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Poisson Algebras, deformations and resolutions of singularities

Final Report Summary - POISSONALGEBRAS (Poisson algebras, deformations and resolutions of singularities)

The general topics of this project were Poisson algebras, their quantisations and their resolutions of singularities. Poisson brackets were invented by Simeon-Denis Poisson at the beginning of the 19th century in his study of the three-body problem in classical mechanics. In that century, Lagrange, Jacobi and Lie studied the properties of Poisson brackets and their underlying geometry. In particular, this led them to the discovery of several modern fruitful notions such as the famous Jacobi identity, and the notion of Lie brackets.

In the 20th century, Poisson brackets have led to the notion of Poisson algebras: a Poisson algebra is a commutative associative algebra together with a Poisson bracket; that is, a Lie bracket that satisfies Leibniz's rule (the bracket is a derivation in each argument). The modern formulation of Poisson algebras is due among others to Lichnerowicz, Kirillov and Weinstein. Poisson algebras have connections with many areas of mathematics and physics (differential geometry, Lie groups and representation theory, noncommutative geometry, integrable systems, singularities, quantum field theory, ...) and so, because of its wide range of applications, their study is of great interest for both the mathematicians and theorical physicists. Currently, this subject is one of the most active in both Mathematics and Mathematical Physics.

One way to approach Poisson algebras is via (deformation) quantisation. In Physics, quantisation is the transition from Classical to Quantum Mechanics. Mathematically, (deformation) quantisation is the transition from Poisson algebras (Poisson geometry) to noncommutative algebras (noncommutative geometry, that is, geometry of ``noncommutative spaces''). Roughly speaking, the idea is to use the Poisson bracket in order to deform the commutative product on the Poisson algebra under consideration---the elements of this algebra being the observables of classical mechanics---and obtain a noncommutative product suitable for quantum mechanics. The existence of such deformation quantisations is a long-standing problem. The proof of the Formality Conjecture by Kontsevich led to the existence and classification of deformation quantisations of arbitrary Poisson manifolds. Because of its importance, this result has an impact on many areas of mathematics.

The main aim of the project was to study Poisson algebras, their quantisations and their resolutions of singularities. Significant positive results have been obtained by the researcher during this project. The main achievements are the following.

1. Introduction of new techniques from automaton theory in order to study the representation theory of quantum algebras.
2. Observation and study of a strong link between the torus-orbits of symplectic leaves of Poisson matrix varieties and torus-invariant prime ideals in quantum matrices. Moreover the researcher and his collaborators showed that the latter were also linked to the theory of total positivity in the sense of Lusztig. This new and unexpected connection between totally nonnegative matrices and quantum matrices has allowed the development of new algorithms to study totally nonnegative matrices.
3. Development of a ``Schubert cells" approach to quantum flag varieties. Moreover the researcher and his collaborators gave a geometrical description of quantum Schubert cells.
4. New connections between Poisson cohomology and Hochschild cohomology in the singular case.
5. Study of quantum cluster algebras.

To summarise, this project led to several new and unexpected results that have been published in a wide range of international leading journals. For all these reasons, this project has been truly successful.