Anna Schenfisch (Dept. of Mathematics, MSU)

3/24/23  9:00am

A/bstract:  In this talk, I focus on two topics from my dissertation that have particularly interesting/fun proofs. The first is related to minimum cardinalities of faithful sets of topological descriptors. Given a finite geometric simplicial complex in general position, we have computed finite sets of augmented persistence diagrams corresponding to lower-star filtrations in various directions so that, taken together, the set uniquely corresponds to the complex, i.e., it is faithful. We explore the question of minimal faithful sets by constructing simplicial complexes for which the minimum faithful set is surprisingly large. This highlights stark practical differences between augmented descriptors and (standard) non-augmented descriptors. In the second half of the talk, we shift focus to computing the algebraic $K$-theory of multi-parameter zig-zag persistence modules through an inductive argument, using previous work done in the one-parameter setting as our base case. This involves decomposing cubical (stratified) manifolds corresponding to module parameter spaces into smaller pieces on which we can use Waldhausen additivity to ``glue'’ the $K$-theory of the corresponding persistence modules back together. Throughout the talk, we focus on geometric aspects of each topic, with plenty of pictures along the way.