Contact geometry was born more than two centuries ago in the work of Huygens, Hamilton, Jacobi as a geometric language for optics. It was soon realized that it has applications in many other areas, including non-holonomic mechanics and thermodynamics. One encounters contact geometry in everyday life when parking a car, skating, using a refrigerator, or watching the beautiful play of light in a glass of water. Sophus Lie, Elie Cartan, Darboux and many other great mathematicians devoted a lot of their work to this subject. However, till very recently most of the results were of a local nature. With the birth of symplectic and contact topology in the Eighties, the subject was reborn, and the last decade witnessed a number of breakthrough discoveries. There were found new important interactions with Hamiltonian mechanics, symplectic and sub-Riemannian geometry, foliation theory, complex geometry and analysis, topological hydrodynamics, 3-dimensional topology, and knot theory.
This course will introduce the basic results, examples and constructions in contact geometry in dimension three. We will then explore two trends in current research. First we will discuss the use of convex surfaces in contact geometry. This will allow us to prove many classification results. The second main trend will be the relation between open book decompositions of three manifolds and contact geometry. This is a relatively new result with many applications (and probably many yet to be discovered).
Here are some class notes on convex surfaces:
Here are some useful references:
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