Abstract

Chromatic homotopy theory is the classical subfield of algebraic topology which uses the theory of smooth one-parameter formal groups to organize calculations and the search for large-scale phenomena in stable homotopy theory. There is a formal group associated to any cohomology theory with Chern classes for complex vector bundles and, remarkably, the information can go the other way: under good conditions we can construct the cohomology theory from the formal group. This allows for a very tight connection between the geometry of formal groups and homotopy theory. For example, the moduli stack of formal groups has a height stratification, and thus induces a corresponding stratification in stable homotopy theory known as the chromatic filtration. Current major projects include calculations at a single height, assembling information from different heights, and the investigation of how various constructions in stable homotopy theory pass between heights. If algebraic geometry is the study of schemes equipped with a sheaf of rings, then derived algebraic geometry is the study of schemes with sheaves of rings in some category amenable to homotopy theory, such as structured ring spectra. There are very important derived stacks and schemes that live over the moduli stack of formal groups, such as the moduli stack of generalized elliptic curves and certain Shimura varieties. This has led to a fruitful interplay between derived algebraic geometry and chromatic homotopy theory. Examples include the work of Gross and Hopkins establishing deep connections between various types of duality, the Hopkins-Miller work on topological modular forms, and considerable recent work on Picard and Brauer groups and spaces. This has expanded the field in new and unexpected directions.