The Combinatorics Seminar meets on Friday in Snow 408 from 3-4pm.
Please contact Jeremy Martin if you are interested in speaking.
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.
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.
No seminar (AMS Fall Meeting at U. North Texas)
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.
Stanley's Diamond Conjecture
The Matrix-Tree Theorem for Simplicial Complexes, II
No seminar (Happy Fall Break!)
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.
Marie Meyer (University of Kentucky)
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.
Higher Nerves of Simplicial Complexes
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.
No seminar (Thanksgiving)
TBA (Stop Day)
For seminars from previous semesters, please see the KU Combinatorics Group page.
Last updated Wed 11/15/17