Math 234:  Contact Geometry

MWF 1-1:50pm
Location: MS 7608

Syllabus

Contact manifolds are odd-dimensional siblings of symplectic manifolds and their importance has grown over the last 30 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 a brisk introduction to contact geometry in three dimensions and survey the more recent developments in higher dimensions.

Instructor: Ko Honda
Office: MS 7919
Office Hours: M 10-11:50
E-mail: honda at math dot ucla dot edu.
URL: http://www.math.ucla.edu/~honda

Topics

  1. Introductory notions: contact structures, symplectic geometry, Legendrian submanifolds
  2. In dimension three: Legendrian knots, tight vs. overtwisted dichotomy, convex surface theory, bypasses, open book decompositions
  3. Weinstein and Liouville domains
  4. In higher dimensions: h-principles, loose Legendrian knots, flexible Weinstein manifolds, classification of overtwisted contact structures a la Borman-Eliashberg-Murphy, convex hypersurface theory

Prerequisites

  • Math 225B or equivalent (a good knowledge of differentiable manifolds and homology).  Some knowledge of symplectic geometry is helpful, but not necessary. 
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. Giroux, Convexité en topologie de contact, Comment. Math. Helv. 66 (1991), 637--677.
  2. Honda, On the classification of tight contact structures I, Geom. Topol. 4 (2000), 309--368.
  3. 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. 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. Etnyre, Legendrian and transversal knots, Handbook of knot theory, 105--185, Elsevier B.V., Amsterdam, 2005.
  2. Chekanov, Differential algebra of Legendrian links, Invent. Math. 150 (2002), 441--483.

H-principle:

  1. Eliashberg-Mischachev, Introduction to the h-principle.

Higher-dimensional contact geometry:

  1. Murphy, Loose Legendrian embeddings in higher-dimensional contact manifolds.
  2. Borman-Eliashberg-Murphy, Existence and classification of overtwisted contact structures in all dimensions, Acta Math. 215 (2015), 281--361.

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

Last modified: January 2, 2019.