Bridging Statistics and Sheaves

Over the past 15 years, applied topology ​has brought tools​ from traditionally pure areas of mathematics, such as algebraic topology and category theory​, to​ ​applied mathematics​ and statistics, all while guided by applications to data science and engineering.​ Topological Data Analysis (TDA), which provides principled, data-driven model discovery and inference procedures, and Topological Signal Processing (TSP), which provides robust reparametrization-invariant properties of observed spatial and temporal data, are among the most notable recent developments in this line of scientific pursuit.

Sheaves and cosheaves have proven to be remarkably ​well-suited at unifying various categorical representations found in ​both ​TDA and TSP. ​In order to quantify the behavior of these representations and algorithms on typical data sets, characterization of their behavior on random instances, as well as their continuity and stability, is required. These questions in turn require introducing topologies and metric structures on the inputs, such as datasets or parameters of statistical models, and on the outputs, such as constructible functions or Reeb graphs. These problems are nontrivial and need to be tackled in a way that loses neither theoretical rigor or practical relevance. This is the purpose of the workshop.

The structure of the workshop is intended to allow time for pedagogical lectures in the morning and working groups in the afternoon. Example topics for lectures and working groups include: ​topologies on the space of cosheaves, stability of Mapper via level-set persistence, sampling theory for topological transforms, statistical sufficiency of various TDA descriptors, probability spaces for topological models​, ​geometric measure theory ideas in sheaf theory, random walks over cosheaves, and integral calculus in o-minimal structures.