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EXPLOREMAPS

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Combinatorial methods from enumerative topology to random discrete structures a...

Combinatorial methods from enumerative topology to random discrete structures and compact data representations "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 relatio... ver más

30/06/2013

CNRS

750K€

Presupuesto del proyecto: 750K€

Líder del proyecto

CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE... No se ha especificado una descripción o un objeto social para esta compañía.

TRL 4-5

Fecha límite participación Sin fecha límite de participación.

Financiación concedida El organismo FP7 notifico la concesión del proyecto el día 2013-06-30 No tenemos la información de la convocatoria

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Líder del proyecto

CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE... No se ha especificado una descripción o un objeto social para esta compañía.

TRL 4-5

Presupuesto del proyecto 750K€

Fecha límite de participación Sin fecha límite de participación.

Descripción del proyecto "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."

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