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.

 

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.

                 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: 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.