Instructor: Ko Honda
Office: DRB 230
E-mail: khonda@math.usc.edu
Telephone: 213-740-3785
URL: http://math.usc.edu/~khonda

Topics

1. Introductory notions: contact structures, foliations, Pfaff's theorem, normal forms, Legendrian curves, some symplectic geometry.
   
2. Convex surface theory: characteristic foliations, convex surfaces, bypasses, convex decompositions.
3. Tight vs. overtwisted: Eliashberg's classification of overtwisted contact structures, fillability, classification of tight contact structures for simple spaces, gluing.
4. Legendrian knots
5. Open book decompositions and fibered links.
6. Heegaard Floer homology and contact geometry.
   
   

1. B. Aebischer, et. al., Symplectic Geometry, Progress in Math. 124, Birkhäuser, Basel, Boston and Berlin, 1994.
2. J. Etnyre, Introductory lectures on contact geometry, in the Proceedings of the 2001 Georgia International Topology Conference. ArXiv:math.SG/0111118.
   

1. K. Honda, 3-dimensional methods in contact geometry, to appear in Different Faces of Geometry.
   
2. E. Giroux, Convexité en topologie de contact, Comment. Math. Helv. 66 (1991), 637--677.

1. Y. Eliashberg, Contact 3-manifolds twenty years since J. Martinet's work, Ann. Inst. Fourier (Grenoble) 42 (1992), 165--192.
2. Y. Eliashberg, Classification of overtwisted contact structures on 3-manifolds, Invent. Math. 98 (1989), 623--637.
3. K. Honda, On the classification of tight contact structures I, Geom. Topol. 4 (2000), 309--368.
4. K. Honda, Gluing tight contact structures, Duke Math. J. 115 (2002), 435--478.
   
5. E. Giroux, Structures de contact en dimension trois et bifurcations des feuilletages de surfaces, Invent. Math. 141 (2000), 615--689.

1. J. Etnyre, Legendrian and transversal knots, to appear in the Handbook of Knot Theory. ArXiv:math.SG/0306256.
2. Y. Chekanov, Differential algebra of Legendrian links, Invent. Math. 150 (2002), 441--483.
   

1. E. Giroux, Géométrie de contact: de la dimension trois vers les dimensions supérieures, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), 405--414, Higher Ed. Press, Beijing, 2002.

1. P. Ozsváth and Z. Szabó, Holomorphic disks and topological invariants for closed three-manifolds, ArXiv:math.SG/0101206, to appear in Ann. Math.
2. P. Ozsváth and Z. Szabó, Holomorphic disks and three-manifold invariants: properties and applications, ArXiv:math.SG/0105202, to appear in Ann. Math.
3. P. Ozsváth and Z. Szabó, Heegaard Floer homologies and contact structures, ArXiv:math.SG/0210127.
   
   

Prerequisites

• Knowledge of differentiable manifolds and calculus on differentiable manifolds.
  

There will be homework assigned during the course of the lecture.  You may work with others or consult other textbooks, but the homework you turn in must be written by you, in your own words, and you must cite your sources used and your collaborators!