Résumé des résultats (Abstract)
(Français)
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Geometric Analysis: analysis (mostly, partial differential equations) on spaces which range from the most regular ones (smooth) to the very irregular, or singular, heterogeneous structures, including: smooth spaces with boundary (as the airplane wing), crystals, semiconductors, porous media, propagation and equilibrium states of waves and fields (acoustic, heat, fluid, electromagnetic) in irregular spaces with, or without, obstacles. The non commutative geometry, very recent field of research created by Alain Connes, integral part of this project, is the unifying tool for studying all these spaces and beyond. The present project, involving some of the very top leading specialists and laboratories in the world in these fields, intends to give further major contributions in these directions. The proposed research intends to extend the existing foundational mathematical tools necessary to make these spaces (especially, singular) more accessible to scientific (mathematics, physics, biology) and technological applications. Catastrophe theory is, for example, a chapter of the theory of singularities.
Formation of turbulence around the edges of the airplane wing, the fact that the lightening hits the acuminated objects, are manifestations of the presence of singularities in these spaces. The complexity of the problems encountered in this multidisciplinary study creates a very fertile and challenging field of research. The first foundational problem in the study of singular spaces requires to create the correct analysis necessary for their study. Index theory studies geometrical-analytical properties of spaces which remain invariant under continuous deformations. Foundations of Index theory were layed by Atiyach-Singer on smooth spaces. It is a challenge to extend it to singular spaces. Present project proposes to make breakthrough contributions in these directions. Technological applications, especially in the sector of electronics (solid state physics) are possible.
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