Towards Chromatic Homotopy Theory

The Landweber Exact Functor Theorem

Sonntag, 3. März 2019, von 10:30–18:30 Uhr, Universität Bayreuth

Sunday, 3 March 2019, 10:30 am to 6:30 pm, Bayreuth University


B. Collas, T. Keller, E. Köck (Universität Bayreuth)

Short description

This Bayerische Kleine AG presents a modern formulation of the Landweber Exact Functor Theorem (LEFT) which stands at the intersection of algebraic topology (spectra and cohomology theories), algebraic geometry (moduli stacks, elliptic curves), and number theory/arithmetic geometry (formal group laws). The LEFT provides a general understanding of how ordinary cohomology theory, complex K-theory and Thom cobordism theory are related to isomorphism classes of formal group laws; it is a step towards the the potential description of the homotopy group of spheres via modular forms, chromatic homotopy theory.

In number theory, formal group laws appear in local class fields theory – Lubin-Tate theory – and with elliptic curves over complete DVR's. Over complex manifolds, the first Chern classes of line bundles in singular cohomology and in complex K-theory provide two examples of additive and multiplicative formal group laws. The LEFT deals with the question which formal group laws can be associated to a certain complex oriented cohomology theory; It provides an explicit means of constructing cohomology theories and gives a dictionary between topology (spectra) and algebra (formal group laws).

The goal of this Bayerische Kleine AG is to introduce the material required for the understanding of this result and of further results in chromatic homotopy theory.

Our main references are the lecture notes of J. Lurie and M. Hopkins Speakers will provide explicit formulations of abstract results and will rely on examples and geometric motivations to remain accessible to algebraic topologists and geometers.

Programme (English):
Poster (English)
  • Talk 1: Formal Groups: Lazard ring & Classifications
  • Talk 2: Complex-oriented Cohomology Theories, Spectra & MU
  • Talk 3: The moduli stack M_FG of formal groups & the MU-construction
  • Talk 4: Landweber Exact Functor Theorem & Flatness
  • Talk 5: Landweber Exact Functor Theorem & Periodicity