First Great Plains Combinatorics Conference 2014
Stephen Hartke, University of Nebraska, Lincoln
Computational Methods for Discrete Mathematics
The phenomenal development of computing power and powerful algorithms during the last fifty years has had a profound impact on modern life as well as on science and mathematics. However, computational approaches have not been as heavily utilized in discrete mathematics as in other areas of mathematics, primarily because of two main challenges: “combinatorial explosion'” of the space of objects, and symmetry present from representing isomorphism classes of objects as labeled objects on a computer. One approach is to combine computational techniques of combinatorial optimization (such as integer programming) with techniques for symmetry reduction (such as from combinatorial generation) to search for objects with desired isomorph-invariant properties. I will discuss these techniques and illustrate their use with applications to uniquely Kr-saturated graphs and the Manickam-Miklós-Singhi conjecture.
Jonathan Kujawa, University of Oklahoma, Norman
Deligne's Category Rep(St)
In an elementary and combinatorial way Deligne introduced a category Rep(St) for any complex number t. These “interpolate” between the representations of the various symmetric groups in that when t is a natural number you can naturally recover the category of St-modules from Deligne's construction. That is, to take poetic license, Deligne gave us a way to study modules for a symmetric group on t letters for any complex number t. I will talk about the construction and some joint results with J. Comes.
Jay Schweig, Oklahoma State University, Stillwater
The Projective Dimension of a Simplicial Complex
Given a simplicial complex K, the minimal non-faces of K form a clutter (or hypergraph). We discuss how the combinatorics of this clutter can be used to study the algebraic properties of the associated Stanley-Reisner ideal. In particular, we show a link between domination parameters of the clutter and the projective dimension of the ideal. As corollaries, we are able to bound the homology of K using these domination parameters. (This is joint work with Hailong Dao.)
John Shareshian, Washington University, St. Louis
Chromatic Quasisymmetric Functions
I will describe joint work with Michelle Wachs (University of Miami) in which we define a quasisymmetric refinement of Richard Stanley's chromatic symmetric function, which is a generating function for the proper colorings of a given graph by the positive integers. We have a conjecture describing a connection between the chromatic quasisymmetric functions of certain graphs and the cohomology of certain topological spaces, known as regular semisimple Hessenberg varieties. I will attempt to explain this connection.
Jessica Striker, North Dakota State University, Fargo
Toggle Group Actions and Applications
In this talk, we introduce the toggle group and its action on order ideals of a partially ordered set, or poset. We use the toggle group to model actions on various objects important in combinatorics and statistical physics, including rotation of noncrossing matchings and gyration on fully-packed loops. This is joint work with Nathan Williams.
Nat Thiem, University of Colorado, Boulder
Combinatorial Hopf Structures and Supered Representation Theories
Historically, there have been many theories for what makes a Hopf structure combinatorial. Most experts seem to know one when they see one, but a precise definition seems elusive. Inspired by the representation theory of the symmetric groups, it was originally thought that combinatorial should suggest representation theoretic underpinnings. However, this definition appeared too limited as there were Hopf structures thought to be combinatorial with no known algebraic interpretation. This led to a variety of competing notions, whose technical details this talk will largely avoid. Instead, this talk introduces the fundamental features of a combinatorial Hopf structure and then explores an alternate approach to the representation theory of groups that has given new life to the original representation theoretic conception of combinatoriality. One result is a surprising realization of the representation theory of the finite unitriangular groups, whose usual representation theory is well-known to be wild.