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

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

Many proteins self-assemble into regular, shell-like, polyhedral structures. Protein capsids are useful, both in nature and in the laboratory, as molecular containers for diverse cargo molecules, including proteins, nucleic acids, metal nanoparticles, quantum dots, and low molecular weight drugs. They can consequently serve as delivery vehicles, bioimaging agents, reaction vessels, and templates for the controlled synthesis of novel materials. Here, we will apply our experience with protein design and laboratory evolution to extend the properties of protein containers to create practical, non-viral encapsulation systems for applications in the test tube and in living cells. Specifically, we will adapt the icosohedral cage structures formed by Aquifex aeolicus lumazine synthase (AaLS) to engineer increasingly sophisticated supramolecular complexes for use as delivery vehicles, nanoreactors and, ultimately, as bacterial organelles. Our principal aims are to: (a) tailor AaLS capsids for selective encapsulation of a broad range of macromolecular guests; (b) develop AaLS capsids as delivery vehicles for medical and imaging applications; (c) design simplified, functional mimics of carbon-fixing carboxysomes; (d) evolve redox active organelles for metabolizing aliphatic alcohols; and (e) engineer artificial organelles for the detoxification of polychlorinated phenols. We anticipate that these experiments will lead to a deeper understanding of the principles underlying the construction, function and evolution of natural protein microcompartments. At the same time, they will establish powerful strategies for creating tailored assemblies for practical applications in delivery and catalysis.

1 889 200 €

Start date: 2013-03-01, End date: 2018-12-31

The advent of the Internet and the increased power of modern day computing have dramatically changed the economic landscape. Billions of dollars worth of goods are being auctioned among geographically dispersed buyers; online brokerages are used to find jobs, trade stocks, make travel arrangements, etc. The architecture of these online (trading) platforms is typically rooted in their pre-Internet counterparts, and advances in the theory of market design combined with increased computing capabilities prompt a careful re-evaluation. This proposal concerns the creation of novel, more flexible institutions using an approach that combines theory, laboratory experiments, and practical policy. The first project enhances our understanding of newly designed package auctions by developing equilibrium models of competitive bidding and measuring the efficacy of alternative formats in controlled experiments. The next project studies novel market forms that allow for all-or-nothing trades to alleviate inefficiencies and enhance dynamic stability when complementarities exist. The third project concerns the design of market regulation and procurement contests to create better incentives for research and development. The fourth project addresses information aggregation properties of alternative voting institutions, suggesting improvements for referenda and jury/committee voting. The Internet has also dramatically altered the nature of social interactions. Emerging institutions such as online social networking tools, rating systems, and web-community Q&A services reduce social distances and catalyze opportunities for social learning. The final project focuses on social learning in a variety of settings and on the impact of social networks on behavior. Combined these projects generate insights that apply to a broad array of social and economic environments and that will guide practitioners to the use of better designed institutions.

1 797 525 €

Start date: 2010-01-01, End date: 2015-12-31

Since the pioneering works of Black & Scholes, Merton and Markowitch, sophisticated quantitative methods are being used to introduce more complex financial products each year. However, this exciting increase in the complexity forces the industry to engage in proper risk management practices. The recent financial crisis emanating from risky loan practices is a prime example of this acute need. This proposal focuses exactly on this general problem. We will develop mathematical techniques to measure and assess the financial risk of new instruments. In the theoretical direction, we will expand the scope of recent studies on risk measures of Artzner et-al., and the stochastic representation formulae proved by the principal investigator and his collaborators. The core research team consists of mathematicians and the finance faculty. The newly created state-of-the-art finance laboratory at the host institution will have direct access to financial data. Moreover, executive education that is performed in this unit enables the research group to have close contacts with high level executives of the financial industry. The theoretical side of the project focuses on nonlinear partial differential equations (PDE), backward stochastic differential equations (BSDE) and dynamic risk measures. Already a deep connection between BSDEs and dynamic risk measures is developed by Peng, Delbaen and collaborators. Also, the principal investigator and his collaborators developed connections to PDEs. In this project, we further investigate these connections. Chief goals of this project are theoretical results and computational techniques in the general areas of BSDEs, fully nonlinear PDEs, and the development of risk management practices that are acceptable by the industry. The composition of the research team and our expertise in quantitative methods, well position us to effectively formulate and study theoretical problems with financial impact.

880 560 €

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

This project aims at developing theoretical and empirical research on the structural transformation that accompanies economic development and on the determinants of its success or failure. This transformation involves changes in policies, institutions, and even preferences and social hierarchies. The project consists of four subprojects. Since China represents the most spectacular ongoing episode of economic transition, four of them focus on the Chinese experience, while the remaining ones address general issues in growth and development. The first subproject analyses some puzzling features of China's recent growth experience, such as the coexistence of high growth with increasing capital export, and the falling labour share, with the aid of a theory which emphasises the efficiency gains associated with the reallocation between firms of different productivity. Since changes in income distribution are an important element, and a concern, of the Chinese transition, the three following subprojects focus, respectively, on the crisis of the system of old age insurance, the effects of the rise of the middle class, and the introduction of special economic zones in the 1980s. Two subprojects study different aspects of competition policy in development. The first one focuses on intellectual property right protection, emphasising the link between innovation, technology adoption and human capital accumulation. The second one studies the coordinating role of industrial policy and how its scope changes with development. The last two subprojects focus on culture. The diffusion of preferences and values that foster cooperation rather than conflict is no less important than incentives for technology adoption. Likewise, the rise of an "entrepreneurial spirit" is an engine of growth in the development transition. We plan to study the emergence and cultural transmission of preferences that are conducive to economic growth, and how they interact with the process of structural change.

1 599 996 €

Start date: 2009-01-01, End date: 2014-06-30

The project is concerned with a coordinated approach to the development of of novel chemical strategies for light harvesting by photovoltaic cells and light generation using light emitting electrochemical cells. Both technologies have proof of principle results from the PIs own laboratory and others world-wide. The bulk of efficient dye sensitized solar cells rely on transition metal complexes as the photoactive component as the majority of traditional organic dyes do not possess long term stability under the operating conditions of the devices. LECs based upon transition metal complexes have been shown to possess lifetimes sufficiently long and efficiencies sufficiently high to become a viable alternative technology to OLEDs in the near future. The disadvantages of state of the art devices for both technologies is that they are based upon second or third row transition metal complexes. Although these elements are expensive, the principle difficulties arise from their low abundance, which creates significant issues of sustainability if the technology is to be adopted. The aim of this project is three-fold. Firstly, to further optimise the individual technologies using conventional transition metal complexes, with increases in efficiency leading to lower metal requirements. Secondly, to explore the periodic table for metal-containing luminophores based on first row transition metals, with an emphasis upon copper and zinc containing species. The final aspect is related to the utilization of solar derived electrons for water splitting reactions, allowing the generation of hydrogen and/or reaction products of hydrogen with organic species. This latter aspect is related to the mid-term requirement for liquid fuels, regardless of the primary fuel sources utilized. The program will involve design and synthesis of new materials, device construction and evaluation (in-house and with existing academic and industrial partners) and iterative refinement of structures

2 399 440 €

Start date: 2011-01-01, End date: 2015-12-31

Algebraic geometry is the study of varieties -- the zero sets of polynomial equations in several variables. The subject has a central role in mathematics with connections to number theory, representation theory, and topology. Moduli questions in algebraic geometry concern the behavior of varieties as the coefficients of the defining polynomials vary. At the end of the 20th century, several basic links between the algebraic geometry of moduli spaces and path integrals in quantum field theory were made. The virtual fundamental class plays an essential role in these connections. I propose to study the algebraic cycle theory of basic moduli spaces. The guiding questions are: What are the most important cycles? What is the structure of the algebra of cycles? How can the classes of geometric loci be expressed? The virtual fundamental class and the associated invariants often control the answers. A combination of virtual localization, degeneration, and R-matrix methods together with new ideas from log geometry will be used in the study. Most of the basic moduli spaces in algebraic geometry related to varieties of dimension at most 3 -- including the moduli of curves, the moduli of maps, the moduli of surfaces, and the moduli of sheaves on 3-folds -- will be considered. The current state of the study of the algebraic cycle theory in these cases varies from rather advanced (for the moduli of curves) to much less so (for the moduli of surfaces). There is a range of rich open questions which I will attack: Pixton's conjectures for the moduli of curves, the structure of the ring of Noether-Lefschetz loci for the moduli of K3 surfaces, the holomorphic anomaly equation in Gromov-Witten theory, and conjectures governing descendents for the moduli of sheaves. The dimension 3 restriction is often necessary for a good deformation theory and the existence of a virtual fundamental class.

2 496 055 €

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