Mathematics faculty and graduate students are invited to attend the PhD Defense of Michael Moy.

Also available via Zoom:  https://zoom.us/j/93145805599?pwd=ajJBb2U0TVE3RGJ5N2NzMTZUWUMxQT09

Meeting ID: 931 4580 5599

Passcode: 167448

Advisor:  Dr. Henry Adams

Committee:  Dr. Chris Peterson, Dr. Amit Patel, Dr. Asa Ben-Hur

Title:  Persistence and Simplicial Metric Thickenings

Abstract:   This talk will give an overview of the standard results from the theory of one-dimensional persistence, emphasizing the algebraic theory of persistence modules and the geometric intuition of simplicial metric thickenings, and will study two particular filtrations of simplicial metric thickenings in detail.  I will describe my work to give a self-contained set of proofs of the most well-known results on persistence, including the interval decomposition of persistence modules, existence of barcodes, the isometry theorem, and the stability of persistent homology.  The filtrations of simplicial metric thickenings studied in detail are the Vietoris–Rips and anti-Vietoris–Rips metric thickenings of the circle.  The study of the Vietoris–Rips metric thickenings is motivated by persistent homology and its use in applied topology.  On the other hand, the study of the anti-Vietoris–Rips metric thickenings is motivated by their connections to graph colorings.  In both cases, the homotopy types of these spaces are shown to be odd-dimensional spheres, with dimensions depending on the scale parameters.