Chromatic homotopy and K-theory – ChroK

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 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:

1. Chromatic homotopy theory, which stratifies the (exceedingly complicated) stable homotopy category, first localizing at a prime p, then into chromatic layers indexed by the natural numbers; these layers are associated to localization with respect to the Morava K-theories K(n). There are profound connections with the theory of formal groups and arithmetic, for example modular forms. The methods of higher algebra are essential ingredients to the project, for example allowing the study of local Picard groups and their exotic elements.

2. The algebraic K-theory of highly structured rings permits an interpolation from Quillen’s higher algebraic K-theory of rings to Waldhausen’s A-theory of spaces, and exhibits profound interactions between arithmetic, algebraic geometry and geometric topology. The chromatic red-shift conjectures, at the core of the project, seek to describe the chromatic homotopy-theoretic nature of this interpolation. Higher algebra encodes algebraic structures up to relaxed coherence conditions by means of higher category theory; in particular, it offers a powerful framework for the study of generalizations of Hochschild homology such as factorization homology, for higher traces and for iterated algebraic K-theory.

3. Functorial methods are extremely powerful in modern algebraic topology; for example Goodwillie calculus, both in homotopy theory and in algebra, provides powerful tools. Functor cohomology studies the cohomology of small categories with coefficients in (bi)functors and is linked via Mac Lane homology to topological Hochschild homology and to traces for algebraic K-theory. The theory provides extremely powerful tools in the study of stability of homology for families of groups and for performing explicit computations of the cohomology of reductive group schemes. Families of groups to be studied include congruence subgroups, with connections to excision problems in algebraic K-theory.

The consortium gathers leading experts in each of these fields, as well as bright young researchers and doctoral students. The complementary competences of its members will permit fruitful interactions and rapid progress. The project will stimulate cross-fertilization and international collaboration, placing the French school of algebraic topology at the forefront of research.