Fall 2012

The Combinatorics Seminar meets on Fridays, 3-4 PM, in Snow 408.

Please contact Jeremy Martin if you are interested in speaking.

**Friday 8/24**

No seminar

**Friday 8/31**

Jeremy Martin

*Simplicial Rook Graphs*

__Abstract__: The vertices of the simplicial rook graph
\(SR(d,n)\) are the points in \(\mathbb{N}^d\) whose coordinates add
up to \(n\), with edges corresponding to "rook moves" that
change two coordinates at a time. For example, \(SR(3,3)\) looks like this:

Unexpectedly, the eigenvalues of the adjacency matrix of \(SR(d,n)\) turn out to be integers -- at least that's what the evidence indicates. I'll describe how we proved this for \(d=3\) case, and present what we think are some tantalizing partial results and open problems for the general case.

This is joint work with Jennifer Wagner (Washburn University).

**Friday 9/7**

Ilya Smirnov

*On the chromatic number of the plane*

**Friday 9/14**

Rob Bradford

*Arrow Ribbon Graphs*

**Friday 9/21**

Jeremy Martin

*Cyclotomic and Simplicial Matroids*

__Abstract__: The cyclotomic matroid \(\mu_n\) is represented by the \(n\)th roots
of unity over \(\mathbb{Q}\). I'll talk about how the
number-theoretic properties of \(n\) correspond to the matroid structure of
\(\mu_n\), and what this has to do with simplicial complexes. (This
part is joint work with Vic Reiner, dating back to 2005.) I will also
talk about the recent beautiful result of Reiner and Gregg Musiker
that interprets the coefficients of the cyclotomic polynomial in terms
of simplicial homology.

**Friday 9/28**

No seminar

**Friday 10/5**

No seminar (Fall Break)

**Friday 10/12**

No seminar

**Friday 10/19**

Jeremy Martin

*The Chip-Firing Game*

**Friday 10/26**

Arindam Banerjee

*Algebra and Combinatorics of Edge Ideals*

__Abstract__: I'll discuss some algebraic and combinatorial properties of edge ideals of simple graphs. In particular interplay between homological invariants like regularity projective dimension of the edge ideal and combinatorial structures like cycles and claws in the graph.

**Friday 11/2**

Logan Godkin

*Colorings, Tensions, and Flows of Cell Complexes*

**Friday 11/9**

Stephanie van Willigenburg (University of British Columbia)

*Maximal supports and Schur-positivity among connected skew shapes*

__Abstract__: The Schur-positivity order on skew shapes is
denoted by \(B < A\) if the difference of their respective Schur
functions is a positive linear combination of Schur functions.
It is an open problem to determine those connected
skew shapes that are maximal with respect to this ordering. In
this talk we see that to determine the maximal connected skew
shapes in the Schur-positivity order it is enough to consider
a special class of ribbon shapes. We
also explicitly determine the support for these ribbon shapes.
This is joint work with Peter McNamara and assumes no prior knowledge.

**Friday 11/16**

Felix Breuer (San Francisco State University)

*Combinatorial Applications of Ehrhart Theory, Hypergraph Coloring Complexes and the Ehrhart f*-vector*

__Abstract:__
The Ehrhart function of a set X in Euclidean space counts the number of integer points in the k-th dilate of X. If X is a polytope with integral vertices, the Ehrhart function of X coincides with a polynomial at all positive integers k. This polynomial is called the Ehrhart polynomial of X and its h*- and f*- vectors are coefficient vectors with respect to certain binomial bases of the space of polynomials. Ehrhart theory offers several nice results about these polynomials, from bounds on their coefficients to geometric interpretations of their values at negative integers. In recent years, Ehrhart theory has found a number of applications in combinatorics. The idea is to model combinatorial counting functions as Ehrhart functions of suitable geometric objects and then apply theorems from Ehrhart theory to obtain results.

I will begin this talk by giving an overview of some of these applications of Ehrhart theory in combinatorics, dealing with chromatic polynomials, flow polynomials and tension polynomials of graphs. These examples motivate the study of hypergraph coloring complexes and I will present some results about hypergraph coloring complexes in greater detail. One interesting fact is that these complexes do not, in general, have a non-negative Ehrhart h*-vector, while their f*-vector, on the other hand, is always non-negative. It turns out that this is no accident: Ehrhart f*-vectors of polytopal complexes are always non-negative, even if the complex is non-convex and does not have a unimodular triangulation. Moreover, the f*-coefficients of Ehrhart polynomials have a concrete counting interpretation. An interesting corollary is that this property characterizes Ehrhart polynomials of partial polytopal complexes: A polynomial is the Ehrhart polynomial of some partial polytopal complex if and only if its f*-vector is non-negative.

**Friday 11/23**

No seminar (Thanksgiving)

**Friday 11/30**

Chris Trotter

**M.A. thesis defense**

Title TBA

**Friday 12/7**

TBA

**Thursday 12/13, Friday 12/14**

Math 824 project presentations; times/speakers TBA

Jeremy Martin's home page

Last updated Fri 11/9/12