Weekly Bulletin

We prove that (logarithmic, Nygaard completed) prismatic and (logarithmic) syntomic cohomologies are representable in the category of logarithmic motives. As an application, we immediately obtain Gysin maps for prismatic and syntomic cohomologies, and we precisely identify their cofibers. In the second part of the talk we develop a descent technique that we call saturated descent, inspired by the work of Niziol on log K-theory. Using this, we prove crystalline comparison theorems for log prismatic cohomology, log Segal conjectures and log analogues of the Breuil-Kisin prismatic cohomology, from which we get Gysin maps for the A_{inf} cohomology.

The notion of a John domain was initially introduced in 1961 by Fritz John, and later named after him by Martio and Sarvas. Typically, its study is motivated by its connections to the properties of quasiconformal and quasisymmetric mappings. Moreover, John domains find extensive applications in the theory of Sobolev functions in metric measure spaces and functional analysis, as they represent essentially the sole class of domains that uphold the Sobolev-Poincaré inequality. In this presentation, I will introduce several recent applications of John domains in the theory of the calculus of variations.

Coarse correlated equilibria are generalizations of Nash equilibria which have first been introduced in Moulin et Vial (1978). They include a correlation device which can be interpreted as a mediator recommending strategies to the players, which makes it particularly relevant in a context of market failure. After establishing an existence and approximation results result in a fairly general setting, we develop a methodology to compute mean-field coarse correlated equilibria (CCEs) in a linear-quadratic framework. We identify cases in which CCEs outperform Nash equilibria in terms of both social utility and control levels. Finally, we apply such a methodology to a CO2 abatement game between countries (a slightly modified version of Barrett (1994)). We show that in that model CCEs allow to reach higher abatement levels than the NE, with higher global utility. The talk is based on joint works with F. Cannerozzi (Milan University), F. Cartellier (ENSAE) and M. Fischer (Padua University).