ERC FUNDED PROJECTS

State-of-the-art microscopy and diffraction imaging provides insight into the atomic and sub-atomic structure of matter. They permit determination of the positions of atoms in a crystal lattice or in a molecule as well as the distribution of electrons inside atoms. State-of-the-art time-resolved spectroscopy with femtosecond and attosecond resolution provides access to dynamic changes in the atomic and electronic structure of matter. Our proposal aims at combining these two frontier techniques of XXI century science to make a long-standing dream of scientist come true: the direct observation of atoms and electrons in their natural state: in motion. Shifts in the atoms positions by tens to hundreds of picometers can make chemical bonds break apart or newly form, changing the structure and/or chemical composition of matter. Electronic motion on similar scales may result in the emission of light, or the initiation of processes that lead to a change in physical or chemical properties, or biological function. These motions happen within femtoseconds and attoseconds, respectively. To make them observable, we need a 4-dimensional (4D) imaging technique capable of recording freeze-frame snapshots of microscopic systems with picometer spatial resolution and femtosecond to attosecond exposure time. The motion can then be visualized by slow-motion replay of the freeze-frame shots. The goal of this project is to develop a 4D imaging technique that will ultimately offer picometer resolution is space and attosecond resolution in time.

2 500 000 €

Start date: 2010-03-01, End date: 2015-02-28

The primary purpose of this project is to build on recent spectacular progress in the Langlands program to study the arithmetic properties of automorphic motives constructed in the cohomology of Shimura varieties. Because automorphic methods are available to study the L-functions of these motives, which include elliptic curves and certain families of Calabi-Yau varieties over totally real fields (possibly after base change), they represent the most accessible class of varieties for which one can hope to verify fundamental conjectures on special values of L-functions, including Deligne's conjecture and the Main Conjecture of Iwasawa theory. Immediate goals include the proof of irreducibility of automorphic Galois representations; the establishment of period relations for automorphic and potentially automorphic realizations of motives in the cohomology of distinct Shimura varieties; the construction of p-adic L-functions for these and related motives, notably adjoint and tensor product L-functions in p-adic families; and the geometrization of the p-adic and mod p Langlands program. All four goals, as well as the others mentioned in the body of the proposal, are interconnected; the final goal provides a bridge to related work in geometric representation theory, algebraic geometry, and mathematical physics.

1 491 348 €

Start date: 2012-06-01, End date: 2018-05-31

The amazing rate of progress in the computer technologies and telecommunications presents many new challenges for Optimization Theory. New problems are usually very big in size, very special in structure and possibly have a distributed data support. This makes them unsolvable by the standard optimization methods. In these situations, old theoretical models, based on the hidden Black-Box information, cannot work. New theoretical and algorithmic solutions are urgently needed. In this project we will concentrate on development of fast optimization methods for problems of big and very big size. All the new methods will be endowed with provable efficiency guarantees for large classes of optimization problems, arising in practical applications. Our main tool is the acceleration technique developed for the standard Black-Box methods as applied to smooth convex functions. However, we will have to adapt it to deal with different situations. The first line of development will be based on the smoothing technique as applied to a non-smooth functions. We propose to substantially extend this approach to generate approximate solutions in relative scale. The second line of research will be related to applying acceleration techniques to the second-order methods minimizing functions with sparse Hessians. Finally, we aim to develop fast gradient methods for huge-scale problems. The size of these problems is so big that even the usual vector operations are extremely expensive. Thus, we propose to develop new methods with sublinear iteration costs. In our approach, the main source for achieving improvements will be the proper use of problem structure. Our overall aim is to be able to solve in a routine way many important problems, which currently look unsolvable. Moreover, the theoretical development of Convex Optimization will reach the state, when there is no gap between theory and practice: the theoretically most efficient methods will definitely outperform any homebred heuristics.

2 090 038 €

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

The numerical simulation of complex compressible flow problem is still a challenge nowaday even for simple models. In our opinion, the most important hard points that need currently to be tackled and solved is how to obtain stable, scalable, very accurate, easy to code and to maintain schemes on complex geometries. The method should easily handle mesh refinement, even near the boundary where the most interesting engineering quantities have to be evaluated. Unsteady uncertainties in the model, for example in the geometry or the boundary conditions should represented efficiently.This proposal goal is to design, develop and evaluate solutions to each of the above problems. Our work program will lead to significant breakthroughs for flow simulations. More specifically, we propose to work on 3 connected problems: 1-A class of very high order numerical schemes able to easily deal with the geometry of boundaries and still can solve steep problems. The geometry is generally defined by CAD tools. The output is used to generate a mesh which is then used by the scheme. Hence, any mesh refinement process is disconnected from the CAD, a situation that prevents the spread of mesh adaptation techniques in industry! 2-A class of very high order numerical schemes which can utilize possibly solution dependant basis functions in order to lower the number of degrees of freedom, for example to compute accurately boundary layers with low resolutions. 3-A general non intrusive technique for handling uncertainties in order to deal with irregular probability density functions (pdf) and also to handle pdf that may evolve in time, for example thanks to an optimisation loop. The curse of dimensionality will be dealt thanks Harten's multiresolution method combined with sparse grid methods. Currently, and up to our knowledge, no scheme has each of these properties. This research program will have an impact on numerical schemes and industrial applications.

1 432 769 €

Start date: 2008-12-01, End date: 2013-11-30

Fundamental interactions in nature are well described by quantum gauge fields in 4 space-time dimensions (4d). When the strength of gauge interaction is weak the Feynman perturbation techniques are very efficient for the description of most of the experimentally observable consequences of the Standard model and for the study of high energy processes in QCD. But in the intermediate and strong coupling regime, such as the relatively small energies in QCD, the perturbation theory fails leaving us with no reliable analytic methods (except the Monte-Carlo simulation). The project aims at working out new analytic and computational methods for strongly coupled gauge theories in 4d. We will employ for that two important discoveries: 1) the gauge-string duality (AdS/CFT correspondence) relating certain strongly coupled gauge Conformal Field Theories to the weakly coupled string theories on Anty-deSitter space; 2) the solvability, or integrability of maximally supersymmetric (N=4) 4d super Yang-Mills (SYM) theory in multicolor limit. Integrability made possible pioneering exact numerical and analytic results in the N=4 multicolor SYM at any coupling, effectively summing up all 4d Feynman diagrams. Recently, we conjectured a system of functional equations - the AdS/CFT Y-system – for the exact spectrum of anomalous dimensions of all local operators in N=4 SYM. The conjecture has passed all available checks. My project is aimed at the understanding of origins of this, still mysterious integrability. Deriving the AdS/CFT Y-system from the first principles on both sides of gauge-string duality should provide a long-awaited proof of the AdS/CFT correspondence itself. I plan to use the Y-system to study the systematic weak and strong coupling expansions and the so called BFKL limit, as well as for calculation of multi-point correlation functions of N=4 SYM. We hope on new insights into the strong coupling dynamics of less supersymmetric gauge theories and of QCD.

1 456 140 €

Start date: 2013-11-01, End date: 2018-10-31

"AEROCAT aims at the elucidation of the potential of nanoparticle derived aerogels in catalytic applications. The materials will be produced from a variety of nanoparticles available in colloidal solutions, amongst which are metals and metal oxides. The evolving aerogels are extremely light, highly porous solids and have been demonstrated to exhibit in many cases the important properties of the nanosized objects they consist of instead of simply those of the respective bulk solids. The resulting aerogel materials will be characterized with respect to their morphology and composition and their resulting (electro-)catalytic properties examined in the light of the inherent electronic nature of the nanosized constituents. Using the knowledge gained within the project the aerogel materials will be further re-processed in order to exploit their full potential relevant to catalysis and electrocatalysis. From the vast variety of possible applications of nanoparticle-based hydro- and aerogels like thermoelectrics, LEDs, pollutant clearance, sensorics and others we choose our strictly focused approach (i) due to the paramount importance of catalysis for the Chemical Industry, (ii) because we have successfully studied the Ethanol electrooxidation on a Pd-nanoparticle aerogel, (iii) we have patented on the oxygen reduction reaction in fuel cells with bimetallic aerogels, (iv) and we gained first and extremely promising results on the semi-hydrogenation of Acetylene on a mixed Pd/ZnO-nanoparticle aerogel. With this we are on the forefront of a research field which impact might not be overestimated. We should quickly explore its potentials and transfer on a short track the knowledge gained into pre-industrial testing."

2 194 000 €

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

The purpose of this project is to study basic questions in algebraic and Kähler geometry. It is well known that the structure of projective or Kähler manifolds is governed by positivity or negativity properties of the curvature tensor. However, many fundamental problems are still wide open. Since the mid 1980's, I have developed a large number of key concepts and results that have led to important progress in transcendental algebraic geometry. Let me mention the discovery of holomorphic Morse inequalities, systematic applications of L² estimates with singular hermitian metrics, and a much improved understanding of Monge-Ampère equations and of singularities of plurisuharmonic functions. My first goal will be to investigate the Green-Griffiths-Lang conjecture asserting that an entire curve drawn in a variety of general type is algebraically degenerate. The subject is intimately related to important questions concerning Diophantine equations, especially higher dimensional generalizations of Faltings' theorem - the so-called Vojta program. One can rely here on a breakthrough I made in 2010, showing that all such entire curves must satisfy algebraic differential equations. A second closely related area of research of this project is the analysis of the structure of projective or compact Kähler manifolds. It can be seen as a generalization of the classification theory of surfaces by Kodaira, and of the more recent results for dimension 3 (Kawamata, Kollár, Mori, Shokurov, ...) to other dimensions. My plan is to combine powerful recent results obtained on the duality of positive cohomology cones with an analysis of the instability of the tangent bundle, i.e. of the Harder-Narasimhan filtration. On these ground-breaking questions, I intend to go much further and to enhance my national and international collaborations. These subjects already attract many young researchers and postdocs throughout the world, and the grant could be used to create even stronger interactions.

1 809 345 €

Start date: 2015-09-01, End date: 2021-08-31

Four years ago, the LIGO/Virgo observation of a black-hole binary merger heralded the dawn of gravitational-wave astronomy. The promise of future observations calls for an invigorated effort to underpin the theoretical framework and supply the predictions needed for detecting future signals and exploiting them for astronomical and astrophysical studies. Ampl2Einstein will take ideas and techniques from recent years' dramatic advances in Quantum Scattering Amplitudes, creating new tools for taking their classical limits and using it for gravitational physics. The powerful `square root' relation between gravity and a generalization of electrodynamics known as Yang--Mills theory will play a key role in making this route simpler than direct classical calculation. We will transfer these ideas to classical General Relativity to compute new perturbative orders, spin-dependent observables, and the dependence on the internal structure of merging objects. We will exploit symmetries and structure we find in order to extrapolate to even higher orders in the gravitational theory. We will make such calculations vastly simpler, pushing the known frontier much further in perturbation theory and in complexity of observables. These advances will give rise to a new generation of gravitational-wave templates, dramatically extending the observing power of detectors. They will allow observers to see weaker signals and will be key to resolving long-standing puzzles about the internal structure of neutron stars. We will apply novel technologies developed for scattering amplitudes to bound-state calculations in both quantum and classical theory. Our research will also lead to a deeper understanding of the classical limit of quantum field theory, relevant to gravitational-wave observations and beyond. The transfer of ideas to the new domain of General Relativity will dramatically enhance our ability to reveal new physics encoded in the subtlest of gravitational-wave signals.

2 372 571 €

Start date: 2021-01-01, End date: 2025-12-31

This proposal is devoted to the applications of a new hypoelliptic Dirac operator, whose analytic properties have been studied by Lebeau and myself. Its construction connects classical Hodge theory with the geodesic flow, and more generally any geometrically defined Hodge Laplacian with a dynamical system on the cotangent bundle. The proper description of this object can be given in analytic, index theoretic and probabilistic terms, which explains both its potential many applications, and also its complexity.

1 112 400 €

Start date: 2012-02-01, End date: 2017-01-31