ERC FUNDED PROJECTS

The simulation of Partial Differential Equations (PDEs) is an indispensable tool for innovation in science and technology. Computer-based simulation of PDEs approximates unknowns defined on a geometrical entity such as the computational domain with all of its properties. Mainly due to historical reasons, geometric design and numerical methods for PDEs have been developed independently, resulting in tools that rely on different representations of the same objects. CHANGE aims at developing innovative mathematical tools for numerically solving PDEs and for geometric modeling and processing, the final goal being the definition of a common framework where geometrical entities and simulation are coherently integrated and where adaptive methods can be used to guarantee optimal use of computer resources, from the geometric description to the simulation. We will concentrate on two classes of methods for the discretisation of PDEs that are having growing impact: isogeometric methods and variational methods on polyhedral partitions. They are both extensions of standard finite elements enjoying exciting features, but both lack of an ad-hoc geometric modelling counterpart. We will extend numerical methods to ensure robustness on the most general geometric models, and we will develop geometric tools to construct, manipulate and refine such models. Based on our tools, we will design an innovative adaptive framework, that jointly exploits multilevel representation of geometric entities and PDE unknowns. Moreover, efficient algorithms call for efficient implementation: the issue of the optimisation of our algorithms on modern computer architecture will be addressed. Our research (and the team involved in the project) will combine competencies in computer science, numerical analysis, high performance computing, and computational mechanics. Leveraging our innovative tools, we will also tackle challenging numerical problems deriving from bio-mechanical applications.

2 199 219 €

Start date: 2016-10-01, End date: 2021-09-30

"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

The goal of this project is to study conformally invariant fractal structures from the perspectives of analysis, dynamics, probability, geometry and physics, emphasizing interrelations of these fields. In the last two decades such structures emerged in several areas: continuum scaling limits of 2D critical models in statistical physics (percolation, Ising model); extremal configurations for various problems in complex analysis (multifractal harmonic measures, coefficient growth of univalent maps, Brennan's conjecture); chaotic sets for complex dynamical systems (Julia sets, Kleinian groups). Capitalizing on recent successes, I plan to continue my work in these areas, exploiting their interactions and connections to physics. I intend to achieve at least some of the following goals: * To establish that several critical lattice models have conformally invariant scaling limits, by building upon results on percolation and Ising models and finding discrete holomorphic observables. * To study geometric properties of arising fractal curves and random fields by connecting them to Schramm's SLE curves and Gaussian Free Fields. * To investigate massive scaling limits by describing them geometrically with generalizations of SLEs. * To lay mathematical framework behind relevant physical notions, such as Coulomb Gas (by relating height functions to GFFs) and Quantum Gravity (by identifying limits of random planar graphs with Liouville QGs). * To improve known bounds in several old questions in complex analysis by studying multifractal spectra of harmonic measures. * To estimate extremal behavior of such spectra by using holomorphic motions of (quasi) conformal maps and thermodynamic formalism. * To understand nature of extremal multifractals for harmonic measure by studying random and dynamical fractals. The topics involved range from century old to very young ones. Recently connections between them started to emerge, opening exciting possibilities for new developments in some long standing open problems.

1 278 000 €

Start date: 2009-01-01, End date: 2013-12-31

One of the oldest and most important questions in PDE theory is that of regularity. A classical example is Hilbert's XIXth problem (1900), solved by De Giorgi and Nash in 1956. During the second half of the XXth century, the regularity theory for elliptic and parabolic PDE's experienced a huge development, and many fundamental questions were answered by Caffarelli, Nirenberg, Krylov, Evans, Nadirashvili, Friedman, and many others. Still, there are problems of crucial importance that remain open. The aim of this project is to go significantly beyond the state of the art in some of the most important open questions in this context. In particular, three key objectives of the project are the following. First, to introduce new techniques to obtain fine description of singularities in nonlinear elliptic PDE's. Aside from its intrinsic interest, a good regularity theory for singular points is likely to provide insightful applications in other contexts. A second aim of the project is to establish generic regularity results for free boundaries and other PDE problems. The development of methods which would allow one to prove generic regularity results may be viewed as one of the greatest challenges not only for free boundary problems, but for PDE problems in general. Finally, the third main objective is to achieve a complete regularity theory for nonlinear elliptic PDE's that does not rely on monotonicity formulas. These three objectives, while seemingly different, are in fact deeply interrelated.

1 335 250 €

Start date: 2019-01-01, End date: 2023-12-31