Three major and interrelated subjects are in the background of the meetings:
Poisson brackets were introduced to study the 3-body problem in celestial mechanics. Their study gained renewed interest in the second half of the twentieth century and POISSON GEOMETRY is now an active field of research, stimulated by its connections with harmonic analysis on Lie groups, quantum physics (in particular deformation quantisation), holonomy and integrable systems.
The passage from one level of physical theory to another can often be achieved using the mathematical notion of deformation, in a properly chosen category. Application of this idea to the passage from classical to quantum mechanics lead Flato, Lichnerowicz and Sternheimer in 1976 to create a new domain called DEFORMATION QUANTISATION.
Symmetries have played a major role in the developments of mathematics and physics in the 20th century and lead to the very wide subject of GROUP REPRESENTATIONS.
Spectacular recent developments have occured in the last years in those three areas and their interplay. For instance, let us mention the proofs of existence and classification for deformation quantisation on general Poisson manifolds, based on the formality theorem, given by Kontsevich in 1997 using graphs and by Tamarkin and him using operads in 1999, which gave rise to a wide array of new results and the extension to algebraic varieties; new geometric constructions of representations of noncompact reductive Lie groups with new character formulas by Kashiwara, Schmid and Vilonen; the development of groupoids and algebroids by Moerdijk, Weinstein and others as a formalism to express very general theories.
The school will present in a few series of talks the fundamentals in those subjects and the conference will develop the state of the art, with new perspectives and new results.