Towards a Chromatic Langlands Program
Originally developed as a tool for computing the stable homotopy groups of spheres in topology, chromatic homotopy theory has proven to be highly interdisciplinary, possessing seemingly fundamental connections to number theory and the mathematics of high-energy physics. On the arithmetic side, it can be interpreted as a Brave New class field theory, with geometric models like tmf acting as a spectral version of arithmetic geometry. On the physical side, these complex-oriented cohomology theories act as receivers of "index maps" central to both concrete computations and geometry in quantum field theory. In this talk, I will propose a chromatic Langlands program that unifies notions of ramification, globalization, and equivariance appearing in these three fields. The approach taken involves modular and global equivariance for topological automorphic forms, cyclotomic trace as a description of geometric and algebraic ramification, and chromatic redshift; and, as suggested by the name, should be thought of as a spectral version of the Langlands program. Broadly speaking, the program aims to give a unified description of transchromatic geometry, and is expected to produce new computational tools in stable homotopy theory. Applications outside of topology include a rigorous interpretation of Witten's equivariant index theory and arithmetic geometry over the field with one element.