Cascade of orthogonal roots and some of its old and new applications, e.g., to the structure of certain coadjoint orbits See the summary here

“Geometric quantization” of minimal nilpotent orbits

Complex structures in the quantization of cotangent bundles If M is a Riemannian manifold, the cotangent bundle T*(M) of M is a symplectic manifold, which comes equipped with a natural vertical polarization. On the other hand, the work of Guillemin-Stenzel and Lempert-Szoke gives T*(M) a (local) complex polarization associated to the "adapted complex structure." In the case that M is a

Spinoptics and symplectic physics New features of geometrical optics for spinning light have recently been revealed by physicists using a semi-classical approximation of stationary wave optics. We will show that one can regard this "spinoptics" as an instance of "symplectic physics" heralded by Jean-Marie Souriau. The model of spinning and colored coadjoint orbits of the Euclidean group is generalized to circularly polarized light rays in Riemannian

Recent incarnations of the Cartan three form I will recall a few results from my paper with Kostant: “Symplectic Reduction, BRS Cohomology, and Infinite-Dimensional Clifford Algebras” ANNALS OF PHYSICS 176 499-113 (1987) and discuss some recent developments due to Li-Bland and Meinenken extending these results to the construction and properties of Courant algebroids.

Uniqueness of symplectic structures

A fundamental question in symplectic topology is, when the space of symplectic forms in a given cohomology class is connected.  Little is known about this problem in dimensions greater than two. A longstanding conjecture asserts that the answer to the question is positive for the four torus. In this lecture I will discuss some known results

Real and Kaehler Quantization of Flag Manifolds One ingredient in geometric quantization is a polarization; given different polarizations on a manifold, we can ask how the resulting quantizations depend on the choice of polarization.  Guillemin and Sternberg investigated this question for flag manifolds, which admit both real and Kahler polarizations.  They showed that the two

Primary Spaces We call a Hamiltonian N-space primary if its moment map is onto a single coadjoint orbit. The question has long been open whether such spaces always split as (homogeneous) × (trivial), as an analogy with representation theory might suggest. For instance, Souriau’s barycentric decomposition theorem asserts just this when N is a Heisenberg group. For general N, we give explicit examples which do not split, and show instead that primary

Differential operators on contact manifolds

This is a joint work with Charles Conley.

The space of differential operators on a smooth manifold has a rich geometric structure. We will consider a special situation of differential operators on a contact manifold viewed as a module over the group of all contact diffeomorphisms. In particular, we show how to associate a contact vector field to an arbitrary second order linear

HOMOLOGY, INVARIANCE AND INFORMATION
Jean-Marie Souriau claimed that Groups lead our understanding of the physical world. He insisted that in classical mechanics the fundamental observable quantities are defined by a momentum of the Galilée group, and the mass of a system is defined by a cohomology class of this group. We will begin the presentation by a short review of the impact of group cohomology in Physics today. Then we

Integration Bundles of Closed 2-forms We’ll see, on the very general level of diffeological spaces, that any closed 2-form is always the curvature of a connexion on some principal bundle(s), with structure group the torus of periods of the 2-form. And this is so, whatever the periods are, provided they do not fill the real line. I’ll give a classification.

Family algebras and generalized exponents I shall speak about a new class of associative algebras (so-called family algebras) introduced and investigated in my recent papers. These algebras are related to simple complex Lie algebras (or root systems). A family algebra is a sort of a finite approximation to the enveloping algebra U(G) viewed as a module over its center. It turns out that several important questions about semi-simple algebras and their

A homotopy theory for diffeological spaces (joint with Prof. Dan Christensen) Diffeological spaces are a natural generalization of smooth manifolds, introduced by
J.-M. Souriau in 1980, and further developed mainly by
P. Iglesias-Zemmour. In this talk, I’ll discuss some recent development of a homotopy theory (model category structure) on the category of diffeological spaces, with applications.

Souriau’s contribution to cosmology

Angular momentum and Horn's problem The central configurations of n point masses are those configurations which admit periodic rigid motions when submitted to the Newtonian attraction. For example, while completing in 1772 the first full reduction of the symmetries in the 3-body problem, Lagrange proved that the only non-colinear central configuration of 3 positive masses is the equilateral triangle. Such rigid motions necessarily take

Maximal Tori in Symplectomorphism Groups It is often stated that symplectomorphism groups behave like infinite dimentional analogs of compact Lie groups. To make this analogy fruitful, it is desirable to find good candidates for what should play the role of maximal tori in groups of symplectomorphisms. I will survey recent works that investigate those maximal tori in connection to other natural geometric questions.

system, the initial conditions, identifiable with a riemannian metric on a spatial slice S along with a second symmetric 2-tensor, are subject to constraints. The Poisson bracket relations among the constraint functions suggest that they should be something like the components of a momentum map for a symmetry group action, but this is not the case. It turns out that these relations can be realized in the sections of a Lie algebroid derived from a diffeological groupoid of "evolutions" whose morphisms are pairs of space-like, codimension-one embeddings of S in lorentzian manifolds. 

On the Poisson brackets of the constraints in general relativity This talk will touch on four of the five themes of this conference, with the hope that the fifth may not be far behind.  It is based on joint work with Christian Blohmann and Marco Cézar Fernandes. When the Einstein equations for a lorentzian metric on a space-time M are posed as an initial value problem for a hamiltonian

spaces are always flat bundles over the coadjoint orbit. This provides the missing piece for a full ‘Mackey theory’ of Hamiltonian G-spaces, where G is an overgroup in which N is normal. Part of this work is joint with Patrick Iglesias-Zemmour.

differential operator in a contact-invariant manner.

The first steps of Topological Hamiltonian and Contact Dynamics A topological hamiltonian/contact dynamical system is a uniform limit of a sequences of smooth hamiltonian/contact isotopies  whose hamiltonian/contact hamiltonian functions converge in the Hofer/contact-Hofer norm. The talk will discuss (following the work of Banyaga-Spaeth and Muller-Spaeth) the first

steps in topological hamiltonian/contact dynamical systems: namely that the limiting homeomorphisms of topological hamiltonian/contact homeomorphisms are in 1-1 correspondance with the Hofer/ contect-Hofer  limits of the corresponding hamiltonian/contact hamiltonian functions. In the contact case, we obtain results only on topological quantomorphisms.

The moment mapping and non periodic tilings This talk is based on joint work with Fiammetta Battaglia. The moment mapping, in a suitably singular setting, provides an effective way of studying a number of non periodic tilings from a symplectic viewpoint. We will describe this procedure in the case of Penrose rhombus tilings, Penrose kite and dart tilings, and Ammann tilings.

representations can be formulated, studied and sometimes solved in terms of family algebras. In particular, I discuss the relations between family algebras and generalized exponents introduced by B. Kostant.

and Finslerian three-manifolds. "Symplectic scattering" yields the classic Snel-Descartes laws, and the novel spin Hall effect of light, i.e., a tiny transverse deviation of light rays across a plane interface evidenced by Hosten et al. (2008). Further extension to arbitrarily polarized light yields a presymplectic structure featuring the Berry and Pancharatnam connections. This enables us to describe the geodesic deviation of light rays and the evolution of polarization in a Fermat background. These effects gained experimental confirmation (Blokh et al., 2008). We will then deal with spinoptics in the paraxial approximation. Apart from the Hall shift, it highlights a spin-induced modification of the Snel-Descartes laws across a curved interface that might be tested experimentally. Part of this work is joint with Z. Horvath, P. Horvathy and P. Zhang.

will present new results (obtained with P.Baudot, ISC, Paris) showing that other kinds of homology theories can be used to understand other fundamental quantities like the entropy in statistical mechanics and quantum mechanics. Then I will show how the notions of groups and information are tightly linked. Finally, if time permits, I will present a new approach (developed with A.Berthoz, C-d-F, Paris) showing how the Galilée group organizes the flow of information in the visuo-vestibular system of vertebrates

related to this question and explain Donaldson's geometric flow approach to the uniqueness problem for hyperkaehler four manifolds.  The Donaldson approach is based on an infinite dimensional hyperkaehler moment map.

quantizations are in some sense "the same," but did not give any direct relationship between them.  I will describe a deformation of the complex structure on the flag manifold leading to a "convergence of polarizations" joining the two quantizations.  This is join work with Hiroshi Konno of Tokyo University.

place in an euclidean space E of even dimension 2p and, the initial configuration being given, they are uniquely defined by the choice of a complex structure on this space (Albouy-Chenciner, 1998).  Introducing two moment maps related to Horn problems, one in dimension p and one in dimension 2p, and using a deep combinatorial lemma of Fulton-Fomin-Li-Poon, we answer the following question: fixing an initial central configuration of n bodies in E, what can be said of the mapping which, to each complex structure, associates the spectrum of the angular momentum (transformed into an antisymmetric endomorphism via the euclidean structure) of the associated relative equilibrium ?

Part of this work was done with Hugo Jimenez Perez.

1 - A. Chenciner "The angular momentum of a relative equilibrium", arXiv:1102.0025, final version to appear in D.C.D.S.

2 - A. Chenciner & Hugo Jiménez-Pérez "Angular momentum and Horn's problem", arXiv:1110.5030, submitted to the Moscow mathematical Journal.

3 - S. Fomin, W. Fulton, C.K. Li, Y.T. Poon, "Eigenvalues, singular values, and Littlewood-Richardson coefficients",Amer. J. Math. 127, no. 1, 101--127 (2005).

compact Lie group, the adapted complex is global. Furthermore, it has been shown that in the compact group case, the pairing map between the vertically polarized and complex-polarized Hilbert spaces is unitary.  In this talk, I will describe joint work with Will Kirwin in which we construct a generalization of the adapted complex structure. We assume that, in addition to a Riemannian structure, M carries a "magnetic field," described by a closed 2-form. Our construction starts with the vertical polarization and pushes it forward by the Hamiltonian flow for a charged particle moving on M in the presence of the given magnetic field. By taking an imaginary time in the flow (interpreted by a suitable analytic continuation), we obtain the desired complex polarization. In the case M = S^2, the "magnetic" polarization is global. I will also describe briefly work with Jeff Mitchell on the quantization of T*(S^2) with respect to this new polarization.