono-Weyman, and Chen-Haiman. We then exhibit that our varieties also represent the chromatic (quasi-)symmetric functions of unit interval graphs.

Donghyun Kim (Seoul National University)

Title: Lusztig q weight multiplicities and KR crystals (slides)

Abstract: Lusztig q weight multiplicity is a polynomial in q whose positivity has been verified by linking it to a specific affine Kazhdan-Lusztig polynomial. However, a combinatorial formula beyond type A has not been known until recently.

In 2019, Lee proposed a combinatorial formula for type C using a novel combinatorial concept known as semistandard oscillating tableaux. We will outline the proof of Lee's conjecture and discuss how it can be extended to type B spin weights case.

Based on joint work with Hyeonjae Choi and Seung Jin Lee.

Melissa Sherman-Bennett (University of California, Davis) 

Title: Combinatorics of amplituhedron tilings (slides)

Abstract: Postnikov and Lusztig independently defined the totally nonnegative (TNN) Grassmannian, a semi-algebraic subset of the Grassmannian. The TNN Grassmannian has a natural decomposition into positroid cells, and Galashin-Karp-Lam showed it is a regular CW complex, the "next best thing" to a polytope. In this talk, I will discuss the amplituhedron, a generalization of the TNN Grassmannian introduced by physicists Arkani-Hamed and Trnka. The amplituhedron is the image of the TNN Grassmannian under a linear map; examples include cyclic polytopes, the TNN Grassmannian itself, and bounded complexes of certain hyperplane arrangements. Of primary interest to physicists are tilings of amplituhedra: decompositions inherited from the positroid decomposition of the TNN Grassmannian. I will discuss tilings of m=2 amplituhedra, which display some linear "polytope-like" behavior, and of m=4 amplituhedra, where linearity begins to fail. This is joint work with C. Even-Zohar, T. Lakrec, M. Parisi, R. Tessler and L. Williams.