Mot du directeur

• Emmanuel Ullmo (IHES)

Room: Centre de conférences Marilyn et James Simons

Stable Homotopy Group, Higher Algebra and the Telescope Conjecture

• Tomer Schlank (Hebrew University of Jerusalem)

Room: Centre de conférences Marilyn et James Simons A fundamental motivating problem in homotopy theory is the attempt to the study of stable homotopy groups of spheres. The mathematical object that binds stable homotopy groups together is a spectrum. Spectra are the homotopy theorist abelian groups, they have a fundamental place in algebraic topology but also appear in arithmetic geometry, differential topology, mathematical physics and symplectic geometry. In a similar vein to the way that abelian groups are the bedrock of algebra and algebraic geometry we can take a similar approach of spectra. I will discuss the picture that emerges and how one can use it to learn about the stable homotopy groups of spheres.

Stacks in the p-adic Hodge Theory of Rigid Analytic Spaces

• Arthur-César Le Bras (CNRS, Université de Strasbourg)

Room: Centre de conférences Marilyn et James Simons I would like to explain in this talk how questions in non-abelian p-adic Hodge theory and in the theory of locally analytic representations of p-adic groups lead to consider new geometric objects attached to rigid analytic spaces, which require to go beyond the formalism of diamonds and are naturally defined in the analytic geometry framework developed by Clausen-Scholze. Based on a joint project (very much in progress) with Anschütz, Rodriguez Camargo and Scholze.

Modularity of Abelian Surfaces

• Vincent Pilloni (CNRS, Université Paris-Saclay)

Room: Centre de conférences Marilyn et James Simons We prove the modularity of a positive proportion of abelian surfaces over the rationals. This is joint work in progress with G. Boxer, F. Calegari and T. Gee.

Motives and Ring Stacks

• Peter Scholze (MPIM Bonn)

Room: Centre de conférences Marilyn et James Simons Several interesting cohomology theories can be described through (analytic) ring stacks, e.g. de Rham, Hodge, crystalline, prismatic, Betti, and even etale cohomology under some restrictions on the base. In this talk, I will recall that to any 6-functor formalism one can associate a (presentable) symmetric monoidal ($\infty$,2)-category. Adopting an extreme Tannaka duality-point of view to formulate the result, I will observe that the symmetric monoidal ($\infty$,2)-category associated to the motivic 6-functor formalism classifies (certain) ring stacks. This picture helps to explain why one has to pass to analytic geometry to find such ring stacks. (For example, the algebraic de Rham stack of $A^1$ is not a ring stack of the required form, only the analytic de Rham stack is.)

Mot de clôture

• Dustin Clausen (IHES)

Room: Centre de conférences Marilyn et James Simons Mot de clôture de Dustin Clausen, Professeur permanent et titulaire de la Chaire Jean-Pierre Bourguignon à l'IHES