6 février 2025: Patrick Schnider et Nina Otter

par Nina Otter (INRIA), Patrick Schnider (Basel)

jeudi 6 févr. 2025, 14:00 → 16:30 Europe/Paris

Maurice Fréchet (Institut Henri Poincaré)

14h00 Patrick Schnider  University of Basel et ETH

Linear Embeddings and Separoids

Given a k-dimensional simplicial complex, can you embed it into d-dimensional Euclidean space? In many cases, the answer depends on the type of embedding you want. In this talk, we focus on linear embeddings, which are less studied than topological or piece-wise linear embeddings. We show that some methods that give obstructions to the existence of a topological embedding can be adapted to the linear setting. We also discuss how this relates to the concept of separoids.

 

15h30 Nina Otter  Demos, Inria et Université Paris-Saclay

Distance-from-flat persistent homology transforms

The persistent homology transform (PHT) was introduced in the field of Topological Data Analysis about 10 years ago, and has since been proven to be a very powerful descriptor of Euclidean shapes. The PHT consists of scanning a shape from all possible directions and then computing the persistent homology of sublevel set filtrations of the respective height functions; this results in a sufficient and continuous descriptor of Euclidean shapes. In this talk I will introduce a generalisation of the PHT in which we consider arbitrary parameter spaces and sublevel-set filtrations with respect to any function. In particular, we study transforms, defined on the Grassmannian AG(m,n) of affine subspaces of n-dimensional Euclidean space, which allow to scan a shape by probing it with all possible affine m-dimensional subspaces P, for fixed dimension m, and by then computing persistent homology of sublevel-set filtrations of the function encoding the distance from the flat P. We call such transforms "distance-from-flat" PHTs. I will discuss how these transforms generalise known examples, how they are sufficient descriptors of shapes and finally present their computational advantages over the classical persistent homology transform introduced by Turner-Mukherjee-Boyer. No previous knowledge on the subject is required. In particular, in the talk I will introduce persistent homology and integral transforms. This talk is based on the preprint https://arxiv.org/abs/2412.18452, which is joint work with Adam Onus and Renata Turkeš.