Math 285E:  Contact Geometry


Spring 2014, MWF 2-2:50pm
Location: MS 6201

Syllabus

In this course I would like to discuss certain aspects of modern geometry and topology which have something to do with contact structures.  Contact manifolds are odd-dimensional siblings of symplectic manifolds and their importance has grown over the last 20 plus years. They are related to Gromov-Witten theory, 3- and 4-dimensional topology, TQFT's, categorification, and dynamical systems.  The goal of this course is to give an introduction to contact geometry and explain their relevance to the various Floer-type homology theories such as Heegaard Floer homology, embedded contact homology, Legendrian contact homology and symplectic field theory.

Instructor: Ko Honda
Office: MS 7901
Office Hours: Mon. 10:30-12, 1-2
E-mail: honda at math dot ucla dot edu.
Telephone: 310-825-2143
URL: http://www.math.ucla.edu/~honda

Topics

  1. Introductory notions: contact structures, symplectic geometry, tight vs. overtwisted dichotomy.
  2. Convex surface theory and open book decompositions.
  3. Invariants of Legendrian knots, Legendrian contact homology.
  4. Heegaard Floer homology and the contact invariant.
  5. Symplectic field theory and embedded contact homology.

Prerequisites

  • Math 225B or equivalent (a good knowledge of differentiable manifolds and homology).  Some knowledge of symplectic geometry is helpful, but not necessary. Math 226A and Math 226B are not prerequisites for Math 285E.
Grading
  • TBA
References

Introductory notions:
  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, Topology and geometry of manifolds (Athens, GA, 2001),  81--107, Proc. Sympos. Pure Math., 71, Amer. Math. Soc., Providence, RI, 2003.
  3. K. Honda, Contact geometry notes.
  4. H. Geiges, An introduction to contact topology, Cambridge Studies in Advanced Mathematics, 109. Cambridge University Press, Cambridge, 2008.
  5. D. McDuff and D. Salamon, Introduction to symplectic topology, 2nd edition, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 1998.
Convex surfaces and open book decompositions:
  1. E. Giroux, Convexité en topologie de contact, Comment. Math. Helv. 66 (1991), 637--677.
  2. K. Honda, On the classification of tight contact structures I, Geom. Topol. 4 (2000), 309--368.
  3. 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.
  4. J. Etnyre, Lectures on open book decompositions and contact structures, Floer homology, gauge theory, and low-dimensional topology,  103--141, Clay Math. Proc., 5, Amer. Math. Soc., Providence, RI, 2006.
Legendrian knots:
  1. J. Etnyre, Legendrian and transversal knots, Handbook of knot theory, 105--185, Elsevier B.V., Amsterdam, 2005.
  2. Y. Chekanov, Differential algebra of Legendrian links, Invent. Math. 150 (2002), 441--483.
Heegaard Floer homology and the contact invariant:
  1. P. Ozsváth and Z. Szabó, Holomorphic disks and topological invariants for closed three-manifolds, Ann. of Math. (2) 159 (2004), 1027--1158.
  2. P. Ozsváth and Z. Szabó, Holomorphic disks and three-manifold invariants: properties and applications, Ann. of Math. (2) 159 (2004), 1159--1245.
  3. P. Ozsváth and Z. Szabó, Heegaard Floer homology and contact structures, Duke Math. J. 129 (2005), 39--61.
  4. K. Honda, W. Kazez and G. Matić, On the contact class in Heegaard Floer homology, J. Differential Geom. 83 (2009), 289--311.
Symplectic field theory and embedded contact homology:
  1. Y. Eliashberg, A. Givental and H. Hofer, Introduction to symplectic field theory, GAFA 2000 (Tel Aviv, 1999), Geom. Funct. Anal. 2000, Special Volume, Part II, 560--673.
  2. M. Hutchings, An index inequality for embedded pseudoholomorphic curves in symplectizations, J. Eur. Math. Soc. (JEMS) 4 (2002), 313--361.
  3. M. Hutchings and C. Taubes, Gluing pseudoholomorphic curves along branched covered cylinders I,  J. Symplectic Geom. 5 (2007), 43--137.

WARNING:  The course syllabus provides a general plan for the course; deviations may become necessary. 

Last modified: April 1, 2014.