Gabriel Angelini-Knoll (Case Western Reserve University)

Algebraic K-theory of Morava K-theory

The prime fields in algebra are the finite fields and the rationals, but in homotopy theory there are more prime fields known as Morava K-theory K(n,p) depending on a height n and a prime p. The algebraic K-theory of fields are some of the most fundamental computations in the subject, for example Quillen computed the algebraic K-theory of finite fields and the algebraic K-theory of the rationals can be computed from the algebraic K-theory of the integers and the algebraic K-theory of finite fields using a localization sequence, however the algebraic K-theory of the integers are not completely known. I will discuss two different filtrations that allow us to access the algebraic K-theory of Morava K-theory: the motivic filtration and the May filtration. The first filtration was analyzed in joint work with Hahn and Wilson and the second filtration was studied in joint work with Bayindir. Combining these two approaches, we can compute the orders of quotients of orders of integral algebraic K-theory of (connective) Morava K-theory in all degrees. In the case of the integers, analogous quotients of orders of K-theory groups recover special values of the Riemann zeta function.