Graduate Student

I 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, and have many other applications in algebraic topology and algebraic geometry.

I 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. I will draw many pictures of examples.

(arXiv:2510.12574)