Math 235:  Contact Geometry
  Spring 2026

Syllabus

Contact manifolds are odd-dimensional siblings of symplectic manifolds and play an essential role in modern geometry/topology. They are related to Floer theory and holomorphic curves, 3- and 4-dimensional topology (via Heegaard Floer homology, embedded contact homology, and Seiberg-Witten Floer homology), quantum topology, categorification, and Hamiltonian/Reeb dynamics.  The goal of this course is to give an introduction to contact geometry in three dimensions, survey the recent developments in higher dimensions, and study the recent discovery of the relationship with cluster algebras.

Instructor: Ko Honda
Office Hours: F 1pm or by appointment
E-mail: honda at math dot ucla dot edu
URL: http://www.math.ucla.edu/~honda

Class Meetings:  MWF 2-2:50pm at MS 5137

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 and Giroux correspondence
  3. Weinstein and Liouville domains, Lefschetz fibrations
  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, bypasses, open book decompositions, and Giroux correspondence
  5. Contact homology and Legendrian contact homology
  6. Lagrangian fillings of Legendrians and relation to cluster algebras

Prerequisites

  • Math 225 sequence or equivalent (a good knowledge of differentiable manifolds and homology).  Some knowledge of symplectic geometry is helpful, but not necessary. 
Grading
  • You will form 2-3 person groups (depending on class size) and each group will give a 50-minute talk on a chosen topic. 
References

Introductory notions:
  1. Aebischer, et. al., Symplectic Geometry, Progress in Math. 124, Birkhäuser, Basel, Boston and Berlin, 1994.
  2. 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. Honda, Contact geometry notes.
  4. Geiges, An introduction to contact topology, Cambridge Studies in Advanced Mathematics, 109. Cambridge University Press, Cambridge, 2008.
  5. McDuff-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, Lagrangian fillings of Legendrian knots, relationship to cluster algebras:
  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.
  3. Ekholm-Honda-Kalman, Legendrian knots and exact Lagrangian cobordisms, J. Eur. Math. Soc. (2016).
  4. Williams, Cluster algebras: an introduction, Bull. AMS; also arXiv:1212.6263.
  5. Shende-Treumann-Williams-Zaslow, Cluster varieties from Legendrian knots, Duke Math. J., 168 (2019), 2801–2871.
  6. Casals-Gao, A Lagrangian filling for every cluster seed, Invent. Math (2024).

Higher-dimensional contact geometry:

  1. Eliashberg-Mischachev, Introduction to the h-principle.
  2. Murphy, Loose Legendrian embeddings in higher-dimensional contact manifolds.
  3. Borman-Eliashberg-Murphy, Existence and classification of overtwisted contact structures in all dimensions, Acta Math. 215 (2015), 281--361.
  4. Breen-Christian-Honda-Huang, Convex hypersurface theory in contact topology, arXiv:1907.0602. Current working version (to be updated on arXiv later).
  5. Breen-Honda-Huang, The Giroux correspondence in arbitrary dimensions, arXiv:2307.02317.
WARNING:  The course syllabus provides a general plan for the course; deviations may become necessary.
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Last modified: March 12, 2026.