Events for 11/08/2019 from all calendars

Mathematical Physics and Harmonic Analysis Seminar

Time: 1:50PM - 2:50PM

Location: BLOC 628

Speaker: Gregory Berkolaiko, Texas A&M University

Title: Quantum graphs with a shrinking subgraph and exotic eigenvalues

Abstract: We address the question of convergence of Schroedinger operators on metric graphs with general self-adjoint vertex conditions as lengths of some of graph's edges shrink to zero. We determine the limiting operator and study convergence in a suitable norm resolvent sense. It is noteworthy that, as edge lengths tend to zero, standard Sobolev-type estimates break down, making convergence fail for some graphs. The failure is due to presence of what we call "exotic eigenvalues": eigenvalues whose eigenfunctions increasingly localize on the edges that are shrinking to a point.

We establish a sufficient condition for convergence which encodes an intricate balance between the topology of the graph and its vertex data. In particular, it does not depend on the potential, on the differences in the rates of convergence of the shrinking edges, or on the lengths of the unaffected edges. In some important special cases this condition is also shown to be necessary. Moreover, when the condition fails, it provides quantitative information on exotic eigenvalues.

Before formulating the main results we will review the setting of Schrodinger operators on metric graphs and the characterization of possible self-adjoint conditions, followed by numerous examples where the limiting operator is not obvious or where the convergence fails outright. The talk is based on a joint work with Yuri Latushkin and Selim Sukhtaiev, arXiv:1806.00561 (Adv. Math. 2019) and on work in progress with Yves Colin de Verdiere.

Algebra and Combinatorics Seminar

Time: 3:00PM - 3:50PM

Location: BLOC 628

Speaker: Peter Stiller, Texas A&M University

Title: Edge Erasures and Chordal Graphs with Applications to Data Clustering

Abstract: We prove several results about chordal graphs and weighted chordal graphs by focusing on exposed edges. These are edges that are properly contained in a single maximal complete subgraph. This leads to a characterization of chordal graphs via deletions of a sequence of exposed edges from a complete graph. Most interesting is that in this context the connected components of the edge-induced subgraph of exposed edges are 2-edge connected. We use this latter fact in the weighted case to give a modified version of Kruskal’s second algorithm for finding a minimum spanning tree in a weighted chordal graph. This modified algorithm benefits from being local in an important sense. In recent work with Culbertson, Dochtermann and Guralnik these results have been generalized, leading to a new result on Simon's conjecture concerning the extendable shellability of certain complexes.