Abstract: Toric topology assigns to each finite simplicial complex K a space with a torus action, called the moment-angle complex, whose equivariant topology neatly reflects combinatorial properties of K and homological properties of the Stanley-Reisner ring of K. A central result in the subject identifies the cohomology of the moment-angle complex with the Koszul homology of the corresponding Stanley-Reisner ring. In this talk, I'll describe how cohomology operations induced by the torus action can be used to extend this result to a topological interpretation of the entire minimal free resolution of the Stanley-Reisner ring, and then outline some work in progress which aims to identify the homotopy types of certain moment-angle complexes in terms of this data. (This is all based on joint work with Ben Briggs.)

Abstract: Like other transform methods, generating functions sometimes offer a lens through which an otherwise complicated problem has a surprisingly elegant solution. I'll offer a quick introduction to probability generating functions and show how we can take advantage of some really beautiful theorems to make practical predictions -- like the probability that COVID-19 has driven the Yamagata flu strain to extinction.