Persistent sheaf cohomology
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.
To give an example, these quantities allow us to prove that one cannot continuously deform the surface of a donut into that of a sphere. Some important invariants are given by algebraic structures that are naturally associated to the geometric objects. Two of those are the homology groups and the cohomology rings of a topological space.
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.
We are interested in pursuing this project and welcome applications if you are self-funded or have funding from other sources, including government sponsorships or your employer.
Please contact the supervisor when you want to pursue this project, citing the project title in your email, or find out more about our PhD programme in Mathematics.