Poisson geometry is a classical 19th century subject (going back to the work of Poisson, Lie, and
Hamilton) which has experienced a tremendous development in the past
decades. It is now at the crossroads of several areas of pure and
applied mathematics such as differential geometry, the calculus of
variations, noncommutative algebra, representation theory, geometric
mechanics, symmetric Hamiltonian bifurcation theory, and structure
preserving numerical algorithms.
Its impact on certain areas of theoretical physics and engineering has
also been significant. Direct applications of Poisson geometry can be
found nowadays in string theory, the theory of integrable systems, the
theory of geometric phases, nonlinear control theory, nonholonomic
mechanics, locomotion generation in robotics, and planetary
mission design.
The fundamental
structure
of Poisson geometry goes back to
classical mechanics on phase space. The properties of the Poisson
bracket of functions are abstracted and set on a manifold.
Symplectic manifolds and (pre)duals of Lie algebras are special cases
of Poisson manifolds. The interest for the general case stems from the
fact that this abstract framework is particularly suitable
for dealing with symmetries, for understanding quantization, and for
studying various infinite-dimensional mechanical systems.
The aim of the Poisson semester is to bring
together
leading scientists and young researchers from different areas of
mathematics, physics, and engineering with a common interest in
Poisson geometry, its applications and its related fields which include
- Symmetries and momentum maps,
- Geometric mechanics,
- Lie algebroids and Lie groupoids,
- Dirac geometry and generalized complex
geometry,
- Geometric quantization and deformation
quantization,
- Theory of integrable systems,
- Theory of nonholonomic systems,
- Gerbes and other higher structures.
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