Our project aims at a better mathematical understanding of several models for the evolution of inhomogeneous flows. Through three main lines of research (see below), we will pursue a twofold final objective. First, we want to develop the current theory of regular solutions for several equations for the evolution of fluids, proposing a new approach and developing tools that are likely to be efficient in various areas of PDEs. Second, for a few selected concrete systems that describe flows in the earth environment or in astrophysics, we wish to use this general approach to extract as much information as possible concerning the qualitative behavior of the solutions. Our first line of research is to provide a rigorous and quantitative justification of simplified models derived from the full Navier-Stokes system that are used in some applications. The usual and natural approach often consists in neglecting the terms that are expected to correspond to very small physical effects. Typical examples are given by incompressible models applied in meteorology although air is not incompressible. We plan to investigate the domain of validity of models that are e.g. used in the low Mach number limit asymptotic, in the diffusive limit for radiating flows, or in the semi-classical limit for fluids endowed with internal capillarity. Our second line of research concerns the interface dynamics for mixtures of non-reacting flows. Following a recent work by two members of the project, we propose to use Lagrangian coordinates in order to study whether the `global model', where interfaces are not prescribed in advance, allows to get relevant qualitative informations on the evolution of mixtures. Both for numerical and theoretical purposes, the possibility to use the global model in this situation would be a great improvement,as one just has to solve a PDE in a fixed domain rather than a free boundary problem with complicated transmission conditions. Our third line is to investigate the evolution of fluids in bounded or exterior domains and to compare their dynamics with those of the whole space case. This is a fundamental issue for, in most applications, the fluid domain has nonempty borders, on which (physical) boundary conditions have to be prescribed. The main difficulty we have to face is that this first means to develop new analytical tools so as to solve the fluid equations in domains, with the same accuracy as in the whole space. The common setting between these three lines of research is the striving for a critical functional framework corresponding to the equation under consideration. Indeed, considering only critical norms or related quantities often yields the optimal results for the study of the Initial Value Problem, the derivation of blow-up (or continuation) criteria and the asymptotic properties of the solutions. Since the 80s, with the growing success of Fourier analysis methods, the critical regularity approach has led to substantial progress in the study of evolution equations for fluids. However, Fourier analysis methods collapse if the fluid domain is not the whole space or the torus. To extend the critical regularity approach to the domain case, we will first have to imagine a good substitute to Fourier analysis. This is a very challenging issue, which is obviously also relevant for other topics than equations for fluids. It could be achieved by improving some endpoint parabolic type maximal regularity estimates (needed in our third objective), as well as extending them to the rough coefficients case (for the second objective). In this regard, encouraging preliminary results have been obtained recently by some members of the project.

The overall aim of this project is to make important progress on the interplay between mirror symmetry and singularities of linear differential equations in any dimension. The general objective and the originality of the project consists in profiting from the expertise of the participants in various diverse fields in order to advance each of the research topics. The French (ANR-funded) and German (DFG-funded) partners have complementary expertise in all topics and a long standing experience of collaboration. The following topics will be considered both separately and in connection to each other: - Mirror symmetry as an effective tool for the computation of Gromov-Witten invariants of various kinds of smooth algebraic varieties or orbifolds, - Irregular singularities of linear differential systems in any dimension, either from the point of view of holonomic D-modules or from that of isomonodromy deformations, - Hodge theoretic aspects of such differential systems. One of the original aspects of the project consists in obtaining results in each topic by exhibiting the interplay between these topics through the use of various tools and methods (algebraic geometry, non commutative Hodge theory, singularity theory and D-modules, symplectic geometry) with, in the background, motivations and conjectures formulated by physicists. A central object of interest will be the generalized hypergeometric systems of linear differential equations (GKZ systems) as models for the quantum D-module of toric manifolds or orbifolds. These GKZ systems also provide a large class of examples of holonomic D-modules with irregular singularities, where conjectures and preliminary results can be tested. The understanding of the geometry of different types of moduli spaces like those for isolated hypersurface singularities, for curves, or more generally for stable mappings (entering in the very definition of Gromov-Witten invariants), and for meromorphic connections on vector bundles, is one of the most important motivations of the whole project. Although the first ones are known to be essential for mirror symmetry, a basic question will be to make sense/fully understand the notion of mirror symmetry for the moduli spaces of irregular singular connections on Riemann surfaces. The Stokes phenomenon, which is a fundamental property of irregular singularities of differential equations, is a basic object to be understood in the context of either Gromov-Witten theory or Landau-Ginzburg models and their extensions to singularity theory. Its relationship with Hodge-theoretic properties (in particular their non-commutative aspects) will allow the analysis of moduli spaces of singularities. This project would allow the French and German teams of researchers the full means to realize their scientific goals and collaborate more effectively. The structure of the project motivates that part of the funding we apply for is devoted to financing travel and invitation costs of members of the project and also of some other recognized scholars. However, most part of the funding will be devoted to post-doctoral financial support, in order to fully develop specific directions.

Periods are a class of complex numbers obtained by integrating algebraic differential forms over algebraically defined domains which only involve rational coefficients. Examples include logarithms of integers, multiple zeta values and certain amplitudes in string and quantum field theory. From the modern point of view, periods appear as entries of a matrix of the comparison isomorphism between algebraic de Rham and Betti cohomology of varieties over number fields. Thanks to this interpretation, the theory of motives becomes a powerful tool to predict all algebraic relations among these numbers and, in some favourable cases, to prove them. It should be thought of as a higher analogue of the Galois theory of algebraic numbers. Indeed, all recent breakthroughs in the study of periods, namely Ayoub's theorem (a relative version of the Kontsevich-Zagier conjecture) and Brown's theorem (every multiple zeta value can be written as a linear combination of those having only 2 and 3 as exponents), were the reflection of the emergence of new ideas and techniques in motivic Galois theory. This JCJC project aims at gathering together young researchers working on the theory of periods and motives form different points of view, the interaction of which seems particularly promising. The researchers were selected according to pre-existing collaborations and the perspective of initiating new ones. We plan to tackle, among others, the following questions: 1) Mixed Tate motives – Give geometric constructions of the extensions of Q(0) by Q(n) and relate them to irrationality proofs of zeta values. Find generators of the category of mixed Tate motives over the ring of integers of cyclotomic fields. 2) Motivic Feynman amplitudes – Study the motives associated with Feynman graphs and the coaction conjecture by Brown, Panzer, and Schnetz, according to which motivic Feynman amplitudes are stable under the Galois action. 3) Exponential motives – Establish a Newton above Hodge theorem for the irregular Hodge filtration and the eigenvalues of Frobenius on the de Rham cohomology associated with a smooth variety together with a regular function. 4) Operads, motives, and the Grothendieck-Teichmüller group – Explain the role of periods in the proofs of the formality of the little disks operad. Study the action of the tannakian group of the category of mixed Tate motives over Z on certain operads coming from algebraic topology. Compute the Galois action on the multiple zeta values appearing as "Kontsevich weights" in deformation quantisation. 5) Motivic Galois group in positive characteristic – Pursue the study of de Rham-like realisation functors on motives in positive characteristic. Understand the structure of the derived Hopf algebras obtained by applying the weak tannakian formalism in order to define a motivic Galois group and tackle the variant of the Grothendieck period conjecture in this setting. 6) p-adic periods – Use the recent work of Bhatt-Morrow-Scholze to study integral aspects of p-adic periods (e.g. multiple zeta values). Compare the rational structures on the l-adic cohomology of varieties over number fields arising from algebraic cycles on their reductions modulo p. 7) BCOV invariants of arithmetic Calabi-Yau varieties –Compute the unknown constant in the proof of the BCOV conjecture for the Dwork pencil in terms of special values of the gamma function and explain the result in the spirit of the Gross-Deligne conjecture by exhibiting a motive with complex multiplication. Undertake a similar study in other arithmetic situations.

This project aims at studying, from the mathematical point of view, certain singular asymptotic regimes for Vlasov equations, that is to say for kinetic transport equations without collision. The asymptotic regimes we would like to consider are of two different nature: firstly large time behavior of solutions, in which case one wants to describe the final dynamics and the speed of convergence, secondly certain scaling limits where some physical quantites are seen as small parameters, in which case the aime to exhibit an effective limit equation. The underlying idea is to consider in a joint way these two types of problems.. The point is to exhibit formal links and to rely on them to obtain original new methods. We wish to tackle singular problems where classical methods cannot be used; singularities can come either from the lack of regularity of some limit objects, or from degeneracy coming from the structure of the equations themselves / We wish to focus in particular on the following problems: - Landau damping and quasineutral limit for the Vlasov-Poisson system; - Monokinetic behavior and high friction limits for the Vlasov-Navier-Stokes system.

This project, at the crossroads of algebraic geometry and number theory, aims at investigating some aspects of the relationship between the arithmetico-geometric and cohomological properties of an algebraic variety. The notion of cohomology is somewhat the modern culmination of the old idea of `linearization', which consists in associating to an object with a rich structure (an algebraic variety $X$), families of simpler objects (linear i.e. vector spaces or modules with additional data) supposed to encode a lot of the relevant information about the original object but easier to handle, classify etc. Abelian cohomology has progressively emerged in topology and complex geometries during the first half of the twentieth century. During this period, first, and second cohomology pointed sets also appear with the study of twists in Galois theory. In the fifties and early sixties, the development of category theory and homological algebra provided the conceptual frame to elaborate systematically a cohomological formalism in abelian categories, paving the way to its diffusion in many areas of mathematics. In particular, it is intrinsically connected to the breathtaking development of modern algebraic geometry, yielding to the introduction of various algebraic Weil cohomologies related by comparison isomorphisms. Around the early seventies, these several cohomological avatars were unified in Grothendieck's fascinating conjectural theory of pure motives, which was later included in the wider theory of mixed motives. These theories - partly conjectural - naturally give rise to `conjectures horizon', such as the Hodge and the Tate conjectures, various conjectures on algebraic cycles etc. The purpose of ECOVA is to study a series of problems arising as consequences or particular cases of these `conjectures horizon' and which can be roughly classified following three main directions: - complex and l-adic coefficients: absolute setting (special cases of the Hodge and the Tate conjectures), variational setting (geometric study of the exceptional loci, representations of the fundamental group, l-independence, varietes de Shimura); - integral, finite, adelic coefficients: absolute setting (obstructions to the Hodge and the Tate conjectures), variational setting (representations of the fundamental group, l-independence); - Finer questions on algebraic cycles (Higher Abel-Jacobi maps, integral and birational aspects, study of zero-cycles and rational points, mixed motives). ECOVA gathers young mathematicians working on the various aspects described above of deep long-standing problems in arithmetic geometry so that they could share the technics they handle, acquire new ones, initiate and pursue collaborations. As a result, a particular emphasis will be put on the organization of weekly or monthly working groups, series of invited lectures of post-doctoral level, visit to and invitation of foreign specialists, attending conferences.The 48 months of the project will be structured around three conference-like events: an opening meeting where the researchers involved in the project will present and discuss the mathematical problems they are working on, a mid-project meeting, in the spirit of a mixed summer school/research conference and a closing meeting, with the purpose of presenting the main mathematical achievements of the projects in the frame of a wide-audience research conference.