SEMINAR ON THE APPLICATIONS OF QUIVER REPRESENTATIONS

Organizers: Yariana Diaz, Ryan Kinser (University of Iowa)

The seminar has since concluded. Below you will find a list of the presented talks and their abstracts.

PAST TALKS

November 15, 2021

Title: Interleaving Distances for Reeb Graphs

Speaker: Elizabeth Munch, Michigan State University

Topic: Reeb graphs and other related graphical signatures have extensive use in applications, but only recently has there been intense interest in finding metrics for these objects. The idea is that graphical signatures such as Reeb graphs, merge trees, and contour trees encode data in both a space and a real valued function, and we want to build metrics that are sensitive to this information. In this talk, we will focus on a particular metric for comparing Reeb graphs known as the interleaving distance which is a categorical reformulation of the eponymous metric from persistence modules arising in Topological Data Analysis. These ideas come from viewing the data of a Reeb graph as stored in a sheaf, allowing both for more combinatorial views of the distance, as well as generalizations to other categorical frameworks which fit this model.

November 8, 2021

Title: Künneth theorems in persistent homology

Speaker: Jose Perea, Northeastern University

Topic: The classical Künneth formula in homological algebra provides a link between the homology of a product space and that of its factors. We will show in this talk a collection of similar results for persistent homology. That is, we show how the persistent homology of a filtered product space — different product filtrations lead to different formulae — can be recovered from that of its filtered factors.

*Tuesday, October 26, 2021 – 2:30-3:20pm

(Zoom only, no in-person meeting)

Title: ℓ^p-Metrics on Multiparameter Persistence Modules

Speaker: Michael Lesnick, SUNY Albany

Topic: (Joint work with Håvard Bjerkevik) Motivated both by theoretical and practical considerations in topological data analysis, we generalize the p-Wasserstein distance on barcodes to multiparameter persistence modules. For each p ∈ [1,∞], we in fact introduce two such generalizations d_I^p and d_M^p, such that d_I^∞ equals the interleaving distance and d_M^∞ equals the matching distance. We show that d_M^p ≤ d_I^p for all p ∈ [1,∞], extending an observation of Landi in the p = ∞ case. We observe that the distances d_M^p can be efficiently approximated. Finally, we show that on 1- or 2-parameter persistence modules over prime fields, d_I^p is the universal (i.e., largest) metric satisfying a natural stability property; this result extends a stability theorem of Skraba and Turner for the p-Wasserstein distance on barcodes in the 1-parameter case, and is also a close analogue of a universality property for the interleaving distance given by the second author. In a forthcoming paper, we apply some of these results to study the stability of (2-parameter) multicover persistent homology.

October 25, 2021

Title: Topic: Generic Gelfand-Tsetlin Representations of Nonstandard Quantized Orthogonal Algebras

Speaker: Jordan Disch, Iowa State University

October 18, 2021

Title: Homological approximations in persistence theory II

Speaker: Thomas Brüstle, Université de Sherbrooke

Topic: Continuation of previous talk

October 11, 2021

Title: Homological approximations in persistence theory

Speaker: Benjamin Blanchette, Université de Sherbrooke

Topic: Complete invariants for persistence modules over posets that are not totally ordered are known to be out of reach, and developing incomplete but rich invariants is one of the main goal of current research on the subject. We make a brief summary of such invariants and discuss their strengths and weaknesses, then introduce a unifying way to understand them as special cases of a construction using homological algebra.

September 27, 2021

Title: Third expository talk on “Signed Barcodes for Multi-Parameter Persistence via Rank Decompositions and Rank-Exact Resolutions”

Speaker: Ryan Kinser, University of Iowa

Topic: Third and final of three expository talks on the paper of the same title by Botnan, Oudot, and Oppermann: https://arxiv.org/abs/2107.06800

September 20, 2021

Title: Second expository talk on “Signed Barcodes for Multi-Parameter Persistence via Rank Decompositions and Rank-Exact Resolutions”

Speaker: Yariana Diaz, University of Iowa

Topic: Second of three expository talks on the paper of the same title by Botnan, Oudot, and Oppermann: https://arxiv.org/abs/2107.06800

September 13, 2021

Title: First expository talk on “Signed Barcodes for Multi-Parameter Persistence via Rank Decompositions and Rank-Exact Resolutions”

Speaker: Yariana Diaz, University of Iowa

Topic: This will be the first of three expository talks on the paper of the same title by Botnan, Oudot, and Oppermann: https://arxiv.org/abs/2107.06800