Welcome to the home page of the research project Chromatic Homotopy and K-Theory – ChroK – funded by the French Agence nationale de la Recherche (ANR), running from October 2016 to March 2023. (Originally scheduled to end in September 2021, the project duration was extended due to the pandemic.)
The project builds upon the new foundations of algebraic topology, with the view to fundamental applications notably in algebraic K-theory and in chromatic homotopy theory.
These new foundations have enormous potential throughout mathematics, using higher structures and functorial methods. They have led to the resolution of long-standing problems (for example the Kervaire invariant problem and the cobordism hypothesis) and to the introduction of new theories which underline the profound relationship with geometry, giving rise to a fascinating interplay between manifolds and deep arithmetic questions from algebraic geometry.
The pioneering work of Adams and Quillen in the 1970s revealed deep connections between geometry, homotopy theory and arithmetic: Adams' work on the image of the J homomorphism opened the way to chromatic homotopy theory and Quillen introduced higher algebraic K-theory. The Lichtenbaum-Quillen Conjectures, and their generalizations by Waldhausen, established that algebraic K-theory and chromatic homotopy theory are intimately related.
Ring spectra were introduced in stable homotopy theory for studying multiplicative cohomology theories. Rigid variants of these (highly structured rings) and Lurie's Higher Algebra have provided powerful generalizations of rings, giving stable homotopy theory a crucial role in higher algebraic geometry.
The project is structured in the following interconnected themes:
The consortium unites experts in each of these fields, as well as bright young researchers and doctoral students.
The project has four partners, with members from seven French mathematical institutes.
A list of the members of the project is available here.