Georgia Institute of Technology College of Sciences

We will review the definition of the Chekanov-Eliashberg differentialgraded algebra for Legendrian knots in R^3 and look at examples tounderstand a few of the invariants that come from Legendrian contacthomology.

Quantum topology is a collection of ideas and techniques for studying knots and manifolds using ideas coming from quantum mechanics and quantum field theory. We present a gentle introduction to this topic via Kauffman bracket skein algebras of surfaces, an algebraic object that relates "quantum information" about knots embedded in the surface to the representation theory of the fundamental group of the surface. In general, skein algebras are difficult to compute. We associate to every triangulation of the surface a simple algebra called a "quantum torus" into which the skein algebra embeds. In joint work with Thang Le, we make use of this embedding to give a simple proof of a difficult theorem.

In this introductory talk we present some basic results in ergodic theory, due to Poincare, von Neumann, and Birkhoff. We will also discuss many examples of dynamical systems where the theory can be applied. As motivation for a broad audience, we will go over the connection of the theory with some classical problems in statistical mechanics (part 2 of 3).

We show that an embedding of a (small) ball into a contact manifold is contact if and only if it preserves the (modified) shape invariant. The latter is, in brief, the set of all cohomology classes that can be represented by the pull-back (to a closed one-form) of a contact form by a coisotropic embedding of a fixed manifold (of maximal dimension) and of a given homotopy type. The proof is based on displacement information about (non)-Lagrangian submanifolds that comes from J-holomorphic curve methods (and gives topological invariants), and the construction of a coisotropic torus whose image (under a given embedding that is not contact) admits a transverse contact vector field (i.e. a convex surface in dimension 3). The definition of shape preserving does not involve derivatives and is preserved by uniform convergence (on compact subsets). As a consequence, we prove C^0-rigidity of contact embeddings (and diffeomorphisms). The underlying ideas are adaptations of symplectic techniques to contact manifolds that, in contrast to symplectic capacities, work well in the contact setting; the heart of the proof however uses purely contact topological methods.