Fall 2017

The Combinatorics Seminar meets on Friday in Snow 408 from 3-4pm.

Please contact Jeremy Martin if you are interested in speaking.

**Friday 8/25**

Ken Duna

*Counting Bubbles with Triangles: An Introduction to Simplicial Homology*

__Abstract:__ This talk will be aimed at introducing the notion of a
simpicial complex and the basic combinatorial and topological properties
associated with simplicial complexes. Briefly, a simpicial complex can
be thought of as a topological space obtained by gluing generalized
triangles (points, lines, triangles, tetrahedra, etc...) together. A
basic question in algebraic topology is how to characterize
"holes" or "bubbles" in a topological space.
Generally, the computation of these "holes" (more precisely,
homology classes) is not easy. However, for simplicial complexes, the
computations are reduced to elementary linear algebra.

**Friday 9/1**

Jeremy Martin

*The Matrix-Tree Theorem*

__Abstract:__ The Matrix-Tree Theorem is a classical result that
counts the spanning trees of a graph using linear algebra. It is
central in studying many other area of algebraic graph theory:
sandpiles, electrical networks on graphs, and higher-dimensional
extensions of these ideas to simplicial complexes, all of which will be
coming soon to this seminar.

**Friday 9/8**

No seminar (AMS Fall Meeting at U. North Texas)

**Friday 9/15**

Jeremy Martin

*The Matrix-Tree Theorem for Simplicial Complexes*

__Abstract:__ The definition of a spanning tree can be extended from graphs
to simplicial complexes of arbitrary dimension, using simplicial homology.
The Matrix-Tree Theorem also extends to high dimension, but there's a twist: not
all trees are counted equally.

**Friday 9/22**

Bennet Goeckner

*Stanley's Diamond Conjecture*

**Friday 9/29**

Jeremy Martin

*The Matrix-Tree Theorem for Simplicial Complexes, II*

**Friday 10/6**

Jeremy Martin

*Parking Functions*

**Friday 10/13**

No seminar (Happy Fall Break!)

**Friday 10/20**

Federico Castillo

*Ehrhart Positivity*

__Abstract:__ We say a polytope is Ehrhart positive if its Ehrhart
polynomial has positive coefficients. There are different examples of
polytopes shown to be Ehrhart positive using different techniques. We
will survey some of these results. Through work of Danilov/McMullen,
there is an interpretation of Ehrhart coefficients relating to the
normalized volumes of faces. We try to make this relation more explicit
in the particular case of the regular permutohedron. This is joint work
with Fu Liu.

**Friday 10/27**

Marie Meyer (University of Kentucky)

*Laplacian Simplices*

__Abstract:__ There are many advantageous ways to associate a polytope to a graph \(G\). A recent construction is
to consider the convex hull of the rows of the Laplacian matrix of \(G\) to form what is known as the Laplacian
simplex, \(T_G\). In this talk we focus on properties of \(T_G\) including reflexivity, the integer decomposition
property, and unimodality of the Ehrhart \(h^*\)-vector according to graph type. This is joint work with Ben Braun.

**Friday 11/3**

Ken Duna

*Higher Nerves of Simplicial Complexes*

**Friday 11/10**

No seminar

**Friday 11/17**

Jeremy Martin

*Random Walks and Electrical Networks*

__Abstract:__ A graph can be viewed as an electrical network in which each edge has a current and voltage. There are surprising and
beautiful connections between the theories of (i) graphs as electrical networks and (ii) random walks on graphs; many random variables
associated with a walk can be interpreted electrically. I will describe these connections, drawing largely on Bollobas' graph theory
textbook. The talk should be accessible to (in fact, will be aimed at) graduate students in both combinatorics and topology; no advanced
knowledge of either field is required.

**Friday 11/24**

No seminar (Thanksgiving)

**Friday 12/1**

TBA

**Friday 12/8**

TBA (Stop Day)

For seminars from previous semesters, please see the KU Combinatorics Group page.

Last updated Wed 11/15/17