Speaker:  Ben Reese, University of Kentucky

Title:  Zero-One Schubert Polynomials

Schubert polynomials were first defined by Bernstein-Demazure in 1973-1974, and they arise in several areas of combinatorics and representation theory. We present three different descriptions of these polynomials due to Lascoux-Schutzenberger, Billey-Jockusch-Stanley, and Magyar. We then discuss the Fink-Meszaros-St. Dizier pattern avoidance characterization of Schubert polynomials having zero-one coefficients.  This is motivated by those Schubert polyomials that equal the integer point transform of the generalized permutahedron.

This is a Master exam talk.

Title: Lee-Lee Conjecture on geometric description of c-vectors. 

Abstract: A quiver is a directed graph without oriented 2 cycles and loops.  It has been shown that for an acyclic quiver, the set of non-initial d-vectors (related to the cluster algebra associated to the quiver), the set of positive c-vectors (from a framed quiver), and the set of real Schur roots (from the root system associated to the quiver) coincide. None of these objects are geometric though. In an effort to give a geometric description, Kyu-Hwan Lee and Kyungyong Lee conjectured that the set of roots obtained from a non-self-crossing admissible curve coincide with the set of c-vectors for an acyclic quiver. In this talk, I will describe these objects and talk about this conjecture for the acyclic quivers of finite case.

Title: BOGO Sale

Abstract: This talk will consist of two 20 minute talks which I will be giving at the Joint Math Meetings. Feedback will be solicited.

2:00 - 2:20 Title: k-matching sequences of simplicial complexes.

Abstract: The homotopy type of the matching complex, M_1(G), has been studied for paths, cycles, and trees. In this talk we will generalize 1-matching complex to k-matching complexes, denoted M_k(G) and consider the sequence (M_1(G), M_2(G),…,M_n(G)) up to homotopy for perfect caterpillar graphs.

2:25 - 2:45 Title: A positivity phenomenon in Elser’s Gaussian-cluster percolation model

Abstract: Veit Elser proposed a random graph percolation model in which physical dimension appears as a parameter. From this model, numerical graph invariants els_k(G) , called Elser numbers, naturally arise and Viet Elser conjectured that els_k(G) \geq 0 for all graph $G$ and integers k \geq 2. In this talk we will interpret the Elser numbers as Euler characteristics of nucleus (simplicial) complexes and prove Elser’s conjecture. This is joint work with Galen Dorpalen-Barry, Cyrus Hettle, David Livingston, Jeremy Martin, George Nasr, and Hays Whitlatch.

Title: An introduction to quiver mutations and green sequences.

Abstract: Quiver mutation is an operation one can define on a directed graph that has shown to model the behavior of a large variety of mathematical objects including algebra, topology, physics and number theory. More specifically it arises when studying Teichmuller theory, total positivity of matrices, and even the behavior of high energy particle impacts. We will introduce the process of quiver mutation, and the proceed to explore quivers for a special sequence of mutations called maximal green sequences. The aim of the talk is to discuss recent work that allows one to build maximal green sequences for larger quivers by looking at "component preserving" sequences on induced subquivers. These new techniques have allowed us to construct maximal green sequences for large families of quivers where their existence was previously unknown.

MASTERS EXAM: Pyramid Decompositions and Normaliz

Title: Chromatic symmetric homology for graphs: some new developments
 

Abstract: In his study of the four colour problem, Birkhoff showed that the number of ways to colour a graph with k colours is a polynomial P(k) in k, which he called the chromatic polynomial.  Later, Stanley defined the chromatic symmetric function X, which is a multivariable lift of the chromatic polynomial so that when the first k variables are set to 1, it recovers P(k).  We showed that this can be further lifted to a homological setting so that its bigraded Frobenius character recovers X. In this talk, we survey some new results regarding the strength of the chromatic symmetric homology of a graph, and state some (surprising?) conjectures.  A part of the talk will be devoted to discussing Specht modules for symmetric group over the complex field, and other fields. This is based on joint work with Chandler, Sazdanovic, and Stella.

Title: The E_8 Lattice and it’s Automorphism Group

Abstract: TBA

Title: k-uniform displacement tableaux

Abstract: In this talk we will introduce a peculiar family of tableaux on rectangular partitions, known as k-uniform displacement tableaux. The primary curiosity of this family is the introduction of a rule that governs the distance between two boxes in a partition in which the same symbol occurs. Our main goals will be analyzing the ways of filling a partition using a minimal number of symbols, discussing an algorithm for constructing a new tableau with a minimal number of symbols from a given tableau, and the geometric implications of this work. If time permits we can also discuss generalizations of these concepts.

Title: The Erdos-Lovasz Double-Critical Conjecture

Abstract: A simple, connected graph is said to be double-critical if removing any pair of adjacent vertices lowers the chromatic number of the graph by exactly two. In 1966, Paul Erdos and Laszlo Lovasz proposed the Double-Critical Conjecture which states that the complete graph is the only simple, connected graph that is a double-critical graph. This result has been proven when the chromatic number of a graph is less than six, but is still open for the other cases. In this talk, concepts and results related to this problem will be discussed.

Title: Algebraic Aspects of Lattice Simplices

Abstract: Given a lattice polytope P, there are open problems of interest related to the integer decomposition property, Ehrhart h*-unimodality, and Ehrhart positivity. In this talk, we will survey some recent results in this area, based on various joint works with Rob Davis, Morgan Lane, Fu Liu, and Liam Solus.