The Applied Topology Day 2025 was a success. Next year, we are planning to extend it to a two days event! Stay tuned.
In the filtered setting, new ideas are required, and a key [partially open] problem is: given a graph G on n vertices, and precisely e edges, what is the tight upper bound on betti_k(Flag(G))? The problem of maximizing total persistence feels much more difficult (see also discussion).
Kozlov, Björner, and others have fully understood how to maximize any Z-linear function on either the dimension vector (f-vector) or the vector of Betti numbers (b-vector). It is a linear optimization problem, and the maxima will appear on vertices of the convex hull of all possible graphs.
We provide a filtered complex which is extremal in multiple ways, and we conjecture that it is the unique maximizer of total persistence for H_1. The construction is rather counter-intuitive and our [technical, combinatorial] proof was based on a conjecture formed from computer experiments.
Questions we consider include: How many (off-diagonal) points can you maximally have in the persistence diagram of a data set on n vertices? What is the longest possible bar? What is the maximal total persistence?
In a recent paper with L. Beers (accepted to SoCG '25) we consider simple questions related to Rips complexes and persistent homology. arxiv.org/abs/2502.21294
I'm hiring a PhD student to work on multiparameter persistence. Deadline: March 15. Please feel free to reach out with any questions. workingat.vu.nl/vacancies/ph...
Ok, thanks for the clarification. I must've missed something.
I heard that the acceptance rate for TDA papers at SoCG this year was a fair bit lower than the general acceptance rate. Wouldn't surprise me if it's always like that.
Kongen er tilbake i statsråd.
