Schedule for: 24w5241 - Representation Theory and Topological Data Analysis

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

I will survey some recent results on theoretical and computational aspects of persistent homology (in one parameter) and its use in topological data analysis. I will illustrate various aspects of persistent homology: its structure, which serves as a topological descriptor, its stability with respect to perturbations of the data, its computation on a large scale, and connections to Morse theory. These aspects will be motivated and illustrated by concrete examples and applications, such as: * reconstruction of a shape and its homology from a point cloud, * faithful simplification of contours of a real-valued function, * existence of unstable minimal surfaces, and * identification of recurrent mutations in the evolution of COVID-19.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Afternoon sessions are devoted to discussions in smaller groups. we use this session to establish a first selection of groups.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Unlike one-parameter persistent homology, the absence of a canonical method for quantifying ‘persistence’ in multiparameter persistent homology remains a hurdle in its application. One of the best-known quantifications of persistence for multiparameter persistent homology, or more broadly persistence modules over arbitrary posets, is the rank invariant. Recently, the rank invariant has evolved into the generalized rank invariant by naturally extending the domain of the rank invariant to the collection of all connected subposets of the domain poset. This extension enables us to measure ’persistence’ across a broader range of regions in the indexing poset compared to the rank invariant. Additionally, restricting the generalized rank invariant can enhance computational efficiency, albeit with a potential trade-off in discriminating power. This talk overviews various aspects of the generalized rank invariant: Möbius invertibility, discriminating power, computation, and its relation to other invariants of multiparameter persistence modules.

The aim of this talk is to present some notions of relative homological algebra that are proving to be useful in the developing of the theory of persistence modules. We will follow closely Sections 3, 4 and 5 of the nice paper [BBH}, which in turn follows the track of the theory of relative homological algebra developed by Auslander and Soldberg for artin algebras in [AS], and that was extended to more general settings in [DRSS]. Let $\mathcal{A}$ be an abelian category, and fix a class of objects $\mathcal{X}$. Let $\mathcal{F} _\mathcal{X}$ denote the class of short exact sequences in $\mathcal{A}$ that remain exact when applying the covariant functor $\mathrm{Hom}_\mathcal{A} (X,-)$ for any $X\in \mathcal{X}$. Dually, let $\mathcal{F}^\mathcal{X}$ denote the class of exact sequences that remain exact when applying the contravariant functor $\mathrm{Hom}_\mathcal{A} (-, X)$ for any $X\in \mathcal{X}$. Both $\mathcal{F} _\mathcal{X}$ and $\mathcal{F}^\mathcal{X}$ define exact estructures over $\mathcal{A}$, so one can make relative homological algebra with respect to both of them. Basic problems, in this setting, are to determine the relative projective objects and the relative injective objects, whether such classes of relative projectives/injectives are resolving/corresolving, whether there are minimal resolutions/corresolutions, do we have relative homological invariants? can we compute relative global dimensions?. The answer to such questions, in general, is difficult and we will outline solutions in settings that, according to [BBH], are of interest for persistence theory. $$ $$ References: $$ $$ [AS] M. Auslander and Ø.\ Solberg, Relative homology and representation theory I: relative homology and homologically finite subcategories. Comm. Alg. 21 (1993), no. 9, 2995-3031. $$ $$ [BBH] BENJAMIN BLANCHETTE, THOMAS BRUSTLE, AND ERIC J. HANSON. Exact Structures for Persistence Modules. arXiv:2308.01790 (2023). $$ $$ [DRSS] P. Dräxler, I. Reiten, S. O. Smalø , and Ø. Solberg, with an appendix by B. Keller, Exact categories and vector space categories. Trans. Amer. Math. Soc. 351 (1999), no. 2, 647-682.

A common approach to studying multiparameter persistence modules is to introduce some "invariant" to determine the similarity between two given modules. In this mostly expository talk, we discuss recent research which utilizes techniques from (relative) homological algebra to interpret classical examples of invariants and define new invariants. The Hilbert function/dimension vector, barcode, and (generalizations of) the rank invariant serve as our main examples. If time permits, we will also discuss the relationship between homological invariants and poset embeddings. Portions of this talk are based on joint works with Claire Amiot, Benjamin Blanchette, and Thomas Brüstle.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Group 1 meets in ... Group 2 meets in ...

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

One-parameter persistence modules decompose into indecomposables of a very simple form, and two interleaved (i.e., "similar") modules allow a nice matching between their sets of indecomposable summands. In multiparameter persistence, not only is there no hope of classifying indecomposables, but simple counterexamples show that there is no reasonable matching between the indecomposable summands of similar modules. However, strong bottleneck stability results can be proven for certain nice families of modules, including, for instance, the projective modules. We will describe two lines of work motivated by these results. In one line, one allows "splitting apart" indecomposable summands before looking for a matching, which gives a notion of similarity of approximate decomposition of general modules. In the second line, one first approximates arbitrary modules algebraically by simpler ones (with a resolution) and then uses a stability result for these simpler modules. We will discuss existing multiparameter stability results as well as open questions; these suggest that the obstacles to proving stronger stability results are similar for the two approaches, despite their apparent differences.

In this talk we will explore some of the different methods of comparing finite dimensional algebras. We will start with the simplest: isomorphism and Morita equivalence and we will see that they are far too rigid. A weakening of the Morita theorem leads to the notion of tilting module which is the first step toward derived equivalences. We will see that, for us, this is a much more interesting notion, allowing us to compare algebras and categories that are a priori very different. The concepts presented will be illustrated by many examples and (a few) conjectures.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Persistent homology with multiple parameters can be phrased in more or less equivalent ways in terms of multigraded modules, or sheaves, or functors, or derived categories. All of these descriptions have in common an underlying partially ordered set indexing a family of vector spaces, and this family is interpreted under increasing layers of abstraction. The simplest objects at any level of abstraction are the "indicator" (or "interval", or "spread") objects, which place a single copy of the ground field at every point of an interval in the underlying poset (an intersection of an upset with a downset). Taking the cue from ordinary persistence, where there is just one totally ordered parameter, a large part of multipersistence theory has revolved around relating arbitrary persistent homology modules as closely as possible to indicator objects. To that end, this survey of perspectives from homological algebra and sheaf theory takes a journey starting with relevant definitions of persistence modules and leading to presentations, resolutions, and stratifications in terms of indicator objects. The way is marked by effective data structures, encodings, and finiteness conditions, leading to syzygy theorems and bounds on homological dimensions.

Filtered poset representations began with work of Kleiner and Nazarova--Roiter, using a process called differentiation. For a grid they appear in work of Bauer--Botnan--Opperman--Steen. For a possibly infinite poset, representations are nothing but persistence modules. This framework can be unified with quiver representations using the notion of a species equipped with commutativity conditions, introduced by Simson. The path algebra of a quiver and the incidence algebra of a poset are both recovered using the tensor algebra. As an example, I will discuss work of Igusa--Rock--Todorov on continuous versions of type A quivers.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.