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.

Water masses of given temperature and salinity are advected without much blending over long periods of time in the ocean. At the numerical level, the discretization of the equation creates a spurious mixing (or numerical diffusion) that artificially mixes the water masses. It can be of the same order of magnitude as the physical mixing. This is especially true in climate simulations where the grid is coarse and the time of integration is long. This project is concerned with two different aspects of the spurious mixing. First, we will study a numerical procedure to quantify precisely in space and time the spurious mixing. The method is both different from usual entropy inequality in maths that are limited to first order schemes, and from global diagnosis based on water mass transformation in oceanography. It should allow us to better understand the geography and effects of spurious mixing. It can be applied to any transported quantity or to the evolution of total energy. Second, we will develop antidiffusive advection schemes for the salinity and temperature in the European ocean general circulation model NEMO. This approach is different from the usual « higher order, finer grids » strategy and has been successfully employed for atmospheric pollution and biphasic flows. Preliminary results obtained by the principal investigator on a new second order antidiffusive scheme show a gain of accuracy in the smooth regions and a correct behavior in 2 space dimensions. This is a major improvement compared to the existing first order antidiffusive schemes, which should allow us to go beyond its use in the vertical direction only. In conclusion, this project proposes a mathematical approach on the question of spurious mixing in ocean general circulation models. The team gathers experts on numerical analysis and ocean modeling. We will collaborate closely together with the aim of having a direct contribution to NEMO.

The VR-MARS project represents a support system for urgent healthcare delivery in isolated environments, based on virtual reality and embodied conversational agents (ECA). We hypothesize that these two technologies enable better situational awareness and care coordination between 3 parties: a care provider in an isolated location, a critically ill patient and the control centre on Earth. VR-MARS explore the scientific fields of emergency medicine, human factors and virtual reality. The use case of VR-MARS will be related to space medicine, in particular emergency care during a manned spaceflight to Mars. During these missions, temporal isolation will add to physical isolation, because of delays in communication between the care provider (on Mars) and ground control (on Earth), which will preclude real-time telemedical support. VR-MARS will be built around two simultaneous decision loops which will allow task assignment and synchronisation between the care provider, the ECA and ground control. The ECA will interact with the care provider via augmented reality. Upon request, it will deliver step-by-step guidance on medical protocols, using reassuring verbal tone and cues in order to mitigate the stress of the care providers. As soon as it is available, ground control on Earth will be made aware of the situation on Mars and of the procedures being undertaken by the care provider. This will improve situational awareness on the ground and enable the most optimal decision making in the mid- to long-term. In return, ground control will deliver its recommendation to the care provider via the ECA. Therefore, the ECA will represent the central hub of communication between the two sites. VR-MARS will be tested on two medical scenarios involving a critically ill patient represented by a high-fidelity simulator. Technical and non-technical skills of the care provider will be assessed at two levels: immediate interactions between the care provider and the ECA (for urgent, life-saving decisions) and delayed interactions between the care provider and ground control (for mid- and long-term decisions). With regards to research output and spinoffs, we anticipate that VR-MARS will improve medical care in remote environments, such as humanitarian missions, the combat environment, medical evacuations, expedition medicine, etc.

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.