Reduction in contact geometry.
Reduction of symplectic Lie algebroids by a Lie subalgebroid and a symmetry Lie group.
Singular Poisson reduction of cotangent bundles.
We consider the Poisson reduced space (T* Q)/K, where the action of the compact Lie group K on the configuration manifold Q is of single orbit type and is cotangent lifted to T* Q. Realizing (T* Q)/K as a Weinstein space we determine the induced Poisson structure and its symplectic leaves. We thus extend the Weinstein construction for principal fiber bundles to the case of surjective Riemannian submersions Q → Q/K which are of single orbit type.
Singular Poisson-Kähler geometry of certain adjoint quotients
The Kähler quotient of a complex reductive Lie group relative to the conjugation action carries a complex algebraic stratified Kähler structure which reflects the geometry of the group. For the group SL(n,ℂ), we interpret the resulting singular Poisson-Kähler geometry of the quotient in terms of complex discriminant varieties and variants thereof.
Singular reduction of generalized complex manifolds.
Singularities of momentum maps of integrable Hamiltonian systems with two degrees of freedom.
Spherical gradient manifolds
We study the action of a real-reductive group on a real-analytic submanifold of a Kähler manifold. We suppose that the action of extends holomorphically to an action of the complexified group on this Kähler manifold such that the action of a maximal compact subgroup is Hamiltonian. The moment map induces a gradient map . We show that almost separates the –orbits if and only if a minimal parabolic subgroup of has an open orbit. This generalizes Brion’s characterization of spherical...
Spinc-quantization and the K-multiplicities of the discrete series
Stability under deformations of Hermite-Einstein almost Kähler metrics
On a -dimensional compact symplectic manifold, we consider a smooth family of compatible almost-complex structures such that at time zero the induced metric is Hermite-Einstein almost-Kähler metric with zero or negative Hermitian scalar curvature. We prove, under certain hypothesis, the existence of a smooth family of compatible almost-complex structures, diffeomorphic at each time to the initial one, and inducing constant Hermitian scalar curvature metrics.
Symplectic geometry on symplectic knot spaces.
The degenerate C. Neumann system I: symmetry reduction and convexity
The C. Neumann system describes a particle on the sphere S n under the influence of a potential that is a quadratic form. We study the case that the quadratic form has ℓ +1 distinct eigenvalues with multiplicity. Each group of m σ equal eigenvalues gives rise to an O(m σ)-symmetry in configuration space. The combined symmetry group G is a direct product of ℓ + 1 such factors, and its cotangent lift has an Ad*-equivariant momentum mapping. Regular reduction leads to the Rosochatius system on S ℓ,...
The Gromov width of complex Grassmannians.
The magnetic geodesic flow on a homogeneous symplectic manifold.
The Quantum Birkhoff Normal Form and Spectral Asymptotics
In this talk we explain a simple treatment of the quantum Birkhoff normal form for semiclassical pseudo-differential operators with smooth coefficients. The normal form is applied to describe the discrete spectrum in a generalised non-degenerate potential well, yielding uniform estimates in the energy . This permits a detailed study of the spectrum in various asymptotic regions of the parameters , and gives improvements and new proofs for many of the results in the field. In the completely resonant...
The reduction of the standard k-cosymplectic manifold associated to a regular lagrangian
The aim of the paper is to define a k-cosymplectic structure on the standard k-cosymplectic manifold associated to a regular Lagrangian and to reduce it via Marsden-Weinstein reduction.
The space of clouds in Euclidean space.
Toric structures on near-symplectic 4-manifolds
A near-symplectic structure on a 4-manifold is a closed 2-form that is symplectic away from the 1-dimensional submanifold along which it vanishes and that satisfies a certain transversality condition along this vanishing locus. We investigate near-symplectic 4-manifolds equipped with singular Lagrangian torus fibrations which are locally induced by effective Hamiltonian torus actions. We show how such a structure is completely characterized by a singular integral affine structure on the base of...
Variation of geometric invariant theory quotients