Considering (Lie) groupoids in order to study geometrical objects started with C. Ehresmann followed by his students in the 50's. Since then, Lie groupoids in relation to geometrical objects have appeared in many other branches of mathematics and physics.
In particular, Lie groupoids have appeared to be an efficient tool in the analytic treatment of geometric singularities since the use of the holonomy groupoid by A. Connes in order to get a pseudodifferential calculus and establish an index theorem for foliations in the 80's. Following his ideas, many groupoids have been constructed in order to carry a pseudodifferential calculus adapted to a given singular space in the last decades: manifolds with boundaries, with corners, stratified spaces possibly foliated, Carnot manifolds... Parallel to this, in the context of quantization problems, A. Weinstein introduced the symplectic groupoid of a Poisson manifold in the 80's. This work has witnessed, and still witnesses, very important developments in symplectic geometry in the broad sense (e.g. Poisson geometry, Dirac geometry, Generalized complex geometry). In most of the situations mentioned above lies the question of associating a suitable groupoid with a singular geometrical space. These questions are related to the problem of integrating Lie algebroids into Lie groupoids or by integrating singular foliations into groupoids (fiberwise Lie). Important recent advances were also obtained in the theory of holomorphic foliations. Complex manifolds endowed with holomorphic geometric structures are important models in theoretical physics. In the same time their geometric features are interesting from many mathematical points of view.
The objective of this school is to bring together different mathematicians working on various aspects of Lie groupoids in relation with foliations and singular spaces and their applications in order to offer students a broad panorama of the techniques and problems encountered.
Speakers
Marius Crainic - Utrecht University (Netherlands)
Claire Debord - Université de Paris (France)
Bertrand Deroin - CY Cergy Paris Université (France) - (To be confirmed)
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