Wojciech Chachólski

Realisations of Posets.
Abstract: My presentation is based on an article with the same title coauthored with A. Jin and F. Tombari arXiv:2112.122.
Encoding information in form of functors indexed by the poset of r tuples of real numbers (persistent modules) is attractive for three reasons:
(a) metric properties of the poset are essential to study distances on persistence modules
(b) the poset of r tuples of real numbers has well behaved discrete approximations which are used to provide finite approximations of persistence modules
(c) the mentioned discretizations and approximations have well studied algebraic and homological properties as they can be identified with multi graded modules over polynomial rings.
In my talk I will describe a construction called realisation, that transforms arbitrary posets into posets which satisfy all three requirements above and hence are particularly suitable for persistence methods. Intuitively the realisation associates a continuous structure to a locally discrete poset by filling in empty spaces. For example the realisation of the poset of natural numbers is the poset of non negative reals. I will focus on illustrating how homological techniques, such as Koszul complexes, can be used to study persistent modules indexed by realisations.

Grégory Ginot

Homotopical and sheaf theoretic point of view on multi-parameter persistence.
Abstract: In this talk we will highlight the study of level set persistence through the prism of sheaf theory and a special type of 2-parameter persistence : Mayer-Vietoris systems and a pseudo-sometry between those. This is based on joint work with Berkouk and Oudot.

Rick Jardine:

Thoughts on big data sets.
Abstract: This talk describes work in progress. The idea is to develop methods for analyzing a very large data sets X ⊂ RN in high dimensional spaces. There are well-known pitfalls to avoid, including the inability to computationally analyze TDA constructions for X on account of its size, the “curse of high dimensionality”, and the failure of excision for standard TDA constructions. We discuss the curse of high dimensionality and define a hypercube metric on RN that may lessen its effects. The excision problem for the Vietoris-Rips construction can be addressed by expanding the TDA discussion to filtered subobjects K of Vietoris-Rips constructions. Unions of such subobjects satisfy excision in path components (clusters) and homology groups, by classical results. The near-term goal is to construct, for each data point x, a “computable” filtered subcomplex Kx ⊂V(X) with x ∈ Kx, which would capture spatial local behaviour of the data set X near x. A large (but highly parallelizable) algorithm finds a nearest neighbour, or a set of k-nearest neighbours for a fixed data point x ∈ X. Some variant of this algorithm may lead to a good construction of the local subcomplex Kx.

Ling Zhou

Persistent homotopy groups of metric spaces.
Abstract: 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 treelike 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.