The project CRISIS is conceived to to give rigorous mathematical grounds to the description of heterogeneity effects, singular behaviours and multi-scale processes which occur and interplay in several real world phenomena related to fluid mechanics, and improve their theoretical understanding. We have in mind situations which range from the context of geophysical flows to the one of collective dynamics. Heterogeneities are ubiquitous in nature: they may appear in complex phenomena, as a result of the interaction of different agents and components (e.g. density or temperature variations, presence of a magnetic field), or at the boundary of the physical domain, where interactions with the external world occur. The key observation is that non-homogeneities are often responsible for the emergence of singular structures. The word "singular" refers here to some lack of regularity, typically due to high concentrations and/or fast oscillations, which may involve, for instance, the vorticity of the fluid (think e.g. to typhoons), the fluid density (like in mixtures, in multi-phase flows or in crowd dynamics), or defects in complex fluids (like liquid crystals). In all those cases, some quantity is strongly localised in some region of space, and experiences a jump across the interface delimiting that region. In many situations the interface is simply advected by the flow, but in some cases of interest (arising e.g. in saturation phenomena) it presents a free boundary, meaning that some mass exchanges may happen between the two regions separated by the interface. For applications, it is of upmost importance to give a qualitative description of the dynamics of the interface; yet, the mathematical difficulties for doing that are deep and intricate, since this requires to go beyond the classical well-posedness theory for non-linear PDEs. Another fundamental observation, which stands at the core the project CRISIS, is that heterogeneities may appear also in multi-scale processes, as a result of the interplay between several factors, which contribute to the mean motion by acting at different scales (in time and/or in space). Such a complexity makes the derivation of reduced models necessary, both for theoretical and application purposes. From the physical viewpoint, the importance of the various factors is assessed by dimensional analysis, by introducing some adimensional parameters, whose values may be very large or very small, depending on the specific considered regime. The usual way for obtaining reduced models consists in performing asymptotic expansions with respect to those parameters, and retaining only the leading order terms in the equations; yet, this approach is merely formal. On the contrary, the rigorous justification of those approximations generates several mathematical difficulties: first of all, one is led to facing singular perturbation problems; in addition, new analytical tools are needed to capture, at the level of the limit equations, the multiple scales which interact in the system. In the context outlined above, the primary goals of the project CRISIS are: (G1) to give a better understanding of the effects due to heterogeneities on fluid motion; (G2) to improve the description of the dynamics of interfaces; (G3) to make progress in the understanding of the role of the boundary; (G4) to rigorously derive reduced models for fluid flows in presence of multiple scales. The proposed research plan focuses on several aspects of the mathematical theory of non-homogeneous fluids, and consists of three main working programmes: (WP1) the study of systems of non-homogeneous fluids in a low/critical regularity framework; (WP2) the analysis of certain free-boundary problems; (WP3) the study of singular limits in the context of geophysical flows.

Prions are lethal proteinaceous pathogens with major public-health risks due to their zoonotic and iatrogenic potential. They are composed of aggregated, misfolded conformers of the host-encoded prion protein that progressively deposit in the brain by a self-perpetuating reaction. The underlying molecular mechanisms of replication and tissue dissemination remain mostly elusive. Our objective is to model these processes entirely based on recent advances that prion aggregates are conformationally heterogeneous and dynamic rather than uniform and static. To achieve this, we will map in prion-infected brain the structural diversification-to-bioactivity/neurotoxicity landscape of prion assemblies in a spatiotemporal manner and mathematically build a multiscale model of diversification and lesion spreading. The goal is to generate an open access model capable of predicting the disease progression and identify key elementary process for therapeutics intervention and early diagnostics.

This research program deals with the quality enhancement of digital images. Specifically, we consider the increasingly common situations where several acquisitions of the same scene are available, possibly through a video sequence. An issue of growing importance is to fuse these images in order to get a single enhanced image. Another crucial question concerns the joint restoration of several images of the same sequence. Such an approach has two great advantages. First, going beyond the physical limitations of sensors becomes possible, in terms of dynamic range, resolution or signal- to-noise ratio. Second, the classical limitations of single shot imaging (blur, specular reflections, over or under-exposure, etc.) may be strongly attenuated. As a counterpart, multi-image restoration faces specific difficulties, the toughest of which are related to change detection, motion detection, outliers (aberrant pixels or regions) detection, inpainting, contrast and colour corrections. The proposed work program gathers several classical problems from image processing together with complex issues in image analysis and comparisons. It builds on various mathematical tools, mainly statistical estimation, optimization, stochastic modeling, variational approaches and optimal transport. Within this program, our main goal will be to develop reliable and efficient numerical algorithms for each of the studied restoration modalities.

GeoMod is a Collaborative International Research project between France and Germany. Contemporary model theory studies abstract properties of mathematical structures from the point of view of first-order logic. It tries to isolate combinatorial properties of definable sets such as the existence of certain configurations, or of rank functions, and to use these properties to obtains structural consequences. These may be algebraic or geometric in nature, and can be applied to specific structures such as Berkovich geometry, difference-differential algebraic geometry, additive combinatorics or Erdos geometry. A good example of a combinatorial configuration implying algebraic structure is the group configuration theorem which asserts that certain combinatorial/dimension theoretic patterns are necessarily induced by the existence of a group, and that moreover the structure of the groups which might give rise to this configuration is highly restricted. This result, which itself generalizes the coordinatization theorems of geometric algebra was given its definitive form for stable theories by Hrushovski in his 1986 PhD thesis and hereafter became one of the most powerful tools in geometric stability theory, used to resolve open problems in classification theory, in the proof of the trichotomy theorem for Zariski geometries, and thereby the crucial component of the model theoretic solution of the function field Mordell-Lang and number field Manin-Mumford conjectures. More recently, the group configuration theorem and its avatars have taken center stage in applications to combinatorics, for example in the work of Bays and Breuillard on extensions of the Elekes-Szabó theorem. The model theoretic study of valued fields provides another example of the confluence of “pure” stability theory and “applied” algebraic model theory. Abraham Robinson identified ACVF, the theory of algebraically closed nontrivially valued fields, as the model companion of the theory of valued fields already in 1959, and for most of the next half century the theory maintained an “applied” character distinct from the stability theory of “pure” model theory. However, in order to describe quotients of definable sets by definable equivalence relations (imaginaries) in valued fields, Haskell, Hrushovski and Macpherson were led to the theory of stable domination and the pure and applied strands merged. The deep connections between these approaches to the theory of valued fields further manifested themselves in the Hrushovski-Loeser approach to nonarchimedian geometry, in which spaces of stably dominated types replaced Berkovich spaces. Our project is structured around these three themes: First we aim to strengthen the still fairly recent relations between model theory and combinatorics. Secondly, we aim to develop the model theory of valued fields, a subject which has traditionally been very strong both in France and in Germany, but using the sophisticated tools of geometric stability (or neostability). Finally, we will develop a more abstract study of the geometric and combinatorial configurations which are a fundamental tool in the previous two subjects.

The aim of the proposed Austrian-French joint project "Arithmetic Randomness" (Aléa arithmétique, Zahlen und Zufall) is to make progress on various important open conjectures and questions in analytic, metric, probabilistic and additive number theory. All problems address random-like phenomena in arithmetic problems. The main areas of interest are distributional properties of prime numbers, its relations to number systems and problems in Diophantine approximation. As modern cryptographic systems are heavily based on prime numbers and pseudorandom numbers, it is important to understand the distribution of the underlying arithmetic quantities and to quantify its relation to randomness. One of the most striking open conjectures in this area is Sarnak's conjecture which currently attracts considerable interest. It states that the Möbius function is orthogonal to any sequence that is realized in a deterministic dynamical system. While still open in its full generality, considerable progress has been obtained with respect to special classes of sequences. We mention the solution of the conjecture by Mu¨llner (2017) for automatic sequences which, informally speaking, are sequences that relate the value of its terms to the digits of its base-q representation. Besides other tools, the work is based on the method of Mauduit and Rivat (2010) on the digits of primes. Sarnak's conjecture is closely related to Chowla's conjecture that in turn is the problem to study correlations of the Möbius function. In the last years, various breakthroughs have been obtained in these areas. We mention the results by Green and Tao (2012) concerning the orthogonality of nilsequences, by Bourgain, Sarnak and Ziegler (2013) concerning the disjointness of the horocycle flow, by Frantzikinakis and Host (2018) on the logarithm Sarnak conjecture for zero entropy topological systems with only countably many ergodic measures, by Green and Bourgain (2012-16) concerning the computational complexity of the Möbius function, by Bourgain (2017) on the number of prime numbers with preassigned digits, by Matomäki and Radziwill (2016) on multiplicative functions in short intervals, by Tao (2016) on a logarithmic version of the Chowla conjecture, and by Maynard (2019) about prime numbers with restricted digits. The proofs rely on tools of analytic number theory, geometry of numbers, Diophantine considerations as well as dynamical systems. Probabilistic and statistical aspects of Diophantine approximation play an important role in the study of the pair correlation of sequences, which are generally wide out of reach for deterministic sequences. Several new tools are available in the metric setting and, somewhat surprisingly, these problems have a close connection with the value distribution of the Riemann zeta function in the critical strip.