Persistent sheaf cohomology

Lead Research Organisation: Cardiff University
Department Name: Sch of Mathematics

Abstract

Algebraic topology studies the features of topological spaces which are invariant under continuous deformations, like the number of holes in a surface or the Euler characteristic of a polyhedron. In data analysis and computer science on the other hand geometric objects are often approximated by ""point clouds"", which rises the following question: How can we reconstruct the topological invariants of a closed subset of Euclidean space from a set of possibly inaccurate point samples? There is a way to associate a topological space to such a point cloud, which is based on a threshold radius and is called the Vietoris-Rips complex. Persistent Homology captures the development of the homology groups of these complexes for all values of the threshold radius in so-called persistence diagrams, which allow conclusions about the invariants of the original space.

The project is about a fairly recent development: the application of sheaf theory to topological data analysis. Sheaves are used to described data distributed over a topological space and to solve ""local to global"" problems. The project will contribute to the development of the theoretical groundwork for these applications by generalising results in persistent cohomology, like stability theorems, to the sheaf theoretic context.

Publications

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Studentship Projects

Project Reference Relationship Related To Start End Student Name
EP/N509449/1 01/10/2016 30/09/2021
1941653 Studentship EP/N509449/1 01/10/2017 31/03/2021 Alvaro Torras Casas