Counting orbits via stable homotopy

Abstract: The rings studied in algebra tend to be obtained from the integers by applying various operations such as introducing fractions, taking quotients, adjoining roots, ... In this sense ordinary algebra takes place "under the integers." In contrast, cobordism theory produces a plethora of rings "over the integers," with the proviso that the notion of a ring had to be extended to incorporate homotopies. One viewpoint on algebraic topology is that it is concerned with the study of these rings through methods that are analogous to those of ordinary algebra. In the late 20th century, this led to the discovery of "higher" analogues of the ordinary prime numbers, which are known as Morava K-theories.

In this talk, I will explain how these theories can be used to give a partial answer to a question of Arnold's in symplectic dynamics, about the minimal number of fixed points of a Hamiltonian diffeomorphism. This is joint work with Blumberg, and I will attempt to assume no prior background in either subject.