The project is organized around four important topics in fluid mechanics: free surfaces and interfaces, boundary layers, vortex dynamics and fluid-structure interactions. The mathematical and the physical-environmental motivations of the project are connected to the events of the 2013 Mathematics of Panet Earth program which will also suggest new directions of research. The four topics are closely interconnected because they often coexist in the same physical situation and because the mathematical tools (such as multiscale analysis, asymptotic expansions, stability theory) that are needed to analyze them are quite similar. Our main directions of research will be: -Free surfaces and interfaces. We are mostly interested in situations which are singular, either because of the lack of smoothness (examples are wave breaking, the description of shorelines and the influence of rough topographies in shallow water models) or because of the presence of small parameters (examples are continuous but sharp stratification in two fluid models, multiscale models that describe the energy spectrum in wave breaking and compressible fluids with free surface at low Mach number). We expect improvements in the modelling and numerical simulations of these phenomena through the derivation of more accurate asymptotic models. We also plan to develop suitable mathematical tools in order to handle these singular situations rigorously. -Boundary layers. We are interested both in the construction of boundary layers expansions and the study of their stability properties. For the first aspect, we shall study the construction of boundary layers in degenerate situations, for example in the presence of rough boundaries or in situations where boundary layers of different sizes need to be connected (this is crucial to understand oceanic circulation). We shall also study the well-posedness of the Prandtl type equations that arise in oceanics models. For the second aspect, we plan to make progress in the understanding of instabilities in boundary layers either in the classical inviscid limit of the incompressible Navier-Stokes equation with Dirichlet boundary condition by addressing the question of the destabilizing effect of viscosity or in slightly regularized situations like some critical Navier conditions or the alpha-models equations -Vortex dynamics. We shall study both perfect and viscous incompressible fluids using mainly the vorticity equation. Our interest lies in singular domains (flow around rough obstacles for example) or in singularly pertubed domains (flow around small obstacles). In the two-dimensional case, the question of understanding the large time behaviour of perfect and viscous fluids will be also adressed. Another important direction of research will be the study of vortex filaments , the most challenging question being the rigorous understanding of the motion of vortex filaments in the vanishing viscosity limit (the expected asymptotic model is the binormal flow). -Fluid-structure interactions. We first plan to get a better understanding of qualitative properties of the fluid-structure interactions on the most simple models (incompressible fluids with rigid bodies). Typical questions that will be addressed are the uniqueness of weak solutions in 2D for viscous fluids and the study of the smoothness of particles trajectories. Some singular limits like vanishing viscosity limit, vanishing particles limit and mean field limit will be also studied. Finally we plan to make progress in the understanding of more complete models that take into account for example deformable solids or compressible fluids.

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.

Recently, new methods have appeared in representation theory and quantum topology, based on the notion of categorical representations of Kac-Moody algebras. They have already had remarkable applications. In representation theory, categorifications give a proof of important particular cases of Broué’s conjecture on modular representations of finite reductive groups, they give a better understanding of the Lusztig conjecture on modular representations of algebraic groups, they yield a proof of a character formula for simple modules in the category O of cyclotomic rational double affine Hecke algebras. In topology, categorifications give new knots invariants which are richer than the Reshetikhin-Turaev invariants, such as Khovanov homology, which permit, in particular, to detect the trivial knot. New types of categorical representations, such as categorical representations of Heisenberg algebras, have also appeared but are not well understood yet. Inspired by physicists’ works on gauge theory, a new approach to knots invariants which is based on the representation theory of double affine Hecke algebras has been proposed. On the other hand, representation theory of quantum groups has taken a proeminent role in computing the equivariant quantum cohomology or K-theory of some standard varieties such as the flag manifolds or the Nakajima’s quiver varieties. Although the relation between quantum cohomology of flag manifolds and Toda lattices appeared at the very begining of the theory in Givental’s works, the relations between Bethe algebras, double affine Hecke algebras, symplectic duality and equivariant quantum cohomology or K-theory is still unclear. The aim of the project is to put together specialists of categorical methods in different domains of representation theory with topologists and geometers so that all the members of the team will profit of mutual interactions. This project will gather 10 mathematicians from 9 universities and it will give them the possibility to organize meetings, collaborations and invitations in order to progress in the understanding of these new structures. In order to promote this research area among young mathematicians, we’ll propose a PhD position. We’ll investigate potential applications of categorifications to finite dimensional representations of quantum affine algebras or to fusion data associated with finite reductive groups, as well as to invariants of knots and 3-manifolds in quantum topology or to equivariant quantum cohomology and K-theory of flag varieties and quiver varieties in geometry.

FATOU is a project in pure mathematics, in the field of dynamical systems. Its purpose is the study of phase space and parameter spaces of holomorphic dynamical systems in several complex variables. Recent developpements in the field suggest that those dynamical systems can have fundamentally different properties than holomorphic dynamical systems in one complex variable, such as wandering Fatou components or open sets of bifurcations in parameter spaces. The ambition of this project is to consider these problems under a new scope. Classically, both the phase and parameter spaces can be described as the union of a stable part and a chaotic part. It has been a major problem in modern dynamics to study to precisely how we can dynamicaly describe this dichotomy. In the context of holomorphic dynamical systems, the description of both parameter and dynamical spaces is somehow more tricky. As examples, one may cite for example the filtration of the chaotic part of the dynamics of an endomorphism f of P^k given by its Green currents and, similarly, the filtration of the set of rational maps of the Riemann sphere which are not structurally-stable on their Julia set induced by the bifurcation currents. The main idea of this project is to gather young researchers from dynamics in several complex variables, and from parameter spaces in one complex variable dynamics to study both the topological dynamics of holomorphic and birational maps of P^k and parameter space phenomena for holomorphic families of rational maps of P^k. More precisely, we wish to : - study the structure of the omega-limit set of wandering Fatou components of endomorphisms of P^k and identify sufficient arithmetic or metric conditions ensuring the absence of wandering domains, - enlight unexpectedly sophisticated dynamical behavior for birational maps, - understand new phenomena responsible for open sets bifurcations and those responsible for stronger bifurcation phenomena in families of endomorphisms of P^k. The parent of the project is Amiens and the team consists in 7 young researchers having permanent positions and who are active in the field. Another objective of the project is to undertake actions promoting this field of research among a general audience (in particular university students) and to promote a fast homogenization of mathematical knowledge inside the group. For this, we will organize two internal meetings every year, which also will help us stay informed about the state of the art, and report about work in progress by the members. An international conference will be organized at the end of the project to review the most recent advances in the topic. The acronym FATOU was chosen as a reference to Pierre Fatou, one of the historical important researchers in the field in the early twentieth century.

Many models exist that describe the emission of greenhouse gases such as CO2 and N2O from soils. However, these global models need improvements to yield more accurate predictions. Indeed these models are ignoring important microscopic aspects of soils, in particular their high level of heterogeneity at the microbial habitat and pore scale, caused by soil structure which can lead to a spatial disconnection between soil carbon and nitrogen, oxygen and the microrganisms. Micro-scale processes that occur within the pores in soils affect phenomena at much larger spatial and temporal scales. New inputs and parameters are needed for the soil compartment in the global “circulation” models used by climatologists to predict future climate patterns. Most microbial degradation models developed in soil science use empirical functions, also called “reduction functions”. They take into account the different environmental factors that affect microbial functions such as biodegradation, denitrification or nitrification. Among these different factors, those linked to temperature and water content are conventionally used and accepted. However this type of approach cannot describe well the complex interactions that occur between processes. Therefore, such interactions need to be better represented in biogeochemical models for more reliable simulations. A recent alternative approach to the simulation of microbial degradation of organic matter is the "Bottom-Up" approach, based on an explicit description of the soil pore space at the small scale, that of the microbial habitats, and of the processes taking place therein. Innovative modeling tools have been developed at scales directly relevant to microorganisms. Emergence can be captured from the diversity of scenarios that can be run from these models and that would have been much more laborious to carry out experimentally. In parallel with the development of these 3D sophisticated models, technological advances have been made in the 3D visualization at the microscale. The Bottom-Up approach faces limitations due to the computational cost of describing the 3D heterogeneities of the soil at the µm scale to produce output at the centimeter column scale. Upscaling methods have been applied in the area of hydrology mainly to upscale water or solute transport properties taking into consideration the porous structure. However averaging methods used in soil physics or hydrology eliminate information that, in some situations like those involving microorganisms, appears essential. One of the challenges is to find a way to bring the micro-heterogeneities registered at the µ-scale to the soil profile using modeling and especially models of intermediate complexity between pore scale 3D models and existing field models. Revisiting upscaling methodology for soil microbial functions are essential to build more accurate soil models of microbial functions. Our previous MEPSOM project (ANR, 2009-2013) showed how soil physical characteristics control the decomposition of organic substrates. It has developed a suite of methods and models to visualize in 3D soil heterogeneity at scales relevant for microorganisms. The goal of this new project is now to go further by using the 3D models resulting from Mepsom to upscale heterogeneities identified at the scale of microhabitats to the soil profile scale. In Soilµ-3D project, MEPSOM’s 3D models will pass the baton to simpler models able to run at the field scale for a better prediction of organic matter decomposition, nitrous oxide emission and organic pollutants impacted by climate and environmental changes. The general question we intend to answer in the proposed research is whether information on the spatial heterogeneity of soils at the microscale can be used to predict the processes observed at the macroscale in soils.