Day 1 of ATMCS 6 is now (mostly) over. Small groups of applied topologists are roaming the streets of Vancouver looking for sights, food or drink, while the less hardy of us have already eaten and retired to our rooms for a quiet night, or a night full of last-minute preparations.
Vin de Silva talked about his currently ongoing research into interesting new perspectives on persistence stability theorems and foundational models for persistence modules. One thing that really caught my attention was the idea of metric certificates: many metrics are defined as the supremum or infimum over a range of potential comparison points. The certificates idea summarizes all these approaches under a common header – a comparison point is a certificate, and the metric is produced by optimizing across certificates.
This is put to use to produce extension theorems of the form
If \(A\) is a subspace of \(B\), and there is a 1-Lipschitz map \(A \to M\), then we can construct a 1-Lipschitz map \(B \to M\).
These theorems turn out to hold in a bunch of situations, and to be highly relevant for persistent homology.
Amit Patel talked about his work on Quillen 2-categories and their relationship to persistent homology. It turns out there are ways of talking about persistent homology that pull in some sheaf-theoretic perspectives and naturally produce a Quillen 2-category that encodes much of the structure.
Sarah Day talked about Conley index theory and symbolic dynamics; with some research geared towards using symbolic dynamics approximations of dynamical systems to discover models and pick out cycles and stable behaviors. Within this project, Conley indices turn out to be useful tools.
Tamal Dey talked about the Graph induced complex, and work with Fengtao Fan and Yusu Wang on data sparsification by building topological models and simplifying the computational complexity of generating topological inferences.
I’ll see if I can recruit volunteers from the audience to keep a stream of conference updates flowing here.