Chromatic Algebraic Geometry

ongoing

Stable homotopy theory provides a powerful framework for the study of algebra in the homotopy coherent setting. Its central notion is that of a spectrum, which is the homotopical generalization of an abelian group. Spectra appear naturally in a wide range of mathematical fields from number theory to differential topology, as they encode in a highly structured fashion various fundamental invariants, such as the algebraic K-theory groups of a ring and the cobordism classes of manifolds. The tools of stable homotopy theory have thus found remarkable applications in both algebra and topology, as well as in symplectic geometry, mathematical physics and more. The prevailing, and highly successful, paradigm in stable homotopy theory is the chromatic approach, which gives rise to an infinite family of “intermediate” characteristics interpolating between 0 and p, allowing one to implement local-to-global methods in homotopical settings. In this proposal, I shall describe several interrelated projects, with two unifying themes: The study of the intermediate characteristics using higher semiadditivity, and the import of ideas and tools from algebraic geometry to study both the local and global aspects of the chromatic approach. The preliminary results already provide significant advances on some of the central problems in stable homotopy theory, such as the chromatic redshift conjecture in algebraic K-theory, the construction of E∞-orientations, and the chromatic splitting conjecture. Furthermore, they facilitate partial computations of the Galois, Picard and Brauer groups in the intermediate characteristics.