ERC FUNDED PROJECTS

"The goal of this project is to achieve breakthroughs in a few fundamental questions in 2D statistical physics, using techniques from complex analysis, probability, dynamical systems, geometric measure theory and theoretical physics. Over the last decade, we significantly expanded our understanding of 2D lattice models of statistical physics, their conformally invariant scaling limits and related random geometries. However, there seem to be serious obstacles, preventing further development and requiring novel ideas. We plan to attack those, in particular we intend to: (A) Describe new scaling limits by Schramm’s SLE curves and their generalizations, (B) Study discrete complex structures and use them to describe more 2D models, (C) Describe the scaling limits of random planar graphs by the Liouville Quantum Gravity, (D) Understand universality and lay framework for the Renormalization Group Formalism, (E) Go beyond the current setup of spin models and SLEs. These problems are known to be very difficult, but fundamental questions, which have the potential to lead to significant breakthroughs in our understanding of phase transitions, allowing for further progresses. In resolving them, we plan to exploit interactions of different subjects, and recent advances are encouraging."

1 995 900 €

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

"Developing rechargeable batteries with larger storage capacity, higher output power, faster charge/discharge time, and longer calendar lifetime could significantly impact the economical and environmental future of the European Union. New generations of lithium-ion batteries (LIBs) based on nanostructured electrodes are perfect candidates to supply all-electrical vehicles and favor the usage of renewable energies instead of fossil fuels. Hence, the global LIB revenue is expected to expand from $11 billion in 2011 up to $50 billion in 2020. The goal of this project is therefore to provide an advanced simulation and optimization platform to design LIBs with improved performance and increase the competitiveness of Europe in this domain. The proposed computer aided design (CAD) tool must satisfy three key requirements in order to reach this ambitious objective: (i) computational efficiency, (ii) results accuracy, and (iii) automated predictability. Massively parallel computing has been identified as the enabling technology to handle the first requirement. The second one will be addressed by implementing a state-of-the-art device operation model relying on a multi-scale resolution of the battery electrodes, a detailed description of the electron and ion motions, a material parametrization derived from ab-initio quantum transport techniques, and a validation of the approach through comparisons with experimental measurements. Finally, to meet the last requirement, the operation model will be coupled to a genetic algorithm optimizer capable of automatically predicting the LIB configuration that best matches pre-defined performance targets. The resulting CAD tool will be released as an open source package so that the entire battery community can benefit from it."

1 492 800 €

Start date: 2013-10-01, End date: 2018-09-30

"In recent years, the research focus in signal processing has shifted away from the classical linear paradigm, which is intimately linked with the theory of stationary Gaussian processes. Instead of considering Fourier transforms and performing quadratic optimization, researchers are presently favoring wavelet-like representations and have adopted ”sparsity” as design paradigm. Our ambition is to develop a unifying operator-based framework for signal processing that would provide the ``sparse"" counterpart of the classical theory, which is currently missing. To that end, we shall specify and investigate sparse stochastic processes that are continuously-defined and ruled by differential equations, and construct corresponding wavelet-like sparsifying transforms. Our hope is to be able to rigorously connect non-quadratic regularization and sparsity-constrained optimization to well-defined continuous-domain statistical models. We also want to develop a novel Lie-group formalism for the design of steerable, signal-adapted wavelet transforms with improved invariance and sparsifying properties, both in 2-D and 3-D. We shall use these tools to define new reversible image representations in terms of singular points (contours and keypoints) and to develop novel algorithms for 3-D biomedical image analysis. In close collaboration with imaging scientists, we shall apply our framework to the development of new 3-D reconstruction algorithms for emerging bioimaging modalities such as fluorescence deconvolution microscopy, digital holography microscopy, X-ray phase-contrast microscopy, and advanced MRI."

2 106 994 €

Start date: 2011-04-01, End date: 2016-03-31

"The project lies at the crossroads between three different topics in Mathematics: moduli spaces of flat connections on surfaces in Differential Geometry and Topology, the Kashiwara-Vergne problem and Drinfeld associators in Lie theory, and combinatorics of planar networks in the theory of Total Positivity. The time is ripe to establish deep connections between these three theories. The main factors are the recent progress in the Kashiwara-Vergne theory (including the proof of the Kashiwara-Vergne conjecture by Alekseev-Meinrenken), the discovery of a link between the Horn problem on eigenvalues of sums of Hermitian matrices and planar network combinatorics, and intimate links with the Topological Quantum Field Theory shared by the three topics. The scientific objectives of the project include answering the following questions: 1) To find a universal non-commutative volume formula for moduli of flat connections which would contain the Witten’s volume formula, the Verlinde formula, and the Moore-Nekrasov-Shatashvili formula as particular cases. 2) To show that all solutions of the Kashiwara-Vergne problem come from Drinfeld associators. If the answer is indeed positive, it will have applications to the study of the Gothendieck-Teichmüller Lie algebra grt. 3) To find a Gelfand-Zeiltin type integrable system for the symplectic group Sp(2n). This question is open since 1983. To achieve these goals, one needs to use a multitude of techniques. Here we single out the ones developed by the author: - Quasi-symplectic and quasi-Poisson Geometry and the theory of group valued moment maps. - The linearization method for Poisson-Lie groups relating the additive problem z=x+y and the multiplicative problem Z=XY. - Free Lie algebra approach to the Kashiwara-Vergne theory, including the non-commutative divergence and Jacobian cocylces. - Non-abelian topical calculus which establishes a link between the multiplicative problem and combinatorics of planar networks."

2 148 211 €

Start date: 2014-02-01, End date: 2019-01-31