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
- Introductory notions: contact
structures, symplectic geometry, tight vs. overtwisted
dichotomy.
- Convex surface theory and open book
decompositions.
- Invariants of Legendrian knots,
Legendrian contact homology.
- Heegaard Floer homology and the
contact invariant.
- 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
References
Introductory notions:
- B. Aebischer, et. al., Symplectic Geometry, Progress in
Math. 124, Birkhäuser, Basel, Boston and Berlin, 1994.
- 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.
- K. Honda, Contact
geometry
notes.
- H. Geiges, An introduction to contact topology,
Cambridge Studies in Advanced Mathematics, 109.
Cambridge University Press, Cambridge, 2008.
- 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:
- E. Giroux, Convexité
en
topologie de contact, Comment. Math. Helv. 66
(1991), 637--677.
- K. Honda, On
the classification of tight contact structures I,
Geom. Topol. 4 (2000), 309--368.
- 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.
- 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:
- J. Etnyre, Legendrian
and
transversal knots, Handbook of knot theory,
105--185, Elsevier B.V., Amsterdam, 2005.
- Y. Chekanov, Differential
algebra
of Legendrian links, Invent. Math. 150
(2002), 441--483.
Heegaard Floer homology and the contact invariant:
- P. Ozsváth and Z. Szabó, Holomorphic disks and
topological invariants for closed three-manifolds,
Ann. of Math. (2) 159 (2004), 1027--1158.
- P. Ozsváth and Z. Szabó, Holomorphic disks and
three-manifold invariants: properties and
applications, Ann. of Math. (2) 159 (2004),
1159--1245.
- P. Ozsváth and Z. Szabó, Heegaard Floer homology
and contact structures, Duke Math. J. 129
(2005), 39--61.
- 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:
- 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.
- M. Hutchings, An
index
inequality for embedded pseudoholomorphic curves in
symplectizations, J. Eur. Math. Soc. (JEMS) 4
(2002), 313--361.
- 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. |