Tom Braden

Title: Complexes for low-degree intersection cohomology and rigidity of polytopes and matroids
Abstract: Several important families of polynomials arise as intersection cohomology Poincare' polynomials of combinatorially defined vareties: examples include the toric g-polynomial of convex polytopes and the "matroid KL polynomials" defined by Elias, Proudfoot and Wakefield. I will focus on the linear and quadratic coefficients of these polynomials, where the intersection cohomology can be computed by small chain complexes which are exact in all but one place.  For the toric g-polynomial, these complexes are related to affine independence of the vertices and to the rigidity of the vertices, edges and two-faces.  I will describe similar complexes for linear and quadratic coefficients of KL polynomials of matroids, which could be thought of as suggesting a possible "rigidity theory" for matroids.  I will explain how these complexes arise, and discuss what is known about their exactness properties. These results are part of joint work with June Huh, Jacob Matherne, Nicholas Proudfoot and Botong Wang.

Hai Long Dao

Title: Regularity, singularities and $h$-vector of graded algebras

Abstract: Let $R$ be a standard graded algebra over a field. We investigate how the singularities of $Spec R$ or $Proj R$ affect the $h$-vector of $R$, which is the coefficients of the numerator of its  Hilbert series. The most concrete consequence of our work asserts that  if $R$ satisfies Serre's condition $(S_r)$ and have reasonable singularities (Du Bois on the punctured spectrum or $F$-pure), then $h_0,dots, h_rgeq 0$. Furthermore the multiplicity of $R$ is at least $h_0+h_1+dots +h_{r-1}$.  We also prove that equality in many cases forces $R$ to be Cohen-Macaulay. This is joint work with Linquan Ma and Matteo Varbaro. 

Jang Soo Kim

Title: Enumeration of bounded lecture hall tableaux

Abstract: Recently Corteel and Kim introduced lecture hall tableaux in their study of multivariate little q-Jacobi polynomials. In this talk, we enumerate bounded lecture hall tableaux. We show that their enumeration is closely related to standard and semistandard Young tableaux. We also show that the number of bounded lecture hall tableaux is the coefficient of the Schur expansion of s_lambda(m+y_1,...,m+y_n). To prove this result, we use two main tools: non-intersecting lattice paths and bijections. In particular we use ideas developed by Krattenthaler to prove bijectively the hook content formula. This is joint work with Sylvie Corteel.

Maitreyee Kulkarni

Title: Infinite friezes and triangulations of an annulus.
Abstract: In this talk I will introduce a combinatorial object called a frieze and describe its relations to triangulations and to representations of certain quivers. We show that each periodic infinite frieze determines a triangulation of an annulus in a unique way. We also study associated module categories and determine an invariant of friezes in terms of modules. This is joint work with Karin Baur, Ilke Canakci, Karin Jacobsen, and Gordana Todorov.

Jae-Hoon Kwon

Title: RSK correspondence, quantum affine algebras and crystals

Abstract: In this talk, we give a new representation theoretic interpretation of the celebrated RSK correspondence in algebraic combinatorics. We show that RSK map is an isomorphism between two realizations of Kashiwara's crystals associated to finite-dimensional representations of quantum affine algebras. This talk is based on joint works with Il-Seung Jang, and Masato Okado.
 

Seung Jin Lee

Title: Back-stable Schubert calculus

Abstract: Back-stable Schubert calculus concerns various (co)homology of infinite flag variety and infinite Grassmannian. In this talk, after reviewing basics of Schubert calculus and definition of infinite flag variety, we define related functions stemmed from back-stable Schubert calculus such as back-stable limit of double Schubert polynomials and double Stanley symmetric functions. Then we discuss the monomial positivity of back-stable double Schubert polynomials and positivity of equivariant Edelman-Greene coefficients. There is no known manifestly positive formula for the second positivity, hence if time permits we discuss special cases of this positivity. This is a joint work with Thomas Lam and Mark Shimozono.

Jacob Matherne (IAS)

Title: Logarithmic concavity of Schur and related polynomials
Abstract: Schur polynomials are the characters of finite-dimensional irreducible representations of the general linear group. We will discuss both a continuous and discrete kind of log-concavity for Schur polynomials. We will also mention relations to representation theory and results/conjectures about related polynomials. This is joint work with June Huh, Karola Mészáros, and Avery St. Dizier.

Nick Proudfoot

Title: The contraction category of graphs
Abstract: We study modules over the contraction category of graphs, which include homology groups of configuration spaces and intersection homology groups of reciprocal planes. By proving results about the generators of these modules, we obtain information about the asymptotic growth of Betti numbers of configuration spaces and Kazhdan-Lusztig coefficients. We also describe how to use modules over this category to compute the cohomology of the outer automorphism group of a free group. This is joint work with Eric Ramos.

Botong Wang

Title: Lyubeznik numbers of irreducible projective varieties

Abstract: Lyubeznik numbers are invariants of singularities that are defined algebraically, but has topological interpretations. In positive characteristics, it is a theorem of Wenliang Zhang that the Lyubeznik numbers of the cone of a projective variety do not depend on the choice of the projective embedding. Recently, Thomas Reichelt, Morihiko Saito and Uli Walther related the problem with the failure of Hard Lefschetz theorem for singular varieties. And they constructed examples of reducible complex projective varieties whose Lyubeznik numbers depend on the choice of projective embeddings. I will discuss their works and a generalization to irreducible projective varieties.