| XVIIIth
Oporto Meeting on Geometry, Topology and Physics |
||
| 9th to 12th July 2009 | ||
Talks 2: |
Courses | Introductory courses | Invited talks | talks | 1 | 2 | 3 |
|
- Variational
principles and
Poisson brackets for complex fluids and systems (Francois Gay-Balmaz,
EPFL, Section de Mathematique, Lausanne).
Abstract:
As
opposed to the case of (simple) fluids, the dynamics complex fluids
depends also on variables called order parameters that describe the
macroscopic variations of the internal structure of the fluid parcels.
These macroscopic variations may form observable patterns, as seen in
liquid crystals. Other examples of perfect complex fluids include
superfluids, Yang-Mills magnetofluids and spin-glasses.
From a series of papers by Holm, Marsden, Ratiu, and Weinstein, it is known that the techniques of Euler-Poincare and Lie-Poisson reductions for semidirect products provide a unified approach for fluids and plasmas and lead to new models.
In this talk I will present an extension of this framework to the case of complex fluids and systems, by using the new theory of affine Euler-Poincare and affine Lie-Poisson reductions with cocycles.
As a consequence of the Lagrangian approach, the variational formulation of the equations is determined. On the Hamiltonian side, the associated Poisson brackets and their symplectic leaves are obtained by reduction of a canonical cotangent bundle.
From a series of papers by Holm, Marsden, Ratiu, and Weinstein, it is known that the techniques of Euler-Poincare and Lie-Poisson reductions for semidirect products provide a unified approach for fluids and plasmas and lead to new models.
In this talk I will present an extension of this framework to the case of complex fluids and systems, by using the new theory of affine Euler-Poincare and affine Lie-Poisson reductions with cocycles.
As a consequence of the Lagrangian approach, the variational formulation of the equations is determined. On the Hamiltonian side, the associated Poisson brackets and their symplectic leaves are obtained by reduction of a canonical cotangent bundle.
Abstract:
A known caveat about the
concept of Lie algebroid is that Lie algebroids, unlike Lie algebras,
do not have an adjoint representation. Various solutions have been
proposed; ultimately Crainic and Arias Abad [1] found a good candidate
for an adjoint representation by considering superrepresentations
instead (which they call "representations up to homotopy").
Unfortunately, their construction depends on a non-canonical choice.
In this talk, we show that there is an intrinsic object that avoids making that choice: VB-algebroids. A VB-algebroid can be defined as a "vector-bundle-object in the category of Lie algebroids", and the concept was originally introduced by Mackenzie [2] independently of the problem of Lie algebroid representations. One could say that a VB-algebroid is to a superrepresentation of Lie algebroids what a linear map is to a matrix.
This is joint work with Rajan Mehta (pdf)
In this talk, we show that there is an intrinsic object that avoids making that choice: VB-algebroids. A VB-algebroid can be defined as a "vector-bundle-object in the category of Lie algebroids", and the concept was originally introduced by Mackenzie [2] independently of the problem of Lie algebroid representations. One could say that a VB-algebroid is to a superrepresentation of Lie algebroids what a linear map is to a matrix.
This is joint work with Rajan Mehta (pdf)
- Seildel
Morphism on the Hamiltonian Group of a Cartesian Product (Andres Pedroza,
Universidad de Colima, Mexico).
Abstract:
The Seidel
homomorphism is a map from the fundamental group of the group of
Hamiltonain diffeomorphisms,$\textup{Ham}(M,\omega)$, to the quantum
homology ring $QH_*(M;\Lambda)$. Using this homomorphism we give a
sufficient condition for when a nontrivial loop $\psi$ in
$\textup{Ham}(M,\omega)$ determines a nontrivial loop
$\psi\times\textup{id}_N$ in $\textup{Ham}(M\times
N,\omega\oplus\eta)$, where $(N,\eta)$ is a closed symplectic manifold
such that $\pi_2(N)=0$. Recently R. Leclercq, generalized this result
by removing the topological constraint on N.
Abstract:
VB-algebroids are "vector
bundle objects in the category of Lie algebroids". As Gracia-Saz
will have explained in his talk, there is a sense in which
VB-algebroids should be thought of as generalized Lie algebroid
representations. Motivated by this perspective, we construct
characteristic classes associated to VB-algebroids. In the case
of the VB-algebroid that plays the role of the adjoint representation,
these characteristic classes agree with those of Crainic and
Fernandes. Finally, we classify VB-algebroids satisfying a
regularity condition, and in the process a new characteristic class
emerges.