ERC FUNDED PROJECTS

"Computer vision is concerned with the automated interpretation of images and video streams. Today's research is (mostly) aimed at answering queries such as ""Is this a picture of a dog?"", (classification) or sometimes ""Find the dog in this photo"" (detection). While categorisation and detection are useful for many tasks, inferring correct class labels is not the final answer to visual recognition. The categories and locations of objects do not provide direct understanding of their function i.e., how things work, what they can be used for, or how they can act and react. Such an understanding, however, would be highly desirable to answer currently unsolvable queries such as ""Am I in danger?"" or ""What can happen in this scene?"". Solving such queries is the aim of this proposal. My goal is to uncover the functional properties of objects and the purpose of actions by addressing visual recognition from a different and yet unexplored perspective. The main novelty of this proposal is to leverage observations of people, i.e., their actions and interactions to automatically learn the use, the purpose and the function of objects and scenes from visual data. The project is timely as it builds upon the two key recent technological advances: (a) the immense progress in visual recognition of objects, scenes and human actions achieved in the last ten years, as well as (b) the emergence of a massive amount of public image and video data now available to train visual models. ACTIVIA addresses fundamental research issues in automated interpretation of dynamic visual scenes, but its results are expected to serve as a basis for ground-breaking technological advances in practical applications. The recognition of functional properties and intentions as explored in this project will directly support high-impact applications such as detection of abnormal events, which are likely to revolutionise today's approaches to crime protection, hazard prevention, elderly care, and many others."

1 497 420 €

Start date: 2013-01-01, End date: 2018-12-31

The goal of this project is to investigate two conjectures in arithmetic geometry pertaining to the geometry of projective varieties over finite and number fields. These two conjectures, formulated by Tate and Grothendieck in the 1960s, predict which cohomology classes are chern classes of line bundles. They both form an arithmetic counterpart of a theorem of Lefschetz, proved in the 1940s, which itself is the only known case of the Hodge conjecture. These two long-standing conjectures are one of the aspects of a more general web of questions regarding the topology of algebraic varieties which have been emphasized by Grothendieck and have since had a central role in modern arithmetic geometry. Special cases of these conjectures, appearing for instance in the work of Tate, Deligne, Faltings, Schneider-Lang, Masser-Wüstholz, have all had important consequences. My goal is to investigate different lines of attack towards these conjectures, building on recent work on myself and Jean-Benoît Bost on related problems. The two main directions of the proposal are as follows. Over finite fields, the Tate conjecture is related to finiteness results for certain cohomological objects. I want to understand how to relate these to hidden boundedness properties of algebraic varieties that have appeared in my recent geometric proof of the Tate conjecture for K3 surfaces. The existence and relevance of a theory of Donaldson invariants for moduli spaces of twisted sheaves over finite fields seems to be a promising and novel direction. Over number fields, I want to combine the geometric insight above with algebraization techniques developed by Bost. In a joint project, we want to investigate how these can be used to first understand geometrically major results in transcendence theory and then attack the Grothendieck period conjecture for divisors via a number-theoretic and complex-analytic understanding of universal vector extensions of abelian schemes over curves.

1 222 329 €

Start date: 2016-12-01, End date: 2021-11-30

"During the past twenty years, we have witnessed profound technological changes, summarised under the terms of digital revolution or entering the information age. It is evident that these technological changes will have a deep societal impact, and questions of privacy and security are primordial to ensure the survival of a free and open society. Cryptology is a main building block of any security solution, and at the heart of projects such as electronic identity and health cards, access control, digital content distribution or electronic voting, to mention only a few important applications. During the past decades, public-key cryptology has established itself as a research topic in computer science; tools of theoretical computer science are employed to “prove” the security of cryptographic primitives such as encryption or digital signatures and of more complex protocols. It is often forgotten, however, that all practically relevant public-key cryptosystems are rooted in pure mathematics, in particular, number theory and arithmetic geometry. In fact, the socalled security “proofs” are all conditional to the algorithmic untractability of certain number theoretic problems, such as factorisation of large integers or discrete logarithms in algebraic curves. Unfortunately, there is a large cultural gap between computer scientists using a black-box security reduction to a supposedly hard problem in algorithmic number theory and number theorists, who are often interested in solving small and easy instances of the same problem. The theoretical grounds on which current algorithmic number theory operates are actually rather shaky, and cryptologists are generally unaware of this fact. The central goal of ANTICS is to rebuild algorithmic number theory on the firm grounds of theoretical computer science."

1 453 507 €

Start date: 2012-01-01, End date: 2016-12-31

Boundary layer theory is a large component of fluid dynamics. It is ubiquitous in Oceanography, where boundary layer currents, such as the Gulf Stream, play an important role in the global circulation. Comprehending the underlying mechanisms in the formation of boundary layers is therefore crucial for applications. However, the treatment of boundary layers in ocean dynamics remains poorly understood at a theoretical level, due to the variety and complexity of the forces at stake. The goal of this project is to develop several tools to bridge the gap between the mathematical state of the art and the physical reality of oceanic motion. There are four points on which we will mainly focus: degeneracy issues, including the treatment Stewartson boundary layers near the equator; rough boundaries (meaning boundaries with small amplitude and high frequency variations); the inclusion of the advection term in the construction of stationary boundary layers; and the linear and nonlinear stability of the boundary layers. We will address separately Ekman layers and western boundary layers, since they are ruled by equations whose mathematical behaviour is very different. This project will allow us to have a better understanding of small scale phenomena in fluid mechanics, and in particular of the inviscid limit of incompressible fluids. The team will be composed of the PI, two PhD students and three two-year postdocs over the whole period. We will also rely on the historical expertise of the host institution on fluid mechanics and asymptotic methods.

1 267 500 €

Start date: 2015-09-01, End date: 2020-08-31

"The purpose of this project is to use the ubiquitous nature of certain combinatorial topological objects called maps in order to unveil deep connections between several areas of mathematics. Maps, that describe the embedding of a graph into a surface, appear in probability theory, mathematical physics, enumerative geometry or graph theory, and different combinatorial viewpoints on these objects have been developed in connection with each topic. The originality of our project will be to study these approaches together and to unify them. The outcome will be triple, as we will: 1. build a new, well structured branch of combinatorics of which many existing results in different areas of enumerative and algebraic combinatorics are only first fruits; 2. connect and unify several aspects of the domains related to it, most importantly between probability and integrable hierarchies thus proposing new directions, new tools and new results for each of them; 3. export the tools of this unified framework to reach at new applications, especially in random graph theory and in a rising domain of algebraic combinatorics related to Tamari lattices. The methodology to reach the unification will be the study of some strategic interactions at different places involving topological expansions, that is to say, places where enumerative problems dealing with maps appear and their genus invariant plays a natural role, in particular: 1. the combinatorial theory of maps developped by the "French school" of combinatorics, and the study of random maps; 2. the combinatorics of Fermions underlying the theory of KP and 2-Toda hierarchies; 3; the Eynard-Orantin "topological recursion" coming from mathematical physics. We present some key set of tasks in view of relating these different topics together. The pertinence of the approach is demonstrated by recent research of the principal investigator."

1 086 125 €

Start date: 2017-03-01, End date: 2022-02-28

A contact structure on an odd dimensional manifold in a maximally non integrable hyperplane field. It is the odd dimensional counterpart of a symplectic structure. Contact and symplectic topology is a recent and very active area that studies intrinsic questions about existence, (non) uniqueness and rigidity of contact and symplectic structures. It is intimately related to many other important disciplines, such as dynamical systems, singularity theory, knot theory, Morse theory, complex analysis, ... Legendrian submanifolds are a distinguished class of submanifolds in a contact manifold, which are tangent to the contact distribution. These manifolds are of a particular interest in contact topology. Important classes of Legendrian submanifolds can be described using generating families, and this description can be used to define Legendrian invariants via Morse theory. Other the other hand, Legendrian contact homology is an invariant for Legendrian submanifolds, based on holomorphic curves. The goal of this research proposal is to study the relationship between these two approaches. More precisely, we plan to show that the generating family homology and the linearized Legendrian contact homology can be defined for the same class of Legendrian submanifolds, and are isomorphic. This correspondence should be established using a parametrized version of symplectic homology, being developed by the Principal Investigator in collaboration with Oancea. Such a result would give an entirely new type of information about holomorphic curves invariants. Moreover, it can be used to obtain more general structural results on linearized Legendrian contact homology, to extend recent results on existence of Reeb chords, and to gain a much better understanding of the geography of Legendrian submanifolds.

710 000 €

Start date: 2009-11-01, End date: 2014-10-31

One of the main challenges in condensed matter physics is to understand strongly correlated quantum systems. Our purpose is to approach this issue from the point of view of rigorous mathematical analysis. The goals are twofold: develop a mathematical framework applicable to physically relevant scenarii, take inspiration from the physics to introduce new topics in mathematics. The scope of the proposal thus goes from physically oriented questions (theoretical description and modelization of physical systems) to analytical ones (rigorous derivation and analysis of reduced models) in several cases where strong correlations play the key role. In a first part, we aim at developing mathematical methods of general applicability to go beyond mean-field theory in different contexts. Our long-term goal is to forge new tools to attack important open problems in the field. Particular emphasis will be put on the structural properties of large quantum states as a general tool. A second part is concerned with so-called fractional quantum Hall states, host of the fractional quantum Hall effect. Despite the appealing structure of their built-in correlations, their mathematical study is in its infancy. They however constitute an excellent testing ground to develop ideas of possible wider applicability. In particular, we introduce and study a new class of many-body variational problems. In the third part we discuss so-called anyons, exotic quasi-particles thought to emerge as excitations of highly-correlated quantum systems. Their modelization gives rise to rather unusual, strongly interacting, many-body Hamiltonians with a topological content. Mathematical analysis will help us shed light on those, clarifying the characteristic properties that could ultimately be experimentally tested.

1 056 664 €

Start date: 2018-01-01, End date: 2022-12-31

Statistical physics is a theory allowing the derivation of the statistical behavior of macroscopic systems from the description of the interactions of their microscopic constituents. For more than a century, lattice models (i.e. random systems defined on lattices) have been introduced as discrete models describing the phase transition for a large variety of phenomena, ranging from ferroelectrics to lattice gas. In the last decades, our understanding of percolation and the Ising model, two classical exam- ples of lattice models, progressed greatly. Nonetheless, major questions remain open on these two models. The goal of this project is to break new grounds in the understanding of phase transition in statistical physics by using and aggregating in a pioneering way multiple techniques from proba- bility, combinatorics, analysis and integrable systems. In this project, we will focus on three main goals: Objective A Provide a solid mathematical framework for the study of universality for Bernoulli percolation and the Ising model in two dimensions. Objective B Advance in the understanding of the critical behavior of Bernoulli percolation and the Ising model in dimensions larger or equal to 3. Objective C Greatly improve the understanding of planar lattice models obtained by general- izations of percolation and the Ising model, through the design of an innovative mathematical theory of phase transition dedicated to graphical representations of classical lattice models, such as Fortuin-Kasteleyn percolation, Ashkin-Teller models and Loop models. Most of the questions that we propose to tackle are notoriously difficult open problems. We believe that breakthroughs in these fundamental questions would reshape significantly our math- ematical understanding of phase transition.

1 499 912 €

Start date: 2018-09-01, End date: 2023-08-31

Designers draw extensively to externalize their ideas and communicate with others. However, drawings are currently not directly interpretable by computers. To test their ideas against physical reality, designers have to create 3D models suitable for simulation and 3D printing. However, the visceral and approximate nature of drawing clashes with the tediousness and rigidity of 3D modeling. As a result, designers only model finalized concepts, and have no feedback on feasibility during creative exploration. Our ambition is to bring the power of 3D engineering tools to the creative phase of design by automatically estimating 3D models from drawings. However, this problem is ill-posed: a point in the drawing can lie anywhere in depth. Existing solutions are limited to simple shapes, or require user input to “explain” to the computer how to interpret the drawing. Our originality is to exploit professional drawing techniques that designers developed to communicate shape most efficiently. Each technique provides geometric constraints that help viewers understand drawings, and that we shall leverage for 3D reconstruction. Our first challenge is to formalize common drawing techniques and derive how they constrain 3D shape. Our second challenge is to identify which techniques are used in a drawing. We cast this problem as the joint optimization of discrete variables indicating which constraints apply, and continuous variables representing the 3D model that best satisfies these constraints. But evaluating all constraint configurations is impractical. To solve this inverse problem, we will first develop forward algorithms that synthesize drawings from 3D models. Our idea is to use this synthetic data to train machine learning algorithms that predict the likelihood that constraints apply in a given drawing. In addition to tackling the long-standing problem of single-image 3D reconstruction, our research will significantly tighten design and engineering for rapid prototyping.

1 482 761 €

Start date: 2017-02-01, End date: 2022-01-31

We propose to radically extend the frontiers of two major themes in computing, parallelism and dynamism, and develop a novel paradigm of computing: dynamic-parallelism. To this end, we will follow two lines of research. First, we will develop techniques for extracting efficiency and high performance from parallel programs written in high-level programming languages. Second, we will develop the dynamic-parallelism model, where computations can respond to a wide variety of dynamic changes to their data automatically and efficiently, by developing novel abstractions (calculi), high-level programming-language constructs, and compilation techniques. The research will culminate in a language that extends the C programming language with support for parallel and dynamic-parallel programming. The proposal is motivated by urgent needs driven by the advent of multicore chips, which is making parallelism mainstream, and the increasing ubiquity of software, which requires applications to operate on highly dynamic data. These advances demand parallel and highly dynamic software, which remains too difficult and labor intensive to develop. The urgency is further underlined by the increasing data and problem sizes---online data grows exponentially, doubling every few years---that require similarly powerful advances in performance. The proposal will achieve profound impact by dramatically simplifying the development of high-performing dynamic and dynamic-parallel software. As a result, programmer productivity and software quality including correctness, reliability, performance, and resource (e.g., time and energy) consumption will improve significantly. The proposal will not only open new research opportunities in parallel computing, programming languages, and compilers, but also in other fields where parallel and dynamic problems abound, e.g., algorithms, computational biology, geometry, graphics, machine learning, and software systems.

1 076 570 €

Start date: 2013-06-01, End date: 2018-05-31

Recent hardware developments from the medical device manufacturers have made possible non-invasive/in-vivo acquisition of anatomical and physiological measurements. One can cite numerous emerging modalities (e.g. PET, fMRI, DTI). The nature (3D/multi-phase/vectorial) and the volume of this data make impossible in practice their interpretation from humans. On the other hand, these modalities can be used for early screening, therapeutic strategies evaluation as well as evaluating bio-markers for drugs development. Despite enormous progress made on the field of biomedical image analysis still a huge gap exists between clinical research and clinical use. The aim of this proposal is three-fold. First we would like to introduce a novel biomedical image perception framework for clinical use towards disease screening and drug evaluation. Such a framework is expected to be modular (can be used in various clinical settings), computationally efficient (would not require specialized hardware), and can provide a quantitative and qualitative anatomo-pathological indices. Second, leverage progress made on the field of machine learning along with novel, efficient, compact representation of clinical bio-markers toward computer aided diagnosis. Last, using these emerging multi-dimensional signals, we would like to perform longitudinal modelling and understanding the effects of aging to a number of organs and diseases that do not present pre-disease indicators such as brain neurological diseases, muscular diseases, certain forms of cancer, etc. Such a challenging and pioneering effort lies on the interface of medicine (clinical context), biomedical imaging (choice of signals/modalities), machine learning (manifold representations of heterogeneous multivariate variables), discrete optimization (computationally efficient inference of higher-order models), and bio-medical image inference (measurement extraction and multi-modal fusion of heterogeneous information sources).

1 500 000 €

Start date: 2011-09-01, End date: 2016-08-31

Forests, of which globally 70% are managed, play a particularly important role in the global carbon cycle. Recently, forest management became a top priority on the agenda of the political negotiations to mitigate climate change because forest plantations may remove atmospheric CO2 and if used for energy production, the wood is a substitute for fossil fuel. However, this political imperative is at present running well ahead of the science required to deliver it. Despite the key implications of forest management on: 1) the carbon-energy-water balance, and 2) production, recreation and environmental protection, there are no integrated studies of its effects on the Earth s climate. The overall goal of DOFOCO is to quantify and understand the role of forest management in mitigating climate change. Specifically, I want to challenge the current focus on the carbon cycle and replace it with a total climate impact approach. Hence, the whole forest management spectrum ranging from short rotation coppice to old-growth forests will be analyzed for its effects on the water, energy and carbon cycles. Climate response of forest will be quantified by means of albedo, evapotranspiration, greenhouse gas sources and sinks and their resulting climate feedback mechanisms. The anticipated new quantitative results will be used to lay the foundations for a portfolio of management strategies which will sustain wood production while minimizing climate change impacts. DOFOCO is interdisciplinary and ground breaking because it brings together state-of-the art data and models from applied life and Earth system sciences; it will deliver the first quantitative insights into how forest management strategies can be linked to climate change mitigation.

1 296 125 €

Start date: 2010-02-01, End date: 2015-10-31

"Our aim is to built on recent combinatorial and algorithmic progress to attack a series of deeply connected problems that have independantly surfaced in enumerative topology, statistical physics, and data compression. The relation between these problems lies in the notion of ""combinatorial map"", the natural discrete mathematical abstraction of objects with a 2-dimensional structures (like geographical maps, computer graphics' meshes, or 2d manifolds). A whole new set of properties of these maps has been uncovered in the last few years under the impulsion of the principal investigator. Rougly speaking, we have shown that classical graph exploration algorithms, when correctly applied to maps, lead to remarkable decompositions of the underlying surfaces. Our methods resort to algorithmic and enumerative combinatorics. In statistical physics, these decompositions offer an approach to the intrinsec geometry of discrete 2d quantum gravity: our method is here the first to outperform the celebrated ""topological expansion of matrix integrals"" of Brezin-Itzykson-Parisi-Zuber. Exploring its implications for the continuum limit of these random geometries is our great challenge now. From a computational geometry perspective, our approach yields the first encoding schemes with asymptotically optimal garanteed compression rates for the connectivity of triangular or polygonal meshes. These schemes improve on a long series of heuristically efficient but non optimal algorithms, and open the way to optimally compact data structures. Finally we have deep indications that the properties we have uncovered extend to the realm of ramified coverings of the sphere. Intriguing computations on the fundamental Hurwitz's numbers have been obtained using the ELSV formula, famous for its use by Okounkov et al. to rederive Kontsevich's model. We believe that further combinatorial progress here could allow to bypass the formula and obtaine an elementary explanation of these results."

750 000 €

Start date: 2008-07-01, End date: 2013-06-30

We propose to lay the theoretical foundations and design efficient computational methods for analyzing, quantifying and exploring relations and variability in structured data sets, such as collections of geometric shapes, point clouds, and large networks or graphs, among others. Unlike existing methods that are tied and often limited to the underlying data representation, our goal is to design a unified framework in which variability can be processed in a way that is largely agnostic to the underlying data type. In particular, we propose to depart from the standard representations of objects as collections of primitives, such as points or triangles, and instead to treat them as functional spaces that can be easily manipulated and analyzed. Since real-valued functions can be defined on a wide variety of data representations and as they enjoy a rich algebraic structure, such an approach can provide a completely novel unified framework for representing and processing different types of data. Key to our study will be the exploration of relations and variability between objects, which can be expressed as operators acting on functions and thus treated and analyzed as objects in their own right using the vast number of tools from functional analysis in theory and numerical linear algebra in practice. Such a unified computational framework of variability will enable entirely novel applications including accurate shape matching, efficiently tracking and highlighting most relevant changes in evolving systems, such as dynamic graphs, and analysis of shape collections. Thus, it will permit not only to compare or cluster objects, but also to reveal where and how they are different and what makes instances unique, which can be especially useful in medical imaging applications. Ultimately, we expect our study to create to a new rigorous, unified paradigm for computational variability, providing a common language and sets of tools applicable across diverse underlying domains.

1 499 845 €

Start date: 2018-01-01, End date: 2022-12-31

Public key cryptography is the backbone of internet security. Yet it is very likely that within the next few decades some government or corporate entity will succeed in building a general-purpose quantum computer that is capable of breaking all of today's public key protocols. Lattice cryptography, which appears to be resilient to quantum attacks, is currently viewed as the most promising candidate to take over as the basis for cryptography in the future. Recent theoretical breakthroughs have additionally shown that lattice cryptography may even allow for constructions of primitives with novel capabilities. But even though the progress in this latter area has been considerable, the resulting schemes are still extremely impractical. The central objective of the FELICITY project is to substantially expand the boundaries of efficient lattice-based cryptography. This includes improving on the most crucial cryptographic protocols, some of which are already considered practical, as well as pushing towards efficiency in areas that currently seem out of reach. The methodology that we propose to use differs from the bulk of the research being done today. Rather than directly working on advanced primitives in which practical considerations are ignored, the focus of the project will be on finding novel ways in which to break the most fundamental barriers that are standing in the way of practicality. For this, I believe it is productive to concentrate on building schemes that stand at the frontier of what is considered efficient -- because it is there that the most critical barriers are most apparent. And since cryptographic techniques usually propagate themselves from simple to advanced primitives, improved solutions for the fundamental ones will eventually serve as building blocks for practical constructions of schemes having advanced capabilities.

1 311 688 €

Start date: 2015-10-01, End date: 2020-09-30

The future of the computing technology relies on fast access, transformation, and exchange of data across large-scale networks such as the Internet. The design of software systems that support high-frequency parallel accesses to high-quantity data is a fundamental challenge. As more scalable alternatives to traditional relational databases, distributed data structures (DDSs) are at the basis of a wide range of automated services, for now, and for the foreseeable future. This proposal aims to improve our understanding of the theoretical foundations of DDSs. The design and the usage of DDSs are based on new principles, for which we currently lack rigorous engineering methodologies. Specifically, we lack design procedures based on precise specifications, and automated reasoning techniques for enhancing the reliability of the engineering process. The targeted breakthrough of this proposal is developing automated formal methods for rigorous engineering of DDSs. A first objective is to define coherent formal specifications that provide precise requirements at design time and explicit guarantees during their usage. Then, we will investigate practical programming principles, compatible with these specifications, for building applications that use DDSs. Finally, we will develop efficient automated reasoning techniques for debugging or validating DDS implementations against their specifications. The principles underlying automated reasoning are also important for identifying best practices in the design of these complex systems to increase confidence in their correctness. The developed methodologies based on formal specifications will thus benefit both the conception and automated validation of DDS implementations and the applications that use them.

1 300 000 €

Start date: 2016-05-01, End date: 2021-04-30

This proposal aims at generalising the central decidability results of Büchi and Rabin to more general settings, and understanding the consequences of those extensions both at theoretical and applicative levels. The original results of Büchi and Rabin state the decidability of monadic second-order logic (monadic logic for short) over infinite words and trees respectively. Those results are of such importance that Rabin's theorem is also called the `Mother of all decidability results'. The primary goal of this project is to demonstrate that it is possible to go significantly beyond the expressiveness of monadic logic, while retaining similar decidability results. We are considering extensions in two distinct directions. The first consists of enriching the logic with the ability to speak in a weak form about set cardinality. The second direction is an extension of monadic logic by topological capabilities. Those two branches form the core of the proposal. The second aspect of this proposal is the study of the `applicability' of this theory. Three tasks are devoted to this. The first task is to precisely determine the cost of using the more complex techniques we will be developing rather than using the classical theory. The second task will be devoted to the description and the study of weaker formalisms allowing better complexities, namely corresponding temporal logics. Finally, the last task will be devoted to the study of related model checking problems. The result of the completion of this program would be twofold. At a theoretical level, new deep results would be obtained, and new techniques in automata, logic and games would be developed for solving them. At a more applicative level, this program would define and validate new directions of research in the domain of the verification of open systems and related problems.

931 760 €

Start date: 2011-01-01, End date: 2015-12-31

"The aim of this project of 5 years is to create a research group on geometric control methods in PDEs with the arrival of the PI at the CNRS Laboratoire CMAP (Centre de Mathematiques Appliquees) of the Ecole Polytechnique in Paris (in January 09). With the ERC-Starting Grant, the PI plans to hire 4 post-doc fellows, 2 PhD students and also to organize advanced research schools and workshops. One of the main purpose of this project is to facilitate the collaboration with my research group which is quite spread across France and Italy. The PI plans to develop a research group studying certain PDEs for which geometric control techniques open new horizons. More precisely the PI plans to exploit the relation between the sub-Riemannian distance and the properties of the kernel of the corresponding hypoelliptic heat equation and to study controllability properties of the Schroedinger equation. In the last years the PI has developed a net of high level international collaborations and, together with his collaborators and PhD students, has obtained many important results via a mixed combination of geometric methods in control (Hamiltonian methods, Lie group techniques, conjugate point theory, singularity theory etc.) and noncommutative Fourier analysis. This has allowed to solve open problems in the field, e.g., the definition of an intrinsic hypoelliptic Laplacian, the explicit construction of the hypoelliptic heat kernel for the most important 3D Lie groups, and the proof of the controllability of the bilinear Schroedinger equation with discrete spectrum, under some ""generic"" assumptions. Many more related questions are still open and the scope of this project is to tackle them. All subjects studied in this project have real applications: the problem of controllability of the Schroedinger equation has direct applications in Nuclear Magnetic Resonance; the problem of nonisotropic diffusion has applications in models of human vision."

785 000 €

Start date: 2010-05-01, End date: 2016-04-30

I intend to cross ressources of holomorphic curves techniques and traditional topological methods to study some fundamental questions in symplectic and contact geometry such as: - The Weinstein conjecture in dimension greater than 3. - The construction of new invariants for both smooth manifolds and Legendrian/contact manifolds, in particular, try to define an analogue of Heegaard Floer homology in dimension larger than 3. - The link, in dimension 3, between the geometry of the ambient manifold (especially hyperbolicity) and the dynamical/topological properties of its Reeb vector fields and contact structures. - The topological characterization of odd-dimensional manifolds admitting a contact structure. A crucial ingredient of my program is to understand the key role played by open book decompositions in dimensions larger than three. This program requires a huge amount of mathematical knowledges. My idea is to organize a team around Ghiggini, Laudenbach, Rollin, Sandon and myself, augmented by two post-docs and one PhD student funded by the project. This will give us the critical size to organize a very active working seminar and to have a worldwide attractivity and recognition. I also plan to invite one confirmed researcher every year (for 1-2 months), to organize one conference and one summer school, as well as several focused weeks.

887 600 €

Start date: 2012-01-01, End date: 2016-12-31

The present proposal is concerned with the analysis of geometric non-linear wave equations, such as the Einstein equations, as well as coupled systems of wave and kinetic equations such as the Vlasov-Maxwell and Einstein-Vlasov equations. We intend to pursue three main lines of research, each of them concerning major open problems in the field. I) The dynamics in a neighbourhood of the Anti-de-Sitter space with various boundary conditions. This is a fundamental open problem of mathematical physics which aims at understanding the stability or instability properties of one of the simplest solutions to the Einstein equations. On top of its intrinsic mathematical interest, this question is also at the heart of an intense research activity in the theoretical physics community. II) Non-linear systems of wave and kinetic equations. We have recently found out that the so-called vector field method of Klainerman, a fundamental tool in the study of quasilinear wave equations, in fact possesses a complete analogue in the case of kinetic transport equations. This opens the way to many new directions of research, with applications to several fundamental systems of kinetic theory, such as the Einstein-Vlasov or Vlasov-Maxwell systems, and creates a link between two areas of PDEs which have typically been studied via different methods. One of our objectives is to develop other potential links, such as a general analysis of null forms for relativistic kinetic equations. III) The Einstein equations with data on a compact manifold. The long time dynamics of solutions to the Einstein equations arising from initial data given on a compact manifold is still very poorly understood. In particular, there is still no known stable asymptotic regime for the Einstein equations with data given on a simple manifold such as the torus. We intend to establish the existence of such a stable asymptotic regime.

1 071 008 €

Start date: 2017-02-01, End date: 2022-01-31

Chemical protein synthesis is an indispensable method in chemical and synthetic biology. However, at the present moment, it is laborious and involves multiple optimization and purification steps. High-throughput approaches for total synthesis of combinatorial libraries of custom-modified protein variants are needed. To change the situation, the work will be carried out in two directions: (1) implementation of microfluidic techniques for automation, miniaturization and multiplexing of experimental steps involved in the total synthesis of proteins, and (2) design and synthesis of novel catalytic proteins for efficient enzyme-assisted peptide ligations under denatured conditions. This innovative research technology will allow robust chemical synthesis of protein libraries with (100–10,000)-compounds with natural and unnatural modifications, bearing variety of post-translational modifications and also protein-like biopolymers. In this project, the new methodology will be validated by chemical synthesis of library of phosphorylated analogues of high mobility group protein A (HMGA), which is involved in gene-transcription and cancer development. Other potential future applications include protein design, biological problems where post-translational modifications play a crucial role (ranging from the ‘histone code’ hypothesis to understanding long-term memory) and functional annotation of newly discovered genes.

1 500 000 €

Start date: 2017-03-01, End date: 2022-02-28

Contemporary cryptography, with security relying on the factorisation and discrete logarithm problems, is ill-prepared for the future: It will collapse with the rise of quantum computers, its costly algorithms require growing resources, and it is utterly ill-fitted for the fast-developing trend of externalising computations to the cloud. The emerging field of *lattice-based cryptography* (LBC) addresses these concerns: it resists would-be quantum computers, trades memory for drastic run-time savings, and enables computations on encrypted data, leading to the prospect of a privacy-preserving cloud economy. LBC could supersede contemporary cryptography within a decade. A major goal of this project is to enable this technology switch. I will strengthen the security foundations, improve its performance, and extend the range of its functionalities. A lattice is the set of integer linear combinations of linearly independent real vectors, called lattice basis. The core computational problem on lattices is the Shortest Vector Problem (SVP): Given a basis, find a shortest non-zero point in the spanned lattice. The hardness of SVP is the security foundation of LBC. In fact, SVP and its variants arise in a great variety of areas, including computer algebra, communications (coding and cryptography), computer arithmetic and algorithmic number theory, further motivating the study of lattice algorithms. In the matter of *algorithm design*, the community is quickly nearing the limits of the classical paradigms. The usual approach, lattice reduction, consists in representing a lattice by a basis and steadily improving its quality. I will assess the full potential of this framework and, in the longer term, develop alternative approaches to go beyond the current limitations. This project aims at studying all computational aspects of lattices, with cryptography as the driving motive. The strength of LattAC lies in its theory-to-practice and interdisciplinary methodological approach

1 414 402 €

Start date: 2014-01-01, End date: 2018-12-31

Time-series of multimodal medical images offer a unique opportunity to track anatomical and functional alterations of the brain in aging individuals. A collection of such time series for several individuals forms a longitudinal data set, each data being a rich iconic-geometric representation of the brain anatomy and function. These data are already extraordinary complex and variable across individuals. Taking the temporal component into account further adds difficulty, in that each individual follows a different trajectory of changes, and at a different pace. Furthermore, a disease is here a progressive departure from an otherwise normal scenario of aging, so that one could not think of normal and pathologic brain aging as distinct categories, as in the standard case-control paradigm. Bio-statisticians lack a suitable methodological framework to exhibit from these data the typical trajectories and dynamics of brain alterations, and the effects of a disease on these trajectories, thus limiting the investigation of essential clinical questions. To change this situation, we propose to construct virtual dynamical models of brain aging by learning typical spatiotemporal patterns of alterations propagation from longitudinal iconic-geometric data sets. By including concepts of the Riemannian geometry into Bayesian mixed effect models, the project will introduce general principles to average complex individual trajectories of iconic-geometric changes and align the pace at which these trajectories are followed. It will estimate a set of elementary spatiotemporal patterns, which combine to yield a personal aging scenario for each individual. Disease-specific patterns will be detected with an increasing likelihood. This new generation of statistical and computational tools will unveil clusters of patients sharing similar lesion propagation profiles, paving the way to design more specific treatments, and care patients when treatments have the highest chance of success.

1 499 894 €

Start date: 2016-09-01, End date: 2021-08-31

The purpose of this proposal is to investigate new connections which have emerged in the recent years between problems from statistical mechanics, namely two-dimensional exactly solvable models, and a variety of combinatorial problems, among which: the enumeration of plane partitions, alternating sign matrices and related objects; combinatorial properties of certain algebro-geometric objects such as orbital varieties or the Brauer loop scheme; or finally certain problems in free probability. One of the key methods that emerged in recent years is the use of quantum integrability and more precisely the quantum Knizhnik--Zamolodchikov equation, which itself is related to many deep results in representation theory. The fruitful interaction between all these ideas has led to many advances in the last few years, including proofs of some old conjectures but also completely new results. More specifically, loop models are a class of statistical models where the PI has made significant progress, in particular in relation to the so-called Razumov--Stroganov conjecture (now Cantini--Sportiello theorem). New directions that should be pursued include: further applications to enumerative combinatorics such as proofs of various open conjectures relating Alternating Sign Matrices, Plane Partitions and their symmetry classes; a full understanding of the quantum integrability of the Fully Packed Loop model, a specific loop model at the heart of the Razumov--Stroganov correspondence; a complete description of the Brauer loop scheme, including its defining equations, and of the underlying poset; the extension of the work on Di Francesco and Zinn-Justin on the loop model/6-vertex vertex relation to the case of the 8-vertex model (corresponding to elliptic solutions of the Yang--Baxter equation); the study of solvable tilings models, in relation to generalizations of the Littlewood--Richardson rule, and the determination of their limiting shapes.

840 120 €

Start date: 2011-11-01, End date: 2016-10-31