Descripción del curso
Contact geometry provides a natural language for many phenomena in classical mechanics, geometric optics, and thermodynamics. In the last two decades it has become increasingly apparent that contact manifolds constitute a natural framework for many problems in low-dimensional geometric topology. As hypersurfaces in symplectic 4-manifolds, 3-dimensional contact manifolds provide a natural bridge to 4-manifold topology.
This one-week course presupposes only some of the fundamental concepts from the Differential Geometry course taught in the first semester of the Master's at UAM. I shall give an introduction to some of the most important concepts in contact topology, but I also aim at giving the audience a glimpse of the far-reaching topological applications of the theory.
One simple application is a contact geometric proof of the Whitney-Graustein theorem, which says that immersions of the circle in the plane are classified by the rotation number. Deeper applications to knot theory will be sketched.
The main focus of the course will be on 3-dimensional contact manifolds and special types of knots in such manifolds.
Programa del Curso
1. Legendrian curves in the standard contact 3-space; front and Lagrangian
projection; the rotation number of a Legendrian immersed circle; a contact geometric proof of the Whitney-Graustein theorem.
2. Integrable vs. non-integrable hyperplane fields (Frobenius theorem); basic
techniques of contact topology: Moser trick, Gray stability, contact Hamiltonians;
Darboux and other neighbourhood theorems.
3. Contact structures on 3-manifolds: Lutz twist and the Lutz-Martinet theorem, tight vs. overtwisted, fillability implies tightness, convex surfaces; Legendrian knots in contact 3-manifolds: the classical invariants and classification
results.
4. Contact Dehn surgery; a surgery presentation of contact 3-manifolds; symplectic caps to fillings; property P for knots.