Topology, Algebraic Geometry, and Dynamics Seminar (TADS): Discontinuities of the Integrated Density of States for Penrose Laplacians
Mar 31, 2023, 1:30 - 2:30 PM
Speaker: May Mei, Denison University
Title: Discontinuities of the Integrated Density of States for Penrose Laplacians
Abstract: The Penrose tiling is among the most popular 2-D model for quasicrystals, materials exhibiting aperiodic order. But the article "the" in front is a misnomer. In this talk, we look at the graph Laplacian associated with four variations from the mutual local derivability class of the Penrose tiling. Two tilings are said to be mutually locally derivable (MLD) if each can be obtained from the other using local rules. From many perspective, two MLD tilings can be thought of as "the same.'' However, local derivability greatly impacts the adjacency relationship, which in turn has ramifications for the graph Laplacian. In each of the four cases, we exhibit locally-supported eigenfunctions, which necessarily cause jump discontinuities in the integrated density of states for these models. Moreover, in several cases we provide concrete lower bounds on this jump. These results suggest a host of questions about spectral properties of the Laplacian on aperiodic tilings, which we will also present.
Time: Friday, March 31, 1:30pm – 2:30pm
Place: Exploratory Hall, room 4208