Current methods in stable homotopy theory

MATH 290A Subject & Catalog Number

Description

A classical problem in homotopy theory is to understand the homotopy groups of spheres, $\pi_{n+k}(S^k)$.  A classical theorem of Freudenthal says that these groups are independent of $k$ once $k>n+1$. The groups one obtains in this stable range are known as the stable homotopy groups of spheres, denoted $\π_n S$. By a theorem of Serre, these groups are finite abelian groups for $n>0$. Understanding the groups $\π_n S$ remains one of the central motivating problems in homotopy theory.

The main approach to this problem is through the theory of spectra. Spectra are mathematical objects analogous to abelian groups in the homotopical world. Over the years, it has become clear that spectra are ubiquitous in mathematics: they play important roles not only in the study of stable homotopy groups of spheres, but also in the classification of manifolds, algebraic K-theory, p-adic geometry, condensed matter physics, algebraic geometry, representation theory, and many other areas.

I will present an introduction to the theory of spectra, using the stable homotopy groups of spheres as a motivating problem. Particular emphasis will be placed on the methods of chromatic homotopy theory as an approach to the study of spectra.

Recommended Prep

Prerequisites: A first-year graduate course in algebraic topology, such as Math 231a, or equivalent background plus basic category theory . Some familiarity with homological algebra would also be very helpful.