Applications of convex integration to symplectic and contact geometry

Nonrational, Nonsimple Convex Polytopes in Symplectic Geometry

In this research announcement we associate to each convex polytope, possibly nonrational and nonsimple, a family of compact spaces that are stratified by quasifolds, i.e., the strata are locally modeled by Rk modulo the action of a discrete, possibly infinite, group. Each stratified space is endowed with a symplectic structure and a moment mapping having the property that its image gives the or...

On Contact and Symplectic Lie Algeroids

In this paper, we will study compatible triples on Lie algebroids. Using a suitable decomposition for a Lie algebroid, we construct an integrable generalized distribution on the base manifold. As a result, the symplectic form on the Lie algebroid induces a symplectic form on each integral submanifold of the distribution. The induced Poisson structure on the base manifold can be represented by m...

Convex Surfaces in Contact Geometry: Class Notes

These are notes covering part of a contact geometry course. They are in very preliminary form. In particular the last few sections have not really been proof read. Hopefully I will be able to come back to these notes in the near future to improve and extend them, but I hope they are useful as is. Section 8 have not been written yet. Section 7 currently contains no proofs and section 1 contains ...

Symplectic, Poisson, and Contact Geometry on Scattering Manifolds

We introduce scattering-symplectic manifolds, manifolds with a type of minimally degenerate Poisson structure that is not too restrictive so as to have a large class of examples, yet restrictive enough for standard Poisson invariants to be computable. This paper will demonstrate the potential of the scattering symplectic setting. In particular, we construct scattering-symplectic spheres and sca...

Applications of Convex and Algebraic Geometry to Graphs and Polytopes