The project ” Regulators and explicit formulae ” aims at drawing together a group of mathematicians working on various aspects of regulator maps : analytic issues, motivic and K- theoretic problems, explicit formulae and reciprocity laws, links with L-functions, geometric interpretations via Arakelov theory. The main themes of the project are: - Motivic cohomology - Arakelov theory and explicit formulae - L-functions of number fields and elliptic curves. Polylogarithms

The goal of our project is to bring together prominent french, but also european mathematicians who work in areas related to measurable group theory, geometric group theory, probability and dynamics. These subjects have known dramatic development in the last fifteen years, mainly thanks to increasing integration between these different mathematical subjects. We also wish, via Post-Doc positions, conferences, and advanced courses, to disseminate the knowledge acquired worldwide, and to stimulate young mathematicians to work in this field, at the intersection of several domain of mathematics, where such a great variety of inspiring and interdisciplinary open problems are waiting to be solved. Our scientific goals deal with various branches of geometric and measured group theory. Our project tackles five main themes of research, and will be organized over four main tasks (actually five if we take into account the organization of scientific events). Let us give a quick account of these subjects. The first theme deals with the interaction between the structure of locally compact groups and its geometric and ergodic properties. It is important to enhance that our main goal here is either to explore specific features of non-discrete groups or to exploit results on locally compact groups to obtain new insights on finitely generated ones. One the coordinators of this project, Romain Tessera has already obtained important results on these topics. Our second theme is related to measured equivalence relations and mainly focuses on two long standing questions: the fixed Price problem and the cost vs first L2 Betti number. These problems originate in a fundamental work of the other coordinator, Damien Gaboriau in 2002. These two problems have strong consequences in Measure Equivalence theory, L2 Invariants theory, von Neumann Algebras and in Percolation on graphs. Instead of a theme, our third part gathers various probabilistic methods in ergodic theory: percolation, invariant random subgroups, and Poisson boundary. These topics form a broad subject which is intimately related to the previous theme, however also coming with their own fascinating open problems. We expect them to shed light on ergodic theory, and vice versa. The fourth theme very interestingly links together many of the previous themes as it combines geometry and measured theory in a very intricate way. Integrable measure equivalence strengthens measure equivalence by taking into account the large-scale geometry of the group via some first moment condition imposed on the coupling. Many important and surprising rigidity results for amenable groups, and for lattices in simple Lie groups of rank 1 have recently brought this subject to the forefront of research in geometric group theory. Finally, our last theme has a more topological dynamics flavor. The idea of soficity takes its origins in the work of Gromov, who aimed to formulate a finite approximation property for groups that encompasses both amenability and residual finiteness. A recent breakthrough in topological dynamics is a definition of entropy invented by Bowen for actions of sofic groups. The notion of topological full group of a topological dynamical system has become prominent recently as it provides a whole new family of finitely generated infinite groups, some being both simple and amenable, or even Liouville. Entropy of the dynamical system might very well be related to geometric properties of the full group. We have chosen each member of this proposal very carefully in order to find a perfect mix between a great variety of expertise and a clear connection to one or more of these themes. We hope to be successful in taking up the challenge of solving some --if not most-- of the many open problems mentioned in this proposal.

Since its introduction in the 70's, the Langlands program has been the object of an incredible number of works by some of the world's most talented mathematicians, with striking arithmetic applications such as the proof of Fermat's Last theorem and the Sato-Tate conjecture. In the past decade or two, beginning with the pioneering work of Breuil, a p-adic version of this program has emerged. By the work of Colmez, the case of the group GL(2,Q_p) is now settled and furthermore Emerton established the compatibility of this p-adic correspondence with the global one. This subject is now at the heart of an increasing number of questions and projects. During the last three years, very rich and deep mathematical objects were invented: the Fargues-Fontaine curve, the notion of perfectoid space, the pro-étale site, infinite level perfectoid Shimura varieties. We believe that these concepts will lead to a revolution in the field, just like deformation techniques for Galois representations and Taylor-Wiles systems did twenty years ago; the best proof of this affirmation is the great number of international conferences and workshops already organized on these topics, attracting an increasing number of mathematicians. These emerging topics are extremely challenging and appeal to a large variety of competencies, which are amply covered by the various members of our group. We can organize the various directions of research into five broad themes: 1) p-adic representation theory of p-adic groups and p-adic Galois representations of p-adic local fields: we aim to go further than the case of the group GL(2,Q_p) settled by Colmez. 2) Infinite level Shimura varieties and Rapoport-Zink spaces: we believe that the Lubin-Tate spaces of infinite level will allow us to reprove Colmez's result for GL(2). They should also give us a window through which to later study the case of GL(n). 3) p-adic families of modular forms: we would like to unify the various points of views and to study some new situations like non-PEL Shimura varieties. 4) The global Langlands program for number fields : the study of coherent cohomology will allow us to explore the relations between non-cohomological algebraic automorphic forms and non regular Galois representations. We also aim to study the completed cohomology using the Hodge-Tate period map and the Harder-Narasimhan stratifications. On the automorphic side we will pursue our program on level 1 automorphic forms. 5) Comparaison theorems for cristalline and syntomic cohomology of adic spaces. The French arithmetic school, since the introduction of the Langlands program and till its more recent developments, has always been at the avant-garde : we mention the proof of the local Langlands correspondance by Laumon-Rapoport-Stuhler, Harris-Taylor and Henniart, the global correspondence for function fields by L. Lafforgue, the proof of the fundamental lemma by Laumon-Ngo, Ngo and Chaudouard-Laumon, the introduction of the p-adic Langlands program by Breuil and Colmez's proof for GL(2,Q_p).The goal of the project is to continue this long tradition of French leadership, providing a necessary support to develop scientific activity inside the group or in cooperation with mathematicians abroad, in a very hot and promising topic. Our group is composed of internationally recognized advanced researchers, bright early stage researchers and promising fresh post doctoral students. As we expect that the subject will develop quickly, the support of the ANR is necessary in order to organize regular workshops, mini-conferences and a big final conference, to travel to attend congresses, to invite world class experts for short stays, to hire post-doctoral students (36 months) and finally to convince the best students and postdocs to develop their projects in our universities.