Applied and Computational Topology
Our research group works in the field of applied and computational topology and geometry. The focus lies on questions regarding the connectivity of data on multiple scales. This is global information, concerning the data as a whole, and inaccessible by standard means of data analysis. Popular methods in this context are persistent homology, Vietoris–Rips complexes, and Delaunay complexes (Alpha shapes).
We consider theoretical questions, such as the stability of persistence barcodes or connections to representation theory, but also computational problems, such as finding the Vietoris–Rips persistence barcode of a set of points. We study the hardness of certain computational problems arising in applied topology, and develop fast code that is widely used in applications (http://ripser.org).
In this project, we study geometric complexes, that is, simplicial complexes constructed from geometric data, and their multiscale topological properties, using homological and Morse-theoretic methods. These complexes are defined for metric spaces, finite or infinite, and their persistent homology enjoys a stability property. This allows for the treatment of the discrete case (finite point clouds) and the continuous case in a unified way.
Ripser is a lean C++ code for the computation of Vietoris–Rips persistence barcodes. It can do just this one thing, but does it extremely well.
The aim of this project is to develop methods for constructing coarse models of the global behavior of a system with complicated dynamics. These models will abstract from individual trajectories and rather provide dynamical information on a discretization of the underlying invariant set in form of a directed graph or a finite state Markov chain. In contrast to existing approaches, our models will incorporate information about cycling motion, generalizing the classic notion of periodic or quasiperiodic dynamics.