ERC FUNDED PROJECTS

Understanding cause-effect relationships between variables is of great interest in many fields of science. However, causal inference from data is much more ambitious and difficult than inferring (undirected) measures of association such as correlations, partial correlations or multivariate regression coefficients, mainly because of fundamental identifiability problems. A main objective of the proposal is to exploit advantages from large-scale heterogeneous data for causal inference where heterogeneity arises from different experimental conditions or different unknown sub-populations. A key idea is to consider invariance or stability across different experimental conditions of certain conditional probability distributions: the invariants correspond on the one hand to (properly defined) causal variables which are of main interest in causality; andon the other hand, they correspond to the features for constructing powerful predictions for new scenarios which are unobserved in the data (new probability distributions). This opens novel perspectives: causal inference can be phrased as a prediction problem of a certain kind, and vice versa, new prediction methods which work well across different scenarios (unobserved in the data) should be based on or regularized towards causal variables. Fundamental identifiability limits will become weaker with increased degree of heterogeneity, as we expect in large-scale data. The topic is essentially unexplored, yet it opens new avenues for causal inference, structural equation and graphical modeling, and robust prediction based on large-scale complex data. We will develop mathematical theory, statistical methodology and efficient algorithms; and we will also work and collaborate on major application problems such as inferring causal effects (i.e., total intervention effects) from gene knock-out or RNA interference perturbation experiments, genome-wide association studies and novel prediction tasks in economics.

2 184 375 €

Start date: 2018-10-01, End date: 2023-09-30

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

This project is devoted to the mathematical analysis of important physical properties of many-body quantum systems. We will be interested in properties of the ground state and low-energy excitations but also of non-equilibrium dynamics. We are going to consider systems with different statistics and in different regimes. The questions we are going to address have a common aspect: correlations among particles play a crucial role. Our main goal consists in developing new tools that allow us to correctly describe many-body correlations and to understand their effects. The starting point of our proposal are ideas and techniques that have been introduced in a series of papers establishing the validity of Bogoliubov theory for Bose gases in the Gross-Pitaevskii regime, and in a recent preprint showing how (bosonic) Bogoliubov theory can also be used to study the correlation energy of Fermi gases. In this project, we plan to develop these techniques further and to apply them to new contexts. We believe they have the potential to approach some fundamental open problem in mathematical physics. Among our most ambitious objectives, we include the proof of the Lee-Huang-Yang formula for the energy of dilute Bose gases and of the corresponding Huang-Yang formula for dilute Fermi gases, as well as the derivation of the Gell-Mann--Brueckner expression for the correlation energy of a high density Fermi system. Furthermore, we propose to work on long-term projects (going beyond the duration of the grant) aiming at a rigorous justification of the quantum Boltzmann equation for fermions in the weak coupling limit and at a proof of Bose-Einstein condensation in the thermodynamic limit, two very challenging and important questions in the field.

1 876 050 €

Start date: 2019-09-01, End date: 2024-08-31

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