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Poisson Algebras, deformations and resolutions of singularities (PoissonAlgebras)
Start date: 01 Oct 2007, End date: 30 Sep 2010 PROJECT  FINISHED 

"The general topics of this proposal are Poisson algebras, their quantisations and their resolutions. Poisson algebras first appeared in the work of Poisson two centuries ago when he was studying the three-body problem in celestial mechanics. Since then, Poisson algebras have been shown to be connected to many areas of mathematics and physics (differential geometry, Lie groups and representation theory, noncommutative geometry, integrable systems, quantum field theory...), and so, because of its wide range of applications, their study is of great interest for both mathematicians and theoretical physicists. Currently, this subject is one of the most active in both mathematics and mathematical physics. One way to approach Poisson algebras is via quantisation. In this context, Poisson algebras are the semiclassical limits of noncommutative algebras. Naturally, this suggests that the underlying geometry of a Poisson algebra should be intimately connected to the noncommutative geometry of the corresponding ""quantum'' noncommutative algebra; the noncommutative geometry of the ""quantum'' spaces is closely related to the geometry of the space of symplectic leaves. The first main aim of this proposal is to gain a better understanding of the link between Poisson algebras and their ""quantum counterparts'', and then, of course, use it to derive some new results on Poisson and ""quantum'' algebras. In the singular case, another way to attack (singular) Poisson algebras is to consider their resolutions of singularities. Roughly speaking, the idea is to attach to a singular Poisson algebra another Poisson algebra that is smooth and that keeps track, at least on the smooth part, of the Poisson structure of the original singular Poisson algebra. The second aim of this project is to study such resolutions; more precisely, we will study the relationship between symplectic singularities and their symplectic resolutions from the point-of-view of representation theory and combinatorics."
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