Séminaires : Séminaire Dérivé

Quillen discovered a correspondence between cohomology theories and formal groups. While Quillen's correspondence is tight enough to be extremely successful in transporting phenomena back and forth, it is not one-to-one: some cohomology theories are missed, as are some formal groups. I will explain how to turn Quillen's correspondence into a one-to-one correspondence by changing the definition of a cohomology theory. From another perspective, this gives a functor-of-points description of Lurie's "derived moduli stack of formal groups", which he specified via charts. This is joint work with Robert Burklund and Ishan Levy.

The goal of this talk is to explain work joint with Scholze where we study a version of algebraic geometry which is built, not out of spectra of commutative rings, but out of spectra of symmetric monoidal higher categories. Unlike traditional algebraic geometry, where the category of affine schemes does not have well behaved gluings, our setup provides an (infinity) topos where every object is, in a sense, affine. This topos contains the usual category of qcqs schemes, but also provides a home to new and interesting objects which cannot be studied with more classical means. We will encounter some of these objects in this talk, with relevance in topological field theory, the theory of motives, and the geometric Langlands program.