Homotopy Theory and its Applications

University Of Washington, Seattle WA

Investigators

Abstract

DMS-0203205 Stephen A. Mitchell The proposed work is centered on K(1)-local homotopy theory and its applications to algebraic K-theory. One goal of the project is to compute the mod 2 homology of the general linear group of a number ring. A broader objective is a systematic study of K(1)-local homotopy theory from the perspective of Iwasawa theory, with a view toward applications to the algebraic K-theory spectra of number rings. One of the most striking features of modern mathematics is its essential unity. Time and again one finds that seemingly unrelated branches of the subject interact in unexpected ways. The interaction of of algebraic topology and number theory is a noteworthy example of this phenomenon. Number theory began as a "discrete" subject, then turned analytic in the 19th century and increasingly geometric in the 20th. In the last thirty years or so, algebraic topology (the study of properties of geometric objects, such as curves and surfaces, that remain invariant under deformation) has entered more and more into number-theoretic questions. The proposed work will explore further the remarkable connections between these two subjects, by analyzing the topology of certain geometric objects that can be associated to classical number-theoretic data.