| XVIIIth
Oporto Meeting on Geometry, Topology and Physics |
||
| 9th to 12th July 2009 | ||
Talks 3: |
Courses | Introductory courses | Invited talks | talks | 1 | 2 | 3 |
|
- On global symmetries of
Poisson-Nijenhuis manifolds (Luca
Stefanini, Penn State University -
Department of Mathematics).
Abstract: Poisson-Nijenhius manifolds were
introduced by Magri and Morosi in 1988 and have been studied ever since
due to their connection to certain bihamiltonian systems and integrable
hierarchies. In this talk, we shall identify the natural global
symmetries of Poisson (almost) Nijenhuis manifolds as Poisson almost
Nijenhuis groupoids, and discuss reduction in this context. Our
approach is essentially algebraic and can be generalized to
Nijenhuis-type structures on Lie algebroids.
- Hamiltonian
Monodromy via Geometric Quantization and Theta Functions (Mauro Spera,
Dipartimento di Informatica - Universita di Verona).
Abstract:
Hamiltonian monodromy is
addressed from the point of view of geometric quantization,
and various differential geometric aspects thereof are dealt with, all
related to holonomies of suitable flat connections. In the
case of completely integrable Hamiltonian systems with two degrees of
freedom, a link is established between monodromy and (2-level)
theta functions, by resorting to the by now classical differential
geometric intepretation of the latter as covariantly constant sections
of a flat connection, via the heat equation. Furthermore, it is shown
that monodromy is tied to the braiding of the Weiestrass roots
pertaining to a Lagrangian torus, when endowed with a natural
complex structure (making it an elliptic curve) manufactured from a
natural basis of cycles thereon. Finally, a new derivation of the
monodromy of the spherical pendulum is provided.
Abstract:
We study Dirac Lie groups,
that is, Lie groups endowed with multiplicative Dirac structures. We
give the definition of this generalization of Poisson Lie groups,
together with first geometric properties of these objects. The notion
of the Lie bialgebra of a Poisson Lie group is expanded to the
situation of a Dirac Lie group. We define Dirac homogeneous spaces of
Dirac Lie groups and give a structure Theorem for a special class of
closed Dirac Lie groups. This is a progress report.
Abstract:
Just as symplectic geometry
is a natural setting for the classical mechanics of point particles,
2-plectic geometry can be used to describe classical strings. Just as a
symplectic manifold is equipped with a closed non-degenerate 2-form, a
"2-plectic manifold" is equipped with a closed non-degenerate 3-form.
The Poisson bracket makes the smooth functions on a symplectic manifold form a Lie algebra. Similarly, any 2-plectic manifold gives a "Lie 2-algebra": the categorified analogue of a Lie algebra, where the usual laws hold only up to isomorphism. We explain these ideas and use them to give a new construction of the "string Lie 2-algebra" associated to a simple Lie group (pdf)
The Poisson bracket makes the smooth functions on a symplectic manifold form a Lie algebra. Similarly, any 2-plectic manifold gives a "Lie 2-algebra": the categorified analogue of a Lie algebra, where the usual laws hold only up to isomorphism. We explain these ideas and use them to give a new construction of the "string Lie 2-algebra" associated to a simple Lie group (pdf)