Categorified independence of l-problems

• Denis-Charles Cisinski (University of Regensbourg)

Room: Amphithéâtre Yvonne Choquet-Bruhat 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).

Derived Hopf rings and motivic homotopy theory

• Tom Bachmann (University of Mainz)

Room: Amphithéâtre Yvonne Choquet-Bruhat 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.