Séminaires : Séminaire Dérivé

The Yoga of motives is born with étale cohomology, as a conceptual way to explain elusive integral aspects of l-adic cohomologies. This has lead to deep independence of l problems that have found positive answers - from Deligne's proof of the Weil conjectures to the proof of Deligne's conjecture on companions over smooth algebraic varieties (Lafforgue, Drinfeld, Esnault and Kerz). The Tate conjecture, together with its variations due to Beilinson and Lichtenbaum remains open though. On the other hand, motives have become a full fledged theory that lead to the proof of the Bloch-Kato conjecture by Rost and Voevodsky. We will formulate categorified version of independence of l conjectures, in the language of motivic sheaves. To our knowledge, they are not equivalent to any of the classical conjectures, but interesting relations can be established. For instance, they follow from the Tate-Beilinson conjecture and imply Deligne's conjecture on companions in full generality (over normal algebraic varieties).

The Ravenel--Wilson Hopf ring is an algebraic object which describes the homology of the spaces comprising the spectrum MU. It is characterized by a universal property in a 1-category. I will report on joint work with M. Hopkins, in which we (a) show that (a slight variant of) the Ravenel--Wilson Hopf ring satisfies a related universal property in an oo-category, and (2) can be used to described the motivic homology of the spaces comprising MGL.

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.