"Homological Tools for Data" (Wednesday)
Abstract: The past fifteen years has witnessed a dramatic burst of applications of topological thinking and theorems in the applied sciences, ranging from statistics to sensor networks, neuroscience, and more, to be surveyed here. Several challenges remain, including: (1) how to compute topological quantities efficiently; (2) how to extend the set of current applications and methods; and, perhaps most importantly, (3) how to educate end-users in the meaning and proper use of homological tools.
This talk will demonstrate why homology is one of the most exciting new tools in applied mathematics.
"New Uses for Sheaf Theory" (Thursday)
Abstract: As ideas from algebraic topology diffuse over into applied domains, we see a recapitulation of the subject's genesis. First, the use of Betti numbers; next, functoriality (cf. persistent homology); then, categorification (current work on stability and interleaving in topological data analysis).
What next? This talk will argue that sheaves and sheaf theory are a good candidate for the next toolbox for applied data science. The talk will give a gentle overview of this (intimidating) subject and provide details of a new class of sheaves useful in inference problems associated with sensor networks.