Objectif "Our aim is to built on recent combinatorial and algorithmic progress to attack a series of deeply connected problems that have independantly surfaced in enumerative topology, statistical physics, and data compression. The relation between these problems lies in the notion of ""combinatorial map"", the natural discrete mathematical abstraction of objects with a 2-dimensional structures (like geographical maps, computer graphics' meshes, or 2d manifolds). A whole new set of properties of these maps has been uncovered in the last few years under the impulsion of the principal investigator. Rougly speaking, we have shown that classical graph exploration algorithms, when correctly applied to maps, lead to remarkable decompositions of the underlying surfaces. Our methods resort to algorithmic and enumerative combinatorics. In statistical physics, these decompositions offer an approach to the intrinsec geometry of discrete 2d quantum gravity: our method is here the first to outperform the celebrated ""topological expansion of matrix integrals"" of Brezin-Itzykson-Parisi-Zuber. Exploring its implications for the continuum limit of these random geometries is our great challenge now. From a computational geometry perspective, our approach yields the first encoding schemes with asymptotically optimal garanteed compression rates for the connectivity of triangular or polygonal meshes. These schemes improve on a long series of heuristically efficient but non optimal algorithms, and open the way to optimally compact data structures. Finally we have deep indications that the properties we have uncovered extend to the realm of ramified coverings of the sphere. Intriguing computations on the fundamental Hurwitz's numbers have been obtained using the ELSV formula, famous for its use by Okounkov et al. to rederive Kontsevich's model. We believe that further combinatorial progress here could allow to bypass the formula and obtaine an elementary explanation of these results." Champ scientifique natural sciencesmathematicspure mathematicstopologynatural sciencesmathematicspure mathematicsgeometrynatural sciencesmathematicspure mathematicsdiscrete mathematicscombinatoricsnatural sciencesphysical sciencestheoretical physics Mots‑clés Enumeration data structures design and analysis of algorithms enumerative topology random discrete structures Programme(s) FP7-IDEAS-ERC - Specific programme: "Ideas" implementing the Seventh Framework Programme of the European Community for research, technological development and demonstration activities (2007 to 2013) Thème(s) ERC-SG-PE1 - ERC Starting Grant - Mathematical foundations Appel à propositions ERC-2007-StG Voir d’autres projets de cet appel Régime de financement ERC-SG - ERC Starting Grant Institution d’accueil CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE CNRS Contribution de l’UE € 750 000,00 Adresse RUE MICHEL ANGE 3 75794 Paris France Voir sur la carte Région Ile-de-France Ile-de-France Paris Type d’activité Research Organisations Contact administratif Julie Zittel (Ms.) Chercheur principal Gilles Schaeffer (Dr.) Liens Contacter l’organisation Opens in new window Site web Opens in new window Coût total Aucune donnée Bénéficiaires (1) Trier par ordre alphabétique Trier par contribution de l’UE Tout développer Tout réduire CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE CNRS France Contribution de l’UE € 750 000,00 Adresse RUE MICHEL ANGE 3 75794 Paris Voir sur la carte Région Ile-de-France Ile-de-France Paris Type d’activité Research Organisations Contact administratif Julie Zittel (Ms.) Chercheur principal Gilles Schaeffer (Dr.) Liens Contacter l’organisation Opens in new window Site web Opens in new window Coût total Aucune donnée