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?

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.