Bridging applied and quantitative topology

• Florian Frick, Chirality and quantifying embeddability

  • The combinatorics of triangulations of a space X provide upper bounds for the topology of the space of embeddings of X into d-dimensional Euclidean space. I will explain the previous sentence and as a consequence present generalizations of classical non-embeddability results. For example, beyond non-embeddability, a first case of interest here is that of chirality: A space is achiral if some embedding is equivalent to its mirror image up to isotopy.
    This is joint work with Michael Harrison.

• Francesca Tombari, What's behind the homotopical decomposition of a simplicial complex

  • Decomposing a simplicial complex by taking a covering of its vertices does not necessarily preserves the homotopy type of the original one. Thus, there is no hope in general to retrieve the homotopy type of the Vietoris-Rips complex of a metric space, just by studying Vietoris-Rips complexes of its subspaces. In this talk, we will investigate this phenomenon in general and present some sufficient conditions for homotopy type preservation by decomposition. We will focus, in particular, on the mathematics behind these decomposability results, which involve the study of homotopy fibres of certain functors and related comma categories. This is a joint work with Wojciech Chachólski, Alvin Jin and Martina Scolamiero.

• Gunnar Carlsson, Deep Learning and TDA

  • I will talk about some ways in which TDA interacts with the Deep Learning methodology. TDA can contribute to explainability as well as to the performance of Deep Learning models.

• Shmuel Weinberger, PH(X^Y) and the geometry of function spaces

  • The homology of function spaces is a central topic in algebraic topology. What about their persistent homology? I will focus on a few examples where geometric considerations shed light.

• Conrad Plaut, Discrete Homotopy Theory and Applications

  • Discrete homotopy theory was originally developed by Valera Berestovskii and Plaut in 2001 as an effort to understand generalized covering spaces of topological groups. Over the last couple of decades the ideas evolved to include uniform spaces and hence metric spaces. The basic idea is to replace traditional paths and homotopies by discrete chains and homotopies, which allows quantification of the fundamental group and other topological features of various kinds of spaces. We will give the basic background needed—noting a further bonus that this approach is much simpler than traditional topological approaches, while (for most spaces!) producing the fundamental group and most regular covering spaces. This approach also is very amenable to counting arguments, especially involving the Gromov-Hausdorff metric. We will mention several applications, including generalized universal covering spaces, finiteness theorems in Riemannian geometry, topology of boundaries of CAT(0) spaces in geometric group theory, and spectra related to the length spectrum of compact Riemannian manifolds. The latter includes “length spectra” when there is no length (e.g. resistance metrics on fractals).

• Ling Zhou, Persistent homotopy groups of metric spaces

  • By capturing both geometric and topological features of datasets, persistent homology has shown its promise in applications. Motivated by the fact that homotopy in general contains more information than homology, we study notions of persistent homotopy groups of compact metric spaces, together with their stability properties in the Gromov-Hausdorff sense. Under fairly mild assumptions on the spaces, we proved that the classical fundamental group has an underlying tree-like structure (i.e. a dendrogram) and an associated ultrametric. We then exhibit pairs of filtrations that are confounded by persistent homology but are distinguished by their persistent homotopy groups. We finally describe the notion of persistent rational homotopy groups, which is easier to handle but still contains extra information compared to persistent homology.

• Claudia Landi, Multi-parameter persistence from the viewpoint of discrete Morse theory

  • Although there is no doubt that multi-parameter persistent homology is a useful tool for the topological analysis of multivariate data, a complete understanding of these modules is still lacking. Issues such as computation, visualization, and interpretation of the output remain difficult to solve. In this talk, I will show how discrete Morse theory can enhance our understanding of multi-parameter persistence by connecting the combinatorial properties of the critical cells of multi-filtered data to the algebraic properties of their persistence modules.

This is joint work with Asilata Bapat, Robyn Brooks, Celia Hacker, and Barbara I. Mahler.

• Žiga Virk, Information encoded in persistence diagrams

  • We will overview several results interpreting parts of persistence diagrams in terms of properties of underlying spaces. The properties in question include homology of a space, shortest 1-dimensional homology basis of a geodesic space, locally shortest loops, systole, homotopy height, subspaces of contraction, proximity properties, rigidity of critical simplices, and more.

• Iris Yoon, Persistent Extension and Analogous Bars: Data-Induced Relations Between Persistence Barcodes

  • A central challenge in topological data analysis is the interpretation of barcodes. The classical algebraic-topological approach to interpreting homology classes is to build maps to spaces whose homology carries semantics we understand and then to appeal to functoriality. However, we often lack such maps in real data; instead, we must rely on a cross-dissimilarity measure between our observations of a system and a reference. We will present a pair of computational homological algebra approaches for relating persistent homology classes and barcodes: persistent extension, which enumerates potential relations between cycles from two complexes built on the same vertex set, and the method of analogous bars, which utilizes persistent extension and the witness complex built from a cross-dissimilarity measure to provide relations across systems. Time permitting, we will demonstrate the use of these methods in studying neural population coding and structure propagation on synthetic and real neuroscience datasets. This is joint work with Robert Ghrist (University of Pennsylvania) and Chad Giusti (University of Delaware)

• Erin Wolf Chambers, Computing optimal homotopies

  • The question of how to measure similarity between curves in various settings has received much attention recently, motivated by applications in GIS data analysis, medical imaging, and computer graphics. Geometric measures such as Hausdorff and Fr\'echet distance have efficient algorithms, but often are not desirable since they do force any deformation based on them to move continuously in the ambient space. In this talk, we'll consider measures that instead are based on a homotopy between the two curves. Such deformations will generally look to minimize some quantity associated with the homotopy, such as total area swept or longest intermediate curve. We will survey several measures based on homotopy which have been introduced and studied in recent years, examining structural properties as well as considering the complexity of the problem or known algorithms to compute it. We will also give an overview of the many remaining open questions connected to this area.

• Mikhail Katz, Extremal Spherical Polytopes and Borsuk's Conjecture

• Alexander Nabutovsky, Isoperimetric inequality for Hausdorff contents and its applications

  • We will discuss the isoperimetric inequality for Hausdorff contents and compact metric spaces in (possibly infinite-dimensional) Banach spaces. We will also discuss some of its implications for systolic geometry, in particular, systolic inequalities of a new type that are true for much wider classes of non-simply connected Riemannian manifolds than Gromov’s classical systolic inequality.
    Joint work with Y. Liokumovich, B. Lishak, and R. Rotman.

• Brittany Terese Fasy, Working with Persistence Diagrams: Some Theory and Some Practical Tips/Tricks

  • In this talk, we investigate how to use persistence diagrams as descriptors in data analysis tasks. In a topological data analysis (TDA) pipeline, we often replace our input data with persistence diagrams that summarize the data. Now that we have these diagrams, what do we do with them? Moreover, if we have too many of them, how do we interpret a large set of persistence diagrams? In some ways, these questions are hard: persistence diagrams don’t lend themselves naturally to common tasks such as averaging and suffer from the curse of dimensionality (since they have infinite doubling dimension). We will discuss the implications of these difficulties, and highlight some recent insights.

• Jürgen Jost, Geometry and Topology of Data

  • We link the basic concept of topological data analysis, intersection patterns of distance balls, with geometric concepts. The key notion is hyperconvexity, and we also explore some variants. Hyperconvexity in turn leads us to a new concept of generalized curvature for metric spaces. Curvature notions originally arose in Riemannian geometry, and the Riemann curvature tensor encodes all invariants of a Riemannian metric. Therefore, curvature notions should also be fundamental for metric geometry. The talk represents joint work with Parvaneh Joharinad.