Abstract: We will begin this talk by showing how to represent the cohomology of some spaces by geometrically nice maps to spaces of cycles on spheres. These maps may be viewed as analogues of differential forms. Moreover, maps between spaces of mod p cycles represent operations on mod p cohomology that generalize the cup product, such as the Steenrod squares and Steenrod powers. These operations are fundamental to certain methods for computing stable homotopy groups.

We will present the first geometric construction of maps between spaces of cycles that represent mod p cohomology operations, with an eye towards applications in quantitative topology. (arXiv:2510.12574)

We will show how to associate to the data of a noncommutative geometry an open quantum system which carries a natural quantum diffusion process. We will consider how this process generalizes the classical Laplacian generated diffusion. We will introduce needed ideas from noncommutative probability, and consider quantum analogs of the Feynman-Kac formulae which we apply to evaluating the spectral action. Finally we highlight some unexplored possibilities which are opened by this mixing of noncommutative probability and geometry.