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

Speaker:  Sophie Morel

                 Princeton University

Title:         Combinatorial proof of a character identity

Abstract:

The calculation of the (intersection) cohomology of a Siegel modular variety includes many difficult character identities (the fundamental lemma, for example). In this lecture, I want to concentrate on the character identity appearing at the infinite place, which involves, among other things, stable discrete series characters and appears to be related in some non-obvious way to an identity of Goresky, Kottwitz and MacPherson. Once we strip away all the Lie group complications, our identity becomes a very elementary statement and can proved directly using the geometry of the Coxeter complex of the symmetric group. The relation with the Goresky-Kottwitz-MacPherson identity also becomes clearer; in particular, neither identity follows from the other, but they should have a common generalization. This is joint work with Richard Ehrenborg and Margaret Readdy.

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.

Title: Mutation of friezes

Abstract: Frieze is an array of positive integers satisfying certain rules.  Friezes of type A were first studied by Conway and Coxeter in 1970's, but they gained fresh interest in the last decade in relation to cluster algebras.   Moreover, the categorification of cluster algebras developed in 2006 yields a new realization of friezes in terms of representation theory of Jacobian algebras.   In this talk, we will discuss friezes of types A and D and their mutations.

Speaker:  Gabor Hetyei, UNC Charlotte

Title:  Alternation acyclic tournaments and the homogeneous Linial arrangement

Abstract:

We define a tournament to be alternation acyclic if it does not

contain a cycle in which descents and ascents alternate. We show that

these label the regions in a homogenized generalization of the Linial

arrangement. Using a result by Athanasiadis, we show that these are

counted by the median Genocchi numbers. By establishing a bijection

with objects defined by Dumont, we show that alternation acyclic

tournaments in which at least one ascent begins at each vertex, except

for the largest one, are counted by the Genocchi numbers of the first

kind. As an unexpected consequence, we obtain a simple model for the

normalized median Genocchi numbers.

Speaker:  Galen Dorpalen-Barry, University of Minnesota

Title:  Whitney Numbers for Cones

Abstract:

An arrangement of hyperplanes dissects space into connected components

called chambers. A nonempty intersection of halfspaces from the

arrangement will be called a cone.  The number of chambers of the

arrangement lying within the cone is counted by a theorem of

Zaslavsky, as a sum of certain nonnegative integers that we will call

the cone's "Whitney numbers of the 1st kind".  For cones inside the

reflection arrangement of type A (the braid arrangement), cones

correspond to posets, chambers in the cone correspond to linear

extensions of the poset, and these Whitney numbers refine the number

of linear extensions.  We present some basic facts about these Whitney numbers,

and interpret them for two families of posets.