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en en 201861 291147-615-4 The Lorentz gas is one of the simplest, most widely used models to study the transport properties of rarified gases in matter. It describes the dynamics of a cloud of non-interacting point particles in an infinite array of fixed spherical scatterers. A major challenge of this...

The Lorentz gas is one of the simplest, most widely used models to study the transport properties of rarified gases in matter. It describes the dynamics of a cloud of non-interacting point particles in an infinite array of fixed spherical scatterers. A major challenge of this research project was to understand the nature of the kinetic transport equation that governs the macroscopic particle dynamics in the limit of low scatterer density (the Boltzmann-Grad limit). Lorentz suggested in 1904 that this equation should be the linear Boltzmann equation. This was confirmed in three celebrated papers by Gallavotti, Spohn, and Boldrighini, Bunimovich and Sinai, under the assumption that the distribution of scatterers is sufficiently disordered. A key finding of this research project, obtained in collaboration with A. Strombergsson (Uppsala University), is that in the case of strongly correlated scatterer configurations (for example the vertex set of a Penrose tiling), the macroscopic dynamics is described by a Markovian random flight process, whose transport equation is in general not the linear Boltzmann equation. The description of the limit process requires the introduction of hidden variables, which depend on the class of scatterer configuration. In the case of a Penrose tiling the limit process is, for example, determined by a random cut-and-project set, which is stationary under the action of a Hilbert modular group. A particularly striking feature in the case of the periodic Lorentz gas is a heavy tail for the distribution of free path lengths, with a diverging second moment. In this case we established, in joint work with B. Toth (Bristol & Budapest), that this divergence leads to a central limit theorem for particle transport in the limit of large times t, with a t log t divergence of the mean-square displacement. This is the first rigorous proof of superdiffusion in the Lorentz gas in dimension three or higher. The key progress in this research project has been achieved through the development and application of sophisticated methods in the theory of homogeneous flows, which are used to "renormalize" the gas dynamics. In particular, we have derived equidistribution theorems for translates of spheres, horospheres and more general embedded submanifolds in various moduli spaces through the application of Ratner's theorem. In addition to applications in kinetic theory, we have exploited these techniques to answer a number of open questions in other research areas, including the theory of Diophantine approximation, pseudo-randomness in the distribution of number-theoretic sequences modulo one, and the distribution of directions in hyperbolic lattices and quasicrystals.

2017-07-24 17:27:04 de,en,es,fr,it,pl pubsum en report de,en,es,fr,it,pl sesam en 3098728 UNIVERSITY OF BRISTOL

UK

en 3725751 en 101168 291147 HFAKT de,en,es,fr,it,pl 845 FP7-IDEAS-ERC http://erc.europa.eu/ 1401 EC 837 FP7 en 12350 ERC-AG-PE1 12781 ERC-AG-PE 16702 ERC-AG 13423 ERC 845 FP7-IDEAS-ERC 1401 EC 837 FP7 de,en,es,fr,it,pl PSE de,en,es,fr,it,pl MST United Kingdom UK 508725800 en en 101168 291147 HFAKT "Since the pioneering work of Maxwell and Boltzmann in the 1860s and 1870s, a major challenge in mathematical physics has been the derivation of macroscopic evolution equations from the fundamental microscopic laws of classical or quantum mechanics. The key idea of the presen... 2012-04-01 2017-03-31 2012-04-01 2017-03-31 2015-08-03 13:53:29 101262 ERC-2011-ADG_20110209 en 1907078 999974262 UNIVERSITY OF BRISTOL

UK www.bristol.ac.uk

en HES en 3195548 en 3195547 United Kingdom UK 508725800 de,en,es,fr,it,pl 845 FP7-IDEAS-ERC http://erc.europa.eu/ 1401 EC 837 FP7 en 12350 ERC-AG-PE1 12781 ERC-AG-PE 16702 ERC-AG 13423 ERC 845 FP7-IDEAS-ERC 1401 EC 837 FP7 de,en,es,fr,it,pl PSE de,en,es,fr,it,pl MST en 179630 The Lorentz gas is one of the simplest, most widely used models to study the transport properties of rarified gases in matter. It describes the dynamics of a cloud of non-interacting point particles in an infinite array of fixed spherical scatterers. More than one hundred... 2016-04-25 15:04:11 en report en sesam de,en,es,fr,it,pl sesam en 201861 The Lorentz gas is one of the simplest, most widely used models to study the transport properties of rarified gases in matter. It describes the dynamics of a cloud of non-interacting point particles in an infinite array of fixed spherical scatterers. A major challenge of this... 2017-07-24 17:27:04 de,en,es,fr,it,pl pubsum en report de,en,es,fr,it,pl sesam en ERC-AG en en 178501 321186-620-2 The goal of this project is to accomplish a leap forward in the knowledge on propagation phenomena in reaction-diffusion equations, in heterogeneous media and/or unconventional diffusion, systems as well as non local interactions. This proposal deals both with the general...

The goal of this project is to accomplish a leap forward in the knowledge on propagation phenomena in reaction-diffusion equations, in heterogeneous media and/or unconventional diffusion, systems as well as non local interactions. This proposal deals both with the general theory of nonlinear Partial Differential Equations (PDEs) of elliptic and parabolic types as well as with the development and study of some specific models. These range from ecology to medicine, mathematical economics and social sciences. We have made progress in all these issues. We have introduced new theoretical notions that have turned out to be mathematically quite rich, and to improve the general knowledge in nonlinear PDEs. They also have proved to be instrumental in understanding more specific models. We have solved important open problems arising in theoretical ecology, such as nonlocal competition or selection-mutation models. We have made progress in the study of several models that we have proposed, and which have a high potential of cross fertilisation with other sciences. One of them is the study of rioting activity: we have devised a new model that is now compared to data. It is promising to have many developments, and to shed a new light on some social phenomena.

2016-05-03 16:35:47 en 2844203 ECOLE DES HAUTES ETUDES EN SCIENCES SOCIALES

FR

en 3530289 en 106803 321186 READI de,en,es,fr,it,pl 845 FP7-IDEAS-ERC http://erc.europa.eu/ 1401 EC 837 FP7 en 12350 ERC-AG-PE1 12781 ERC-AG-PE 16702 ERC-AG 13423 ERC 845 FP7-IDEAS-ERC 1401 EC 837 FP7 de,en,es,fr,it,pl PSE de,en,es,fr,it,pl MST France FR 258902568 en report en sesam de,en,es,fr,it,pl sesam en en 106803 321186 ReaDi Our goal is to accomplish a leap forward in the knowledge on propagation phenomena in reaction-diffusion equations, in heterogeneous media and/or non standard diffusion, systems as well as non local interactions. This proposal deals both with the general theory of nonlinear P... 2013-02-01 2018-01-31 2013-02-01 2018-01-31 2016-03-31 14:29:40 106067 ERC-2012-ADG_20120216 en 1916836 999510117 ECOLE DES HAUTES ETUDES EN SCIENCES SOCIALES

FR

en HES en 3319160 en 3319159 France FR 258902568 de,en,es,fr,it,pl 845 FP7-IDEAS-ERC http://erc.europa.eu/ 1401 EC 837 FP7 en 12350 ERC-AG-PE1 12781 ERC-AG-PE 16702 ERC-AG 13423 ERC 845 FP7-IDEAS-ERC 1401 EC 837 FP7 de,en,es,fr,it,pl PSE de,en,es,fr,it,pl MST en 178501 The goal of this project is to accomplish a leap forward in the knowledge on propagation phenomena in reaction-diffusion equations, in heterogeneous media and/or unconventional diffusion, systems as well as non local interactions. This proposal deals both with the general... 2016-05-03 16:35:47 en report en sesam de,en,es,fr,it,pl sesam en ERC-AG en en 100209 279723 SURFARI This research proposal concerns a fundamental problem in the theory of al- gebraic surfaces which poses one of the most important challenges in order to understand the inner structure of algebraic surfaces beyond the current state of the art. Our research team will investigat... 2011-10-01 2016-09-30 2011-10-01 2016-09-30 2017-07-12 09:06:05 99628 ERC-2011-StG_20101014 en 1905855 999981828 GOTTFRIED WILHELM LEIBNIZ UNIVERSITAET HANNOVER

DE www.uni-hannover.de

en HES en 3704450 en 3704449 Germany DE 221940264 de,en,es,fr,it,pl 845 FP7-IDEAS-ERC http://erc.europa.eu/ 1401 EC 837 FP7 en 12334 ERC-SG-PE1 12771 ERC-SG-PE 11092 ERC-SG 13423 ERC 845 FP7-IDEAS-ERC 1401 EC 837 FP7 de,en,es,fr,it,pl PSE de,en,es,fr,it,pl MST en 197313 For an algebraic surface, the Picard number is arguably the most basic invariant which is not preserved under deformations. Roughly, it measures the quantity of curves lying on the surface, thus providing a deep insight into its inner structures. The project is concerned with... 2017-04-11 11:53:12 de,en,es,fr,it,pl pubsum en report de,en,es,fr,it,pl sesam en ERC-SG en en 197313 279723-615-4 For an algebraic surface, the Picard number is arguably the most basic invariant which is not preserved under deformations. Roughly, it measures the quantity of curves lying on the surface, thus providing a deep insight into its inner structures. The project is concerned with...

For an algebraic surface, the Picard number is arguably the most basic invariant which is not preserved under deformations. Roughly, it measures the quantity of curves lying on the surface, thus providing a deep insight into its inner structures. The project is concerned with the fundamental problem which Picard numbers occur within a given class of surfaces. Or main result is an affirmative solution for complex quintics in IP^3: all Picard numbers from 1 to 45 do actually occur. Key ingredients consist in a novel technique of arithmetic deformations, reductions to K3 surfaces and wild automorphisms. Towards higher degrees, we developed a closed formula for the Picard numbers of Delsarte surfaces with isolated rational double point singularities. For sextics in IP^3, this enables us to realize most small Picard numbers in a rather explicit way. Another project deals with the more refined problem of singular complex K3 surfaces, i.e. those attaining the maximum Picard number 20. Here the difference between field of definition and field of moduli is the critical point. We analyzed the fields of moduli completely and proved that all singular K3 surfaces of class number 2 possess a model defined over Q. The third main stream of research concerns Enriques surfaces. We give a complete classification of Enriques surfaces supporting a maximal root type generated by smooth rational curves. Remarkably, the techniques can be developed quite uniformly for all characteristics, and the underlying moduli spaces behave very nicely. All in all, the project has facilitated many interesting results and powerful novel techniques for the study of the arithmetic of algebraic surfaces and their Picard groups.

2017-04-11 11:53:12 de,en,es,fr,it,pl pubsum en report de,en,es,fr,it,pl sesam en 2924967 GOTTFRIED WILHELM LEIBNIZ UNIVERSITAET HANNOVER

DE

en 3550109 en 100209 279723 SURFARI de,en,es,fr,it,pl 845 FP7-IDEAS-ERC http://erc.europa.eu/ 1401 EC 837 FP7 en 12334 ERC-SG-PE1 12771 ERC-SG-PE 11092 ERC-SG 13423 ERC 845 FP7-IDEAS-ERC 1401 EC 837 FP7 de,en,es,fr,it,pl PSE de,en,es,fr,it,pl MST Germany DE 221940264 en en 101506 278246 GEODYCON "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... 2012-01-01 2016-12-31 2012-01-01 2016-12-31 2015-03-10 18:25:27 99628 ERC-2011-StG_20101014 en 1915121 999854176 UNIVERSITE DE NANTES

FR www.univ-nantes.fr

en HES en 3087047 en 3087046 France FR 258902568 de,en,es,fr,it,pl 845 FP7-IDEAS-ERC http://erc.europa.eu/ 1401 EC 837 FP7 en 12334 ERC-SG-PE1 12771 ERC-SG-PE 11092 ERC-SG 13423 ERC 845 FP7-IDEAS-ERC 1401 EC 837 FP7 de,en,es,fr,it,pl PSE de,en,es,fr,it,pl MST en 199912 The project has been exploring the interactions between the geometry of a space, and the dynamical properties of a special type of evolutive systems on this space: the flow of a Reeb vector field. To give a concrete picture, one can imagine a space filled with water so that... 2017-06-20 17:49:32 de,en,es,fr,it,pl pubsum en report de,en,es,fr,it,pl sesam en ERC-SG en en 100779 277749 EPSILON The scope of this project is to perform an analytical study of the geometric properties of the interafaces arising in the scalar Ginzburg-Landau-Allen-Cahn equation, with particular attention to possible 1D symmetries. Also, we would like to analyze the cases in which the ope... 2012-01-01 2016-12-31 2012-01-01 2016-12-31 2017-06-26 10:27:00 99628 ERC-2011-StG_20101014 en 1907167 999995796 UNIVERSITA DEGLI STUDI DI MILANO

IT www.unimi.it

en HES en 3684694 Italy IT 309758504 en 1907861 999927120 FORSCHUNGSVERBUND BERLIN EV

DE www.fv-berlin.de

en REC en 3684692 en 3684691 Germany DE 221940264 en 1906260 999844864 UNIVERSITA DEGLI STUDI DI ROMA TOR VERGATA

IT www.uniroma2.it

en HES en 3684693 Italy IT 309758504 de,en,es,fr,it,pl 845 FP7-IDEAS-ERC http://erc.europa.eu/ 1401 EC 837 FP7 en 12334 ERC-SG-PE1 12771 ERC-SG-PE 11092 ERC-SG 13423 ERC 845 FP7-IDEAS-ERC 1401 EC 837 FP7 de,en,es,fr,it,pl PSE de,en,es,fr,it,pl MST en 199889 We have developed intense research in the fields of partial differential equations, integro-differential equations, geometric measure theory and calculus of variations. About 70 papers are listed in MathSciNet as authored by the PI in the report period, all published in... 2017-06-20 18:05:08 de,en,es,fr,it,pl pubsum en report de,en,es,fr,it,pl sesam en ERC-SG en en 100398 278940 PSC and LMCF The interplay between Geometry and Analysis has been among the most fruitful mathematical ideas in recent years, the most obvious example being Perelman's proof of Poincare' conjecture. The goal of this proposal is to establish a research group that will make significant prog... 2012-01-01 2016-12-31 2012-01-01 2016-12-31 2016-03-17 17:50:51 99628 ERC-2011-StG_20101014 en 1905566 999993468 IMPERIAL COLLEGE OF SCIENCE TECHNOLOGY AND MEDICINE

UK www.imperial.ac.uk

en HES en 3308627 en 3308626 United Kingdom UK 508725800 de,en,es,fr,it,pl 845 FP7-IDEAS-ERC http://erc.europa.eu/ 1401 EC 837 FP7 en 12334 ERC-SG-PE1 12771 ERC-SG-PE 11092 ERC-SG 13423 ERC 845 FP7-IDEAS-ERC 1401 EC 837 FP7 de,en,es,fr,it,pl PSE de,en,es,fr,it,pl MST en 199878 The goal of the project was to study minimal surfaces and their interacations with the ambient geometry. We pioneered the use of min-max methods to further explore this connection and in doing so not only we solved several open problems in Geometry but we also developed... 2017-06-21 09:30:34 de,en,es,fr,it,pl pubsum en report de,en,es,fr,it,pl sesam en ERC-SG en en 102223 291060 analysisdirac 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 Lap... 2012-02-01 2017-01-31 2012-02-01 2017-01-31 2016-03-18 13:32:16 101262 ERC-2011-ADG_20110209 en 1905574 999907235 UNIVERSITE PARIS-SUD

FR www.u-psud.fr

en HES en 3312452 en 3312451 France FR 258902568 de,en,es,fr,it,pl 845 FP7-IDEAS-ERC http://erc.europa.eu/ 1401 EC 837 FP7 en 12350 ERC-AG-PE1 12781 ERC-AG-PE 16702 ERC-AG 13423 ERC 845 FP7-IDEAS-ERC 1401 EC 837 FP7 de,en,es,fr,it,pl PSE de,en,es,fr,it,pl MST en 199916 The ERC project was to bring together various fields of mathematics: geometry, analysis, probability theory, mathematical physics, and dynamical systems, by showing the potential of a new object introduced by the PI, the hypoelliptic Laplacian, that has the potential to... 2017-06-20 17:46:20 de,en,es,fr,it,pl pubsum en report de,en,es,fr,it,pl sesam en ERC-AG