Abstract for the talk on 25.05.2022 (14:00 h)

Seminar on Nonlinear Algebra

Fabian Lenzen (TU Munich)
Efficient computation of 2-parameter persistent cohomology
25.05.2022, 14:00 h, MPI für Mathematik in den Naturwissenschaften Leipzig, G3 10 (Hörsaal)

Persistent homology is one of the key concepts in topological data analysis and an active area of research in computational topology. It associates to a filtered simplicial complex the system of the homology vector spaces of each complex in the filtration. This collection can be viewed as a k[x]-module, and a common goal is to compute a free presentation of this module, which can be succinctly described by the so-called barcode. A common optimisation scheme in current software exploits the fact that the computation of persistent cohomology, albeit yielding equivalent results, can be carried out far more efficiently.

Analogously, the system of homology vector spaces of a two-parameter filtration can be viewed as a k[x, y]-module. Computing a free presentation of it is more involved, though, and the efficiency of existing implementations lags behind that of one-parameter persistent homology software. This is because optimisations using cohomology cannot be applied straightforwardly anymore, due to the fact that, unlike the one-parameter case, cochain modules are not free anymore.

I will show how cohomology can be used to develop efficient algorithms for two-parameter persistence nevertheless by considering free resolutions of cochain modules instead, using a result that links free resolutions of persistent homology and cohomology.

Physical attendance in seminars is restricted to 15 participants.
If you want to attend this talk, you need to register in advance using this special form. Registrations will be accepted on first come, first served basis. External participants from Leipzig University need to fill in their name, address, email, and phone number.
Please also check our general Corona rules page. Participants must wear a face mask, and they are encouraged to use the corona rapid tests made available by the institute.

18.05.2022, 00:09