ERC FUNDED PROJECTS

The traditional (Riemann) approach to analytic number theory uses the zeros of zeta functions. This requires the associated multiplicative function, say f(n), to have special enough properties that the associated Dirichlet series may be analytically continued. In this proposal we continue to develop an approach which requires less of the multiplicative function, linking the original question with the mean value of f. Such techniques have been around for a long time but have generally been regarded as “ad hoc”. In this project we aim to show that one can develop a coherent approach to the whole subject, not only reproving all of the old results, but also many new ones that appear inaccessible to traditional methods. Our first goal is to complete a monograph yielding a reworking of all the classical theory using these new methods and then to push forward in new directions. The most important is to extend these techniques to GL(n) L-functions, which we hope will now be feasible having found the correct framework in which to proceed. Since we rarely know how to analytically continue such L-functions this could be of great benefit to the subject. We are developing the large sieve so that it can be used for individual moduli, and will determine a strong form of that. Also a new method to give asymptotics for mean values, when they are not too small. We wish to incorporate techniques of analytic number theory into our theory, for example recent advances on mean values of Dirichlet polynomials. Also the recent breakthroughs on the sieve suggest strong links that need further exploration. Additive combinatorics yields important results in many areas. There are strong analogies between its results, and those for multiplicative functions, especially in large value spectrum theory, and its applications. We hope to develop these further. Much of this is joint work with K Soundararajan of Stanford University.

2 011 742 €

Start date: 2015-08-01, End date: 2020-07-31

Our first goal is to develop a new tool to determine brain activity with a high temporal (< 1 msec) and spatial (about 2 mm) resolution with the focus on motor control. High density EEG (up to 256 electrodes) will be used for EEG source localization. Advanced force-controlled robot manipulators will be used to impose continuous force perturbations to the joints. Advanced closed-loop system identification algorithms will identify the dynamic EEG response of multiple brain areas to the perturbation, leading to a functional interpretation of EEG. The propagation of the signal in time and 3D space through the cortex can be monitored: 4D-EEG. Preliminary experiments with EEG localization have shown that the continuous force perturbations resulted in a better signal-to-noise ratio and coherence than the current method using transient perturbations.. 4D-EEG will be a direct measure of the neural activity in the brain with an excellent temporal response and easy to use in combination with motor control tasks. The new 4D-EEG method is expected to provide a breakthrough in comparison to functional MRI (fMRI) when elucidating the meaning of cortical map plasticity in motor learning. Our second goal is to generate and validate new hypotheses about the longitudinal relationship between motor learning and cortical map plasticity by clinically using 4D-EEG in an intensive, repeated measurement design in patients suffering from a stroke. The application of 4D-EEG combined with haptic robots will allow us to discover how dynamics in cortical map plasticity are related with upper limb recovery after stroke in terms of neural repair and using behavioral compensation strategies while performing a meaningful motor tasks.. The non-invasive 4D-EEG technique combined with haptic robots will open the window about what and how patients (re)learn when showing motor recovery after stroke in order to allow us to develop more effective patient-tailored therapies in neuro-rehabilitation.

3 477 202 €

Start date: 2012-06-01, End date: 2017-05-31

Advanced capitalist economies experience large and persistent movements in asset prices that are difficult to justify with economic fundamentals. The internet bubble of the 1990s and the real state market bubble of the 2000s are two recent examples. The predominant view is that these bubbles are a market failure, and are caused by some form of individual irrationality on the part of market participants. This project is based instead on the view that market participants are individually rational, although this does not preclude sometimes collectively sub-optimal outcomes. Bubbles are thus not a source of market failure by themselves but instead arise as a result of a pre-existing market failure, namely, the existence of pockets of dynamically inefficient investments. Under some conditions, bubbles partly solve this problem, increasing market efficiency and welfare. It is also possible however that bubbles do not solve the underlying problem and, in addition, create negative side-effects. The main objective of this project is to develop this view of asset bubbles, and produce an empirically-relevant macroeconomic framework that allows us to address the following questions: (i) What is the relationship between bubbles and financial market frictions? Special emphasis is given to how the globalization of financial markets and the development of new financial products affect the size and effects of bubbles. (ii) What is the relationship between bubbles, economic growth and unemployment? The theory suggests the presence of virtuous and vicious cycles, as economic growth creates the conditions for bubbles to pop up, while bubbles create incentives for economic growth to happen. (iii) What is the optimal policy to manage bubbles? We need to develop the tools that allow policy makers to sustain those bubbles that have positive effects and burst those that have negative effects.

1 000 000 €

Start date: 2010-04-01, End date: 2015-03-31

"Digital 3D models are gaining more and more importance in diverse application fields ranging from computer graphics, multimedia and simulation sciences to engineering, architecture, and medicine. Powerful technologies to digitize the 3D shape of real objects and scenes are becoming available even to consumers. However, the raw geometric data emerging from, e.g., 3D scanning or multi-view stereo often lacks a consistent structure and meta-information which are necessary for the effective deployment of such models in sophisticated down-stream applications like animation, simulation, or CAD/CAM that go beyond mere visualization. Our goal is to develop new fundamental algorithms which transform raw geometric input data into augmented 3D models that are equipped with structural meta information such as feature aligned meshes, patch segmentations, local and global geometric constraints, statistical shape variation data, or even procedural descriptions. Our methodological approach is inspired by the human perceptual system that integrates bottom-up (data-driven) and top-down (model-driven) mechanisms in its hierarchical processing. Similarly we combine algorithms operating on different levels of abstraction into reconstruction and modeling networks. Instead of developing an individual solution for each specific application scenario, we create an eco-system of algorithms for automatic processing and interactive design of highly complex 3D models. A key concept is the information flow across all levels of abstraction in a bottom-up as well as top-down fashion. We not only aim at optimizing geometric representations but in fact at bridging the gap between reconstruction and recognition of geometric objects. The results from this project will make it possible to bring 3D models of real world objects into many highly relevant applications in science, industry, and entertainment, greatly reducing the excessive manual effort that is still necessary today."

2 482 000 €

Start date: 2014-03-01, End date: 2019-02-28

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

The understanding of spatial, social and dynamical organization of large numbers of agents is presently a fundamental issue in modern science. ADORA focuses on problems motivated by biology because, more than anywhere else, access to precise and many data has opened the route to novel and complex biomathematical models. The problems we address are written in terms of nonlinear partial differential equations. The flux-limited Keller-Segel system, the integrate-and-fire Fokker-Planck equation, kinetic equations with internal state, nonlocal parabolic equations and constrained Hamilton-Jacobi equations are among examples of the equations under investigation. The role of mathematics is not only to understand the analytical structure of these new problems, but it is also to explain the qualitative behavior of solutions and to quantify their properties. The challenge arises here because these goals should be achieved through a hierarchy of scales. Indeed, the problems under consideration share the common feature that the large scale behavior cannot be understood precisely without access to a hierarchy of finer scales, down to the individual behavior and sometimes its molecular determinants. Major difficulties arise because the numerous scales present in these equations have to be discovered and singularities appear in the asymptotic process which yields deep compactness obstructions. Our vision is that the complexity inherent to models of biology can be enlightened by mathematical analysis and a classification of the possible asymptotic regimes. However an enormous effort is needed to uncover the equations intimate mathematical structures, and bring them at the level of conceptual understanding they deserve being given the applications motivating these questions which range from medical science or neuroscience to cell biology.

2 192 500 €

Start date: 2017-09-01, End date: 2022-08-31

"Significant parts of our cultural heritage are produced on the Web, yet only insufficient opportunities exist for accessing and exploring the past of the Web. The ALEXANDRIA project aims to develop models, tools and techniques necessary to archive and index relevant parts of the Web, and to retrieve and explore this information in a meaningful way. While the easy accessibility to the current Web is a good baseline, optimal access to Web archives requires new models and algorithms for retrieval, exploration, and analytics which go far beyond what is needed to access the current state of the Web. This includes taking into account the unique temporal dimension of Web archives, structured semantic information already available on the Web, as well as social media and network information. Within ALEXANDRIA, we will significantly advance semantic and time-based indexing for Web archives using human-compiled knowledge available on the Web, to efficiently index, retrieve and explore information about entities and events from the past. In doing so, we will focus on the concurrent evolution of this knowledge and the Web content to be indexed, and take into account diversity and incompleteness of this knowledge. We will further investigate mixed crowd- and machine-based Web analytics to support long- running and collaborative retrieval and analysis processes on Web archives. Usage of implicit human feedback will be essential to provide better indexing through insights during the analysis process and to better focus harvesting of content. The ALEXANDRIA Testbed will provide an important context for research, exploration and evaluation of the concepts, methods and algorithms developed in this project, and will provide both relevant collections and algorithms that enable further research on and practical application of our research results to existing archives like the Internet Archive, the Internet Memory Foundation and Web archives maintained by European national libraries."

2 493 600 €

Start date: 2014-03-01, End date: 2019-02-28

The objective of this proposal is to bring together a local team of young researchers who will work closely with international collaborators to advance the state of the art of Algorithmic Game Theory and open new venues of research at the interface of Computer Science, Game Theory, and Economics. The proposal consists mainly of three intertwined research strands: algorithmic mechanism design, price of anarchy, and online algorithms. Specifically, we will attempt to resolve some outstanding open problems in algorithmic mechanism design: characterizing the incentive compatible mechanisms for important domains, such as the domain of combinatorial auctions, and resolving the approximation ratio of mechanisms for scheduling unrelated machines. More generally, we will study centralized and distributed algorithms whose inputs are controlled by selfish agents that are interested in the outcome of the computation. We will investigate new notions of mechanisms with strong truthfulness and limited susceptibility to externalities that can facilitate modular design of mechanisms of complex domains. We will expand the current research on the price of anarchy to time-dependent games where the players can select not only how to act but also when to act. We also plan to resolve outstanding questions on the price of stability and to build a robust approach to these questions, similar to smooth analysis. For repeated games, we will investigate convergence of simple strategies (e.g., fictitious play), online fairness, and strategic considerations (e.g., metagames). More generally, our aim is to find a productive formulation of playing unknown games by drawing on the fields of online algorithms and machine learning.

2 461 000 €

Start date: 2013-04-01, End date: 2019-03-31

Our field is cryptology. Our overarching objective is to advance significantly the frontiers in design and analysis of high-security cryptography for the future generation. Particularly, we wish to enhance the efficiency, functionality, and, last-but-not-least, fundamental understanding of cryptographic security against very powerful adversaries. Our approach here is to develop completely novel methods by deepening, strengthening and broadening the algebraic foundations of the field. Concretely, our lens builds on the arithmetic codex. This is a general, abstract cryptographic primitive whose basic theory we recently developed and whose asymptotic part, which relies on algebraic geometry, enjoys crucial applications in surprising foundational results on constant communication-rate two-party cryptography. A codex is a linear (error correcting) code that, when endowing its ambient vector space just with coordinate-wise multiplication, can be viewed as simulating, up to some degree, richer arithmetical structures such as finite fields (or products thereof), or generally, finite-dimensional algebras over finite fields. Besides this degree, coordinate-localities for which simulation holds and for which it does not at all are also captured. Our method is based on novel perspectives on codices which significantly widen their scope and strengthen their utility. Particularly, we bring symmetries, computational- and complexity theoretic aspects, and connections with algebraic number theory, -geometry, and -combinatorics into play in novel ways. Our applications range from public-key cryptography to secure multi-party computation. Our proposal is subdivided into 3 interconnected modules: (1) Algebraic- and Number Theoretical Cryptanalysis (2) Construction of Algebraic Crypto Primitives (3) Advanced Theory of Arithmetic Codices

2 447 439 €

Start date: 2017-10-01, End date: 2022-09-30