An Introduction to Chromatic Homotopy Theory - Part 1

We will watch the first two parts of a four part mini-course given by Agnes Beaudry. We might pause to discuss or discuss afterwards.

The video for the first talk is here: https://www.youtube.com/watch?v=ZsHtsx_A8j8

More information is here: https://sites.google.com/colorado.edu/agnesbeaudry/links/echt-may-2019

Abstract:

At the center of homotopy theory is the classical problem of understanding the stable homotopy groups of spheres. Despite its simple definition, this object is extremely intricate; there is no hope of computing it completely. It hides beauty and pattern behind a veil of complexity.

Chromatic homotopy theory gives us a way to see through this veil by using the algebraic geometry of formal groups to organize theory and computations. The simplest case is via K-theory: Bott periodicity gives rise to a repeating family of elements in the stable homotopy groups of spheres via the image of the J homomorphism. It is an insight of Morava that there are higher analogues of K-theory and that they should give rise to higher periodicity in the stable homotopy groups of spheres.

In the 1980s, Ravenel and Hopkins made a series of conjectures describing this connection, most of which were proved in the 1980s and 1990s by Devinatz, Hopkins, Smith and Ravenel. Two of these problems remain open: the chromatic splitting conjecture and the telescope conjecture. The ultimate goal of the course will be to motivate and state these two famous problems.

The course will include a quick reminder of spectra and a brief introduction to complex orientations and localizations. We will discuss periodicity and the chromatic filtration, leading to a statement of the two conjectures. We will also discuss the higher K-theories and the role they play in modern computations. Many topics will only be touched briefly, as my intention is to provide a roadmap of the field to non-experts.

I will assume the knowledge of an advanced course in algebraic topology, and some familiarity with the category of spectra, K-theory, the Adams operations and cobordism.