The Curry-Howard isomorphism between mathematical proofs and computer programs, initially limited to a purely functional notion of computations and intuitionistic logic, has been extended since then to new programming paradigms including computational features such as control operators. Interestingly, the addition of control operators to a programming language corresponds, through Curry-Howard isomorphism, to the addition of the principle of excluded-middle to the logic. This phenomenon is not isolated: adding axioms to a logical system correspond to the addition of computing primitives to a programming language. To be able to reason on more expressive programming languages, it is necessary to find the axioms that the corresponding logic should satisfy. Dually, studying the computational counterpart of a given axiom has shown to be particularly fruitful. We propose to apply this methodology to choice principles, that are ubiquitous in mathematics. The full axiom of choice has been deeply studied in numerous works in mathematics and can be expressed in different ways. Even though these variants are equivalent in a classical system (that is in a logic including the excluded-middle), this is no longer the case in a constructive setting. For instance, under common assumptions, the usual axiom of choice (in terms of quotient sets) is known to entails the law of excluded-middle, itself incompatible with many constructive settings. We can thus affirm that constructive theories do not capture the true nature and full strength of the axiom of choice. One of the mid-term goal of this project is then to wonder: what is the computational interpretation of the axiom of choice? This question actually stands for several restricted form of choices, such as König's lemma, whose importance has been emphasized by the program of reverse mathematics. This project can be seen as a first step to pave the way towards a computational point of view on constructive reverse mathematics.

This project concerns geometric topology, more precisely the study of smooth manifolds of dimension 4 and higher. Building on the recent theory of trisections of smooth 4--manifolds introduced by Gay and Kirby, the project investigates the relationship between this theory and symplectic geometry, and the possible generalization to higher dimensional manifolds. A Heegaard splitting is a decomposition of a compact 3-manifold into two handlebodies glued along a surface, a key notion in the study of 3-manifolds. Gay and Kirby developed an analogous construction for smooth compact 4-manifolds: they define a trisection of such a manifold as a decomposition into three 4-dimensional 1-handlebodies, with conditions on the gluing of the pieces. This project aims at exploring two aspects of trisections. The first objective is to study the relation between trisections and symplectic structures. This is motivated by the induced structures on the boundary of the manifold: a trisection induces an open book decomposition, a symplectic structure induces a contact structure, and the Giroux correspondence establishes a strong relation between open book decompositions and contact structures. The second objective is to generalize the theory of trisections to higher dimensions, studying first a notion of quadrisections for smooth compact 5-manifolds.

This project concerns basic research in fundamental mathematics, more specifically it combines geometry in a broad sense, low dimensional topology, group theory, dynamical systems, and geometric analysis. In recent years, the proof of Thurston's geometrization conjecture has considerably sharpened our understanding of 3-manifolds. Nevertheless, three main questions concerning the structure of a hyperbolic 3-manifold and its fundamental group still remain unanswered. The first two questions, known as the « virtually Haken » and « virtual fibration » conjectures, address the problem of understanding whether any given hyperbolic 3-manifold admits a finite cover containing an essential surface or, even better, admitting a surface fibration over the circle; the third one, Cannon's conjecture, deals with the dynamical characterisation of its fundamental group. These questions can be restated in more algebraic terms. For the first two, this boils down to asking whether a Kleinian group (cocompact or with finite covolume) contains a surface group (i.e. the manifold contains an immersed surface) enjoying certain separability properties (i.e. the immersion lifts to an embedding in some finite cover of the manifold) and possibly having specified limit set (i.e. the surface is virtually a fibre of a fibration over the circle). The third one can be tackled by looking for splittings of the fundamental group into elementary bricks. Motivated by the above questions that show the importance of understanding the subgroup structure of certain groups as well as their splittings, the aim of the present project is to develop techniques to detect the existence of special subgroups (surface subgroups, quasiconvex subgroups) in groups that are relevant in geometry or dynamics (hyperbolic and relatively hyperbolic groups, CAT(0) groups, and convergence groups) and to study their properties or to establish conditions under which certain properties hold. We note that the understanding of the subgroup structure of a group is strongly connected to the understanding of the splittings of the group. The main tools we wish to exploit to tackle these problems are: - Cubulations. We remark that cubulations have been exploited by Wise et al. and provide a setting in which the virtual fibration conjecture holds. - Lp-cohomology. Lp-cohomology was exploited by Bourdon in particular to detect splittings of hyperbolic groups. - Dynamics of the induced action on the boundary. Note that this is also relevant to the boundary characterisation of Kleinian groups and to Cannon's conjecture. Although proving these conjectures appears to be an extremely ambitious task, the hope is that the results obtained while carrying out this project will provide useful insight on the subject. Also, one can expect to understand whether results which are known to be valid for hyperbolic 3-manifolds extend to more general settings (hyperbolic and relatively hyperbolic groups, CAT(0) groups). One of the strong points of the project is that it brings together several mathematicians working in different fields, all relevant to the project, namely combinatorial and geometric group theory, low dimensional topology, hyperbolic geometry, conformal dynamics, etc. Members of the project are also among the authors of some important recent breakthroughs in the subject. A feature of the project will be to encourage members to share their specific expertise and acquire new knowledge at the same time. The project will be structured by biannual meetings having the three following purposes: providing background on each topic, presenting the main open questions and advances obtained, and setting up collaborations on the future tasks. This will be one of the scopes of the « ateliers » that will be organised twice a year.

Coupled fluid-porous flow problems appear in a wide range of environmental settings and industrial applications. However, existing interface conditions are mainly limited to one-dimensional non-inertial flows that significantly restricts the amount of applications which can be accurately simulated. In the project, new generalised and computable macroscale models for multi-dimensional non-inertial and inertial flows in coupled single-phase fluid-porous systems will be developed, validated and analysed. The global dissipation and global well-posedness without restriction on the size of data for the resulting coupled problems with the generalised interface conditions will be proved. A thorough comparison study of the developed models and other models available in the literature will be done. Robust and efficient numerical methods for the derived problems will be developed and analysed. The generalised coupling conditions will be used to simulate several applications.

Rapid adaptation to environmental changes is a key requirement for all living organisms. In the last decades, regulatory RNAs have emerged as tremendous players in such adaptative responses by controlling gene expression. In Escherichia coli and other bacteria, small regulatory RNAs (sRNAs) were shown to play crucial roles in a wide variety of pathways, such as stress response and control of metabolism, up to promoting antibiotic resistance. While the molecular mechanisms of action of sRNAs are increasingly understood, an important question remains as to the specific properties of RNA regulation and their advantage in respect to proteinaceous regulators such as transcription factors. Due to their rapid synthesis and action, sRNAs have been proposed to allow a fast and dynamic control of gene expression that is crucial to adapt to sudden environmental changes. However, kinetics aspects of gene regulation by sRNAs and their consequences on cell physiology are still insufficiently documented, mainly because of technical difficulties to directly observe sRNA regulation and their phenotypic consequences in vivo and in real time. In this ever-rising field, we recently made the key observation that, in response to iron starvation, the sRNA RyhB triggers Escherichia coli phenotypic resistance to a major class of antibiotics. This phenotype is due to the regulation by RyhB of a life essential process, Fe-S cluster biogenesis. Strikingly, we newly found that two other sRNAs, OxyS and FnrS, control Fe-S cluster biogenesis in stress conditions encountered during the infectious process, oxidative stress and anaerobic conditions, respectively. Thanks to our previous work and numerous solid data gathered by our consortium, we propose that regulation of Fe-S clusters biogenesis by this sRNA triad is essentially dynamic and serves to accelerate and provide a rapid adaptation to stress conditions. In this way, dynamic regulation by sRNAs will profoundly affect cell physiology up to modifying bacterial antibiotics resistance. In addition to revealing new regulators of Fe-S clusters biogenesis, the Kinebiotics project directly addresses the important question of the dynamics of sRNA regulation and their consequences on cell physiology. To tackle this question, the Kinebiotics project takes advantage of the multiple expertise of our partnership to build an integrated and multidisciplinary approach. The project combines the extraordinary genetic amenability of E. coli, robust molecular biology approaches, cutting edge microfluidic characterization and powerful mathematical formalism. Altogether, the Kinebiotics project will provide an unprecedented view on how sRNAs modulate gene expression during stress directly correlated with their effect on cell physiology. While challenging, our goal has a high breakthrough potential beyond the microbiology field as it will make a pivotal impact in the understanding of sRNA functions applicable to all living systems.