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

I propose to investigate the Hodge locus of variation of Hodge structures (VHS) and explore its incarnations in various areas of Mathematics. The Hodge locus is a central object in Algebraic and Arithmetic Geometry introduced in the previous century and recently connected with the Zilber–Pink philosophy about typical and atypical intersections. I plan to bring a new and deeper understanding of such an object: only by employing an emerging viewpoint on functional transcendence one can reach the full potential of the Zilber-Pink dichotomy. This step will use techniques from algebro-differential geometry rather than o-minimality. Such non-trivial, yet natural, generalization is crucial for various unexpected and concrete consequences. It particular it exploits the period torsor and Hodge bundle associated the VHS, rather than its associated period domain. A recurrent case of study will be the universal family of smooth hypersurfaces of some degree d. The three main applications are as follows. (1) Most notably I plan to apply such results to re-understand certain special subvarieties of the moduli space of abelian differentials and start a program aimed at replacing dynamical tools by functional transcendence in the opportune foliated bundle. (2) Study representations of complex hyperbolic lattices (both arithmetic and not) and find new rigidity problems. (3) Investigate the relationship between p-adic functional transcendence and Diophantine results on integral points of subvarieties of abelian varieties and moduli spaces. The main novelty is that the proposed viewpoint aims at unifying seemingly unrelated problems and investigate an emerging link between the previously mentioned topics and hyperbolic geometry. The success of the ERC proposal will be crucial to achieve this, since there are various important aspects that can be investigated by postdocs and Ph.D. students with different backgrounds.

Hilbert geometry, defined on any convex body in a real affine space, is a rich source of examples of metric spaces and has had numerous applications since its description by Hilbert in 1895. The members of this consortium are contributing to various generalizations of this concept and its applications to different contexts, of affine spaces over other field than the real numbers. The objective of this project is threefold: - to develop a unified approach to these generalizations: unified definitions, common generalization of Benzecri's results and of notions of volumes; - to explore the interplay between the different contexts, through numerous examples; - to obtain meaningful applications of Hilbert geometry in each specific case. Applications include projects around: - the study of the metrics of minimal entropy for symmetric spaces; - degeneracy of convex projective structures on surfaces; - around the frontier of the set of Anosov representations in complex hyperbolic geometry; - new linear programming algorithms, with Smale's 9th problem in mind.

The GALF project represents an inter-European and transatlantic fundamental research project with the broad objective to study several of the most relevant problems in current Number Theory using p-adic methods. The team consists of internationally renowned and leading experts from Paris, Lille and Bordeaux in France, from Montreal in Canada and from Luxembourg, joining their complementary expertise in the project. A central problem in Number Theory is the relationship between special values of L-functions and fundamental arithmetic invariants such as regulators or Tate-Shafarevich groups. The study of p-adic properties of special cycles on algebraic varieties plays a key role in this theory.The GALF project will investigate various facets of this, including applications of special cycles on Shimura varieties to p-adic analogs of the Birch and Swinnerton- Dyer conjecture, and p-adic aspects of the Plectic Conjecture. Connexions between Plectic Conjecture, Iwasawa theory and Fargues-Fontaine theory shall be examined. Other important GALF research themes to be examined are the role of p-adic and plectic cohomology methods in extending the theory of complex multiplication to other settings like that of real quadratic fields. The Langlands programme is a vast international research effort establishing deep links between various mathematical areas and appearing in various different forms. It relates automorphic forms and representations with Galois representations and hence number theory. The GALF research will target p-adic and mod p aspects of those via Galois deformation techniques. Geometrically defined Hilbert modular forms mod p of parallel and partial weight one play a very special role. In the GALF project, their attached Hecke algebras shall be related to universal deformation rings with a particular focus on the ramification properties at p. A part of the GALF project is to attach an L-function to an overconvergent eigenform mod p of finite slope and to examine the local behavior at p of the attached Galois representation in view of a formulation of a T-adic Main Conjecture. The theory of deformations of Galois representations has a geometric counter part in so-called eigenvarieties. The GALF efforts in this context especially target the local structure of the eigenvariety at classical weight one points. Moreover, properties of companion forms attached to specific weight 1 modular forms, in particular their Fourier coefficients, will be investigated both in the archimedean and the p-adic settings, with applications to explicit Class Field Theory and Kudla's programme.

Classical enumerative geometry is the branch of algebraic geometry concerned with counting numbers of solutions to geometric questions, for instance the number of curves in a variety that intersect given subvarieties in a prescribed manner. This field was revolutionized through ideas of theoretical physics which have led to deep interactions between diverse fields of mathematics such as symplectic topology, tropical geometry, category theory, integrable systems, algebraic geometry, representation theory. The modern developments began with the introduction of the Gromov-Kontsevich moduli spaces of stable pseudo-holomorphic maps into a symplectic manifold and Witten's insight that the intersection numbers on these moduli spaces can be interpreted as partition functions of quantum field theories. This idea led to significant amount of deep conjectures in mathematics, among which are Witten's conjecture and the mirror symmetry conjecture. A generalization of these ideas lies within the realm of real enumerative geometry. Even classically the progress in this field as compared to complex enumerative geometry has been slow since solutions in real geometry can disappear and only a lower bound of their count is foreseeable. Such lower bound was first introduced by Welschinger for rational curves in lower dimensions. A full theory in arbitrary genus based on moduli spaces of symmetric curves was only recently developed by Georgieva and Zinger. The intersection theories on these moduli spaces can be interpreted as partition functions of certain extended quantum field theories and with this perspective all conjectures in the complex setting have their real counterpart and are open for investigation. The main ingredient in the proofs of mirror symmetry rely on our ability to calculate the corresponding invariants and this is one of the main difficulties in the general case. A proposal made by Kontsevich to use tropical curves introduces a combinatorial flavor to the question. The prediction of applications of tropical geometry in enumerative geometry was confirmed by Mikhalkin who established an appropriate correspondence theorem and found a combinatorial algorithm for computation of Gromov-Witten type invariants for toric surfaces. The tropical approach has important applications in real enumerative geometry. In particular, Mikhalkin's correspondence theorem allows one to calculate or estimate Welschinger invariants (real analogs of genus zero Gromov-Witten invariants) in some situations. Mikhalkin's seminal work paved the way to many important applications of tropical geometry in (real) enumerative geometry, and since then a number of generalizations of Mikhalkin's correspondence theorem were proved. An important recent development in complex, real, and tropical enumerative geometries is related to the Göttsche-Shende conjecture. Block and Göttsche proposed to attribute certain polynomial weights to planar tropical curves arising in tropical calculations of Gromov-Witten type invariants. This suggestion was motivated by the study of refined Severi degrees that was initiated by Göttsche and Shende. The Göttsche-Shende conjecture conjecture highlights a special class of smooth real algebraic varieties: those for which the Euler characteristic of the real part coincides with the signature of the complexification and it is of a particular interest to determine which moduli spaces appearing in enumerative problems belong to this class. The aim of this project on one side is to further develop the above mentioned topics and on another to foster interactions between specialists in these different domains, to aide the exchange of ideas and help their transmission from one point of view to another.