EPFL Applied Topology Seminar 2021/22

The seminar will take place both in person and online until further notice. If you have any questions please contact Adélie Garin , Celia Hacker or Kathryn Hess.

Program

Date and place Title Speaker
05.10.21,
13:00
Some perspectives on persistence modules structure Azélie Picot
12.10.21,
13:00
MA B2 485
Topology of random 2-dimensional cubical complexes Erika Roldan
02.11.21,
13:00
Trees to Barcodes and Back Again: Combinatorial and Probabilistic Perspectives Adélie Garin
30.11.21,
13:00
Topology-Driven Diffusion in Structured and Unstructured Data Sets Bastian Rieck
14.12.21,
13:00
TBA Sara Kališnik
18.01.21,
15:00
TBA Matteo Pegoraro
08.02.21,
13:00
TBA Henry Adams

Abstracts

A. Picot: While real persistence modules decomposition is well understood, little can be found in the literature about persistence modules morphisms.
First, we define the matrix of a morphism which depends on the chosen basis of the persistence modules.
There is an equivalence between persistence modules morphisms and persistence modules on Rx{1,2}. This allows to define the decomposition of a morphism thanks to the Crawley-Boevey decomposition theorem. In this talk, we give examples of indecomposable morphisms, which depend strongly on the combinatorics of the barcodes of persistence modules and the matrix. We finally give the precise statement of a general decomposition theorem of a monomorphism.

E. Roldan: We study a natural model of random 2-dimensional cubical complexes which are subcomplexes of an n-dimensional cube, and where every possible square (2-face) is included independently with probability p. Our main result exhibits a sharp threshold p=1/2 for homology vanishing as the dimension n goes to infinity. This is a 2-dimensional analogue of the Burtin and Erdős-Spencer theorems characterizing the connectivity threshold for random graphs on the 1-skeleton of the n-dimensional cube. Our main result can also be seen as a cubical counterpart to the Linial-Meshulam theorem for random 2-dimensional simplicial complexes. However, the models exhibit strikingly different behaviors. We show that if p > 1 – sqrt(1/2) (approx 0.2929), then with high probability the fundamental group is a free group with one generator for every maximal 1-dimensional face. As a corollary, homology vanishing and simple connectivity have the same threshold. This is joint work with Matthew Kahle and Elliot Paquette.

A. Garin: Methods of topological data analysis have been successfully applied in a wide range of fields to provide useful summaries of the structure of complex data sets in terms of topological descriptors, such as persistence diagrams, or barcodes. While there are many powerful techniques for computing topological descriptors, the inverse problem, i.e., recovering the input data from topological descriptors, has proved to be challenging. In this talk, I will consider specifically the inverse problem from barcodes to merge trees. I will describe a connection between the space of barcodes and symmetric groups, and show how to use it to study distributions of neurons modelled as trees, creating a bridge between the field of permutation statistics and TDA. I will then extend this symmetric group connection into a new way to coordinatize the space of barcodes, opening the door to a statistical and probabilistic study of the space of barcodes using a geometric group theory point of view.
This is joint work with B. Brück, J. Curry, J. DeSha, K. Hess, L. Kanari and B. Mallery.

B. Rieck: Diffusion is a general umbrella term for information propagation schemes. In this talk, I will briefly summarise our recent work on leveraging such schemes on structured data sets, i.e. graphs, and unstructured ones, i.e. point clouds. I will demonstrate how topology-driven diffusion schemes can improve classification performance and help in creating new hierarchical summaries of complex data sets.