Symplectic Capacities in Manifolds

Symplectic Topology and Capacities

I am going to talk about symplectic topology, a field that has seen a remarkable development in the past 15 years. Let’s begin with what was classically known – which was rather little. We start with a symplectic form ω, that is, a closed 2-form which is nondegenerate. This last condition means that ω is defined on an even dimensional manifold M and that its wedge with itself n times is a top d...

Symplectic Capacities of Domains in C

SYMPLECTIC TOPOLOGY OF b-SYMPLECTIC MANIFOLDS

A Poisson manifold (M2n, π) is b-symplectic if ∧n π is transverse to the zero section. In this paper we apply techniques of Symplectic Topology to address global questions pertaining to b-symplectic manifolds. The main results provide constructions of: b-symplectic submanifolds à la Donaldson, b-symplectic structures on open manifolds by Gromov’s h-principle, and of b-symplectic manifolds with ...

Symplectic Toric Manifolds

Foreword These notes cover a short course on symplectic toric manifolds, delivered in six lectures at the summer school on Symplectic Geometry of Integrable Hamiltonian Systems, mostly for graduate students, held at the Centre de Recerca Matemàtica in Barcelona in July of 2001. The goal of this course is to provide a fast elementary introduction to toric manifolds (i.e., smooth toric varieties)...

Generalized Symplectic Manifolds

This paper uses a generalization of symplectic geometry, known as n-symplectic geometry and developed by Norris1, to find observables on three-dimensional manifolds. It will be seen that for the cases considered , the n-symplectic observables are derivable from the symplectic observables of C2. The quantization of these observables, as well as those on the frame bundle of R, is also examined.