ERC FUNDED PROJECTS

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 Laplacian with a dynamical system on the cotangent bundle. The proper description of this object can be given in analytic, index theoretic and probabilistic terms, which explains both its potential many applications, and also its complexity.

1 112 400 €

Start date: 2012-02-01, End date: 2017-01-31

"During the past twenty years, we have witnessed profound technological changes, summarised under the terms of digital revolution or entering the information age. It is evident that these technological changes will have a deep societal impact, and questions of privacy and security are primordial to ensure the survival of a free and open society. Cryptology is a main building block of any security solution, and at the heart of projects such as electronic identity and health cards, access control, digital content distribution or electronic voting, to mention only a few important applications. During the past decades, public-key cryptology has established itself as a research topic in computer science; tools of theoretical computer science are employed to “prove” the security of cryptographic primitives such as encryption or digital signatures and of more complex protocols. It is often forgotten, however, that all practically relevant public-key cryptosystems are rooted in pure mathematics, in particular, number theory and arithmetic geometry. In fact, the socalled security “proofs” are all conditional to the algorithmic untractability of certain number theoretic problems, such as factorisation of large integers or discrete logarithms in algebraic curves. Unfortunately, there is a large cultural gap between computer scientists using a black-box security reduction to a supposedly hard problem in algorithmic number theory and number theorists, who are often interested in solving small and easy instances of the same problem. The theoretical grounds on which current algorithmic number theory operates are actually rather shaky, and cryptologists are generally unaware of this fact. The central goal of ANTICS is to rebuild algorithmic number theory on the firm grounds of theoretical computer science."

1 453 507 €

Start date: 2012-01-01, End date: 2016-12-31

Despite our great expressive skills, we humans lack an easy way of communicating the 3D shapes we imagine, and even more so when it comes to dynamic shapes. Over centuries humans used drawing and sculpture to convey shapes. These tools require significant expertise and time investment, especially if one aims to describe complex or dynamic shapes. With the advent of virtual environments one would expect digital modeling to replace these traditional tools. Unfortunately, conventional techniques in the area have failed, since even trained computer artists still create with traditional media and only use the computer to reproduce already designed content. Could digital media be turned into a tool, even more expressive and simpler to use than a pen, to convey and refine both static and dynamic 3D shapes? This is the goal of this project. Achieving it will make shape design directly possible in virtual form, from early drafting to progressive refinement and finalization of an idea. To this end, models for shape and motion need to be totally rethought from a user-centered perspective . Specifically, we propose the new paradigm of responsive 3D shapes – a novel representation separating morphology from isometric embedding – to define high-level, dynamic 3D content that takes form, is refined, moves and deforms based on user intent, expressed through intuitive interaction gestures. Scientifically, while the problem we address belongs to Computer Graphics, it calls for a new convergence with Geometry, Simulation and Human Computer Interaction. In terms of impact, the resulting “expressive virtual pen” for 3D content will not only serve the needs of artists, but also of scientists and engineers willing to refine their thoughts by interacting with prototypes of their objects of study, educators and media aiming at quickly conveying their ideas, as well as anyone willing to communicate a 3D shape This project thus opens up new horizons for science, technology and society.

2 498 116 €

Start date: 2012-04-01, End date: 2017-03-31

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 both smooth manifolds and Legendrian/contact manifolds, in particular, try to define an analogue of Heegaard Floer homology in dimension larger than 3. - The link, in dimension 3, between the geometry of the ambient manifold (especially hyperbolicity) and the dynamical/topological properties of its Reeb vector fields and contact structures. - The topological characterization of odd-dimensional manifolds admitting a contact structure. A crucial ingredient of my program is to understand the key role played by open book decompositions in dimensions larger than three. This program requires a huge amount of mathematical knowledges. My idea is to organize a team around Ghiggini, Laudenbach, Rollin, Sandon and myself, augmented by two post-docs and one PhD student funded by the project. This will give us the critical size to organize a very active working seminar and to have a worldwide attractivity and recognition. I also plan to invite one confirmed researcher every year (for 1-2 months), to organize one conference and one summer school, as well as several focused weeks.

887 600 €

Start date: 2012-01-01, End date: 2016-12-31