Programme:

 

 

Schedule:

Time

Monday

Tuesday

Wednesday

Thursday

Friday

Weekend-HalongBay

9:30-11

(Opening)

Chenciner

Zung

Chenciner

Zung

Chenciner

(1-hour talk)

 

 

 

EXCURSION TO HALONG BAY

Break

 

 

 

 

 

11:30-12:30

Salamon

Gonzalo

Perez-Marco

Gurel

Zaitsev

LUNCH

 

 

 

 FREE AFTERNOON

 

 

14:00-15:00

Vu The Khoi

Latschev

Hamilton

 Rodriguez-Olmos

15:00-16:00

Usher

Le Anh Vu

Le Van Hong

 Solha

Break

 

 

 

 

16:30-17:30

Do Ngo Diep

Sandon

Batoreo

(Closing)

Chaperon

 

Colour's code: In orange: Minicourses (1 hour 30 minutes)

                       In Turquoise-blue: 1-hour talks

Note: There will be a Banquet of Thursday's night.

On Friday talks will start at 10.00 am.

                

 

Minicourses:

 Alain Chenciner (Observatoire de Paris-IMCCE): "The Lagrange reduction of the N-body problem"

Abstract :
In his fondamental "Essai sur le problème des trois corps" (Essay on the 3-body problem), Lagrange, well before Jacobi's "reduction of the node", carries out the first complete reduction of symetries in this problem. Discovering the so-called homographic motions (Euler had treated only the colinear case), he shows that these motions necessarily take place in a fixed plane, a result which is simple only for the "relative equilibria". In order to understand the true nature of this reduction -- and of Lagrange's equations - it is necessary to
consider the n-body problem in an euclidean space of arbitrary dimension. The actual dimension of the  ambiant space then appears as a constraint, namely the the angular momentum bivector's degeneracy. I shall describe the results obtained in a joint paper with Alain Albouy published (in french !) in 1998. For a non homothetic homographic motion to exist, it is necessary that the motion takes place in an even dimensional space E. Two cases are possible: either the configuration is ``central" (that is a critical point of the potential among configurations with a given moment of inertia) and the space E is endowed with an hermitian structure, or it is ``balanced" (that is a critical point of the potential among configuration with a given inertia spectrum) and the motion is a new type, quasi-periodic, of relative equilibrium.
 

References:
J.L. Lagrange, Essai sur le problème des trois corps, Oeuvres volume 6, pages 229-324 (1772)
E. Betti, Sopra il moto di un sistema di un numero qualunque di punti che si attragono o si
respingono tra di loro, Annali di Matematica s.2 t. 8, pages 301-311 (1877)
A. Wintner, The analytical foundations of Celestial Mechanics, Princeton University Press (1941)
A. Albouy & A. Chenciner, Le problème des n corps et les distances mutuelles, Inventiones
mathematicae 131, pages 151-184 (1998)
A. Albouy, Mutual distances in Celestial Mechanics, Lectures at Nankai Institute, Tianjin, Chine, juin 2004


 

 

 

 

 

 

 

 

 

 

Nguyen Tien Zung (Toulouse)    "Entropy in Physics and Mathematics"

Abstract :
In this minicrouse, intended to be accessible also to undergraduate students,
I'll try to explain various manifestiations of the concept of entropy, including
Shannon entropy in information theory (amount of information),
Clausius-Boltzmann entropy in physics (laws of thermodynamics),
Kolmogorov-Sinai entropy in dynamical systems (rate of information), entropy
in geometry problems (e.g. isoperimetric inequalities), etc. In the last part
of this minicourse I'll talk about my little results on the entropy of geometric
structures, which generalizes the notion of geometric entropy of folations introduced
by Ghys--Langevin--Walczak.
 

 

 

 

 

Lecturers:

Marta Batoreo (UC Santa Cruz):"On the Rigidity of the Maslov Index for Coisotropic Submanifolds of
Rational Symplectic Manifolds"

Abstract: A result of V. Ginzburg on the rigidity of the coisotropic
Maslov index asserts that there exists a non-trivial loop (tangent to
the characteristic foliation of a stable coisotropic submanifold of a
symplectically aspherical manifold) with certain bounds on its Maslov
index and area. This theorem also holds for some symplectic manifolds
not necessarily symplectically aspherical. We shall state the theorem
for the rational case and sketch its proof.

 

 

 

Marc Chaperon (Paris VII)" Generalized Hopf bifurcations"

Abstract:Following Thom's principle: "Always look for the organizing center of phenomena", I was able to prove a general "birth lemma" for families of dynamical systems at partially elliptic rest points; as noticed by Santiago López de Medrano, it implies the birth of normally hyperbolic compact invariant manifolds diffeomorphic to all kinds of moment-angle manifolds in generic families; these manifolds can form a family of matrioshkas, providing for example a very simple model for the transition between two periodic regime

 


Alain Chenciner (Observatoire de Paris-IMCCE): "The angular momentum of a relative equilibrium in dimensions higher than 3" abstract

Jesus Gonzalo (Madrid), "Flow dynamics and the existence of contact circles"

Abstract:For flows tangent to plane fields in dimension 3 we define a twisting property, related to the Anosov property. We shall explain how this property gives a simple geometric understanding
of contact circles, which are pairs of contact structures discovered and studied by H. Geiges and the speaker. We shall mention open questions and challenges provided by this connection. For example one can use Seiberg-Witten to exhibit plane fields none of whose tangent flows satisfy the twisting property.

Basak Gurel (Vanderbilt) "Conley conjecture for negative monotone symplectic manifolds"

Abstract: The Conley conjecture, formulated by Conley in 1984, asserts the existence of infinitely many periodic orbits for Hamiltonian diffeomorphisms of tori and was established by Hingston in 2004. Of course, one can expect the conjecture to hold for a much broader class of closed manifolds and this is indeed the case. For instance, by now, it has been proved for all closed, symplectically aspherical manifolds and Calabi-Yau manifolds using symplectic topological methods. Most recently, jointly with Ginzburg, we establish the conjecture for negative monotone, closed symplectic manifolds.

In this talk, based on joint works with Ginzburg, we will examine the question of existence of infinitely many periodic orbits for Hamiltonian diffeomorphisms and outline a proof of the Conley conjecture in the negative monotone case.

 

 

 

Mark Hamilton ( Mount Allison), "Real and Kahler quantization of flag manifolds."

Janko Latschev (Hamburg University) "Algebraic torsion in contact manifolds

AbstractIn this talk, I will report on recent joint work with Chris Wendl
which uses the algebraic formalism of symplectic field theory to
define a hierarchy of obstructions to symplectic filling of and
exact symplectic corbordisms between given contact manifolds.

I will also present 3-dimensional examples to show that these
obstructions are non-trivial and distinct.




 

 

Do Ngoc Diep (Institute of Mathematics, VAST ) "Geometric Quantization of Fields Based on Geometric Langlands
Correspondence"

Abstract:We expose a new procedure of quantization of fields, based on the
Geometric Langlands Correspondence. Starting from fields in the target
space, we first reduce them to the case of fields on
one-complex-variable target space, at the same time increasing the
possible symmetry group $G^L$. Use the sigma model and momentum maps,
we reduce the problem to a problem of quantization of trivial vector
bundles with connection over the space dual to the Lie algebra of the
symmetry group $G^L$. After that we quantize the vector bundles with
connection over the coadjoint orbits of the symmetry group $G^L$. Use
the electric-magnetic duality to pass to the Langlands dual Lie group
$G$. Therefore, we have some affine Kac-Moody loop algebra of
meromorphic functions with values in Lie algebra Lie(G). Use the
construction of Fock space reprsentations to have representations of
such affine loop algebra. And finally, we have the automorphic
representations of the corresponding Langlands-dual Lie groups $G$.
http://www.hindawi.com/journals/ijmms/2009/749631/:

 

 

 

 

 

 

 

Ricardo Perez Marco (Paris XIII)" Total integrability or general non-integrability in Hamiltonian Dynamics"

Miguel Rodríguez-Olmos (Universitat Politecnica de Catalunya) "Symmetric Hamiltonian bifurcations and isotropy"

Abstract:We will use the bundle equations of Roberts et al for symmetric
Hamiltonian systems in order to study some aspects of the stability
and bifurcations of relative equilibria. It will be shown how the
existence of continuous isotropy groups for the symmetry action can
affect these properties with respect to the free case. Notably, we
will discuss how these isotropy groups can induce bifurcations from a
formally stable branch of relative equilibria. This is a joint work
with J. Montaldi.

 


 

 

Dietmar Salamon (ETH-Zurich): "Uniqueness of symplectic structures."

Sheila Sandon (Nantes) "On existence of translated points for contactomorphisms"
 

Abstract:A point p in a contact manifold is called a translated point for a
contactomorphism \phi with respect to a fixed contact form if p and
\phi(p) belong to the same Reeb orbit and if the contact form is preserved
at p. In my talk I will discuss the problem of existence of translated
points, and its relation with the Arnold conjecture, the chord conjecture
and the problem of leafwise coisotropic intersections. If I will have the
time I will also explain how to use generating functions techniques to
study this problem for contactomorphisms of the euclidean space, the
sphere and the projective space.
 

 

 


 

Romero Solha (Univesitat Politecnica de Catalunya) "Foliated cohomology and geometric quantisation of integrable systems with singularities"

Abstract:  This talk shows an attempt to extend some results by Snyaticki, Guillemin and Sternberg in geometric quantisation considering regular fibrations as real polarisations to the singular setting. The real polarisations concerned here are given by integrable systems with nondegenerate singularities (in the Morse-Bott sense). And the definition of geometric quantisation used is the one suggested by Kostant; via higher cohomology groups. The case of nondegenerate singularities was obtained in dimension 2 by Hamilton and Miranda and the completely elliptic case was considered by Hamilton in any dimension. The approach is to combine previous results of Miranda and Presas on a Künneth formula to reduce to the 2-dimensional case with an extension of the results of Rawnsley on the Kostant complex. This talk is based on joint work in progress with Eva Miranda.


 

 

Michael Usher (University of Georgia) "Aperiodic symplectic manifolds"

Abstract: We describe a general construction which, on a very diversefamily of closed manifolds, gives rise to symplectic forms that admit
autonomous Hamiltonian flows with no nontrivial periodic orbits. In particular, our family includes many of the classic examples of
interesting symplectic four-manifolds with b+ > 1. This contrasts with a result of Lu which, when combined with results from
Taubes-Seiberg-Witten theory, shows that such symplectic forms can never exist on manifolds with b+ = 1. All this suggests a number of
open questions, some of which we will discuss.

 

 

Le Anh Vu (Hanoi):       " The Structure of Coadjoint Orbits (K-orbits) of a Class of Real   Solvable Lie Groups"

Abstract: The talk introduces the Kirillov’s bilinear form and the symplectic structure on each K-orbit of  an Lie group. We also introduce an overview of foliations formed by the family of maximal dimensional K-orbits (MD- foliations) of solvable real, simply connected Lie groups such that its K-orbits are either orbits of dimension zero or orbits with maximal dimensions (MD-groups). In addition, the talk gives analytical description or characterization Connes' C*-algebras of these foliations by KK-functors. 
 

 

 

Le Van Hong (Praha)  "Smooth structures  on   stratified  symplectic spaces"slides here

Abstract. We  introduce the notion of a smooth structure   on a
stratified symplectic space. We show that under a mild  condition  many
properties of  a symplectic manifold can be extended to a   symplectic
stratified space provided with a smooth Poisson structure, e.g. the
existence  and  uniqueness of a Hamiltonian flow, the isomorphism between  the
Brylinski-Poisson homology and the de Rham homology,  the existence of a
Leftschetz decomposition on a symplectic  stratified space.:

 

 


 

Vu The Khoi, "ON THE BURNS-EPSTEIN INVARIANTS OF SPHERICAL CR 3-MANIFOLDS"

Abstract.Abstract: In this talk we present a method to compute the Burns-
Epstein invariant of a spherical CR homology sphere, up to an in-
teger, from its holonomy representation.

 

Dmitri Zaitsev (Trinity College) "Dynamics of one-resonant biholomorphisms"

Abstract:We construct a simple formal normal form for holomorphic diffeomorphisms in $C^n$ whose differentials have one-dimensional family of resonances in
 the first $m$ eigenvalues, $m\leq n$ (but more resonances are allowed for other eigenvalues).Next, we provide invariants and give conditions for the existence of basins of attraction.
 Finally, we give applications and examples demonstrating the sharpness of our conditions. This is a joint work with Filippo Bracci.