Access Type

Open Access Dissertation

Date of Award

January 2017

Degree Type

Dissertation

Degree Name

Ph.D.

Department

Mathematics

First Advisor

Andrew Salch

Abstract

In this dissertation, we study the interactions between periodic phenomena in the homotopy groups of spheres and algebraic K-theory of ring spectra. C. Ausoni and J. Rognes initiated a program to study the arithmetic of ring spectra using algebraic K-theory and gave a higher chromatic version of the Lichtenbaum-Quillen conjecture, called the red-shift conjecture, that is expected to govern this arithmetic. This dissertation provides a proof of a special case of a variation on the red-shift conjecture. Specifically, we show that, under conditions on the order of the fields, iterated algebraic K-theory of finite fields detects a periodic family chromatic height two.

To prove that iterated algebraic K-theory of finite fields detects a periodic family of chromatic height two, we compute approximations to iterated algebraic K-theory using the theory of trace methods. We develop a tool for computing higher order topological Hochschild homology (THH) using a filtration of a commutative ring spectrum. We then compute THH of algebraic K-theory of finite fields after smashing with a finite complex and provide initial results towards mod p THH of algebraic K-theory of finite fields. We then detect height two periodic elements in the circle homotopy fixed points of THH and show that periodic families of height two are detected in iterated algebraic K-theory of finite fields after smashing with a finite cell complex.

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