Math 235: Contact Geometry
Spring 2026 — Syllabus
Contact manifolds are odd-dimensional siblings of symplectic manifolds and play an essential role in modern geometry and topology. They relate to Floer theory, holomorphic curves, 3‑ and 4‑dimensional topology (Heegaard Floer homology, embedded contact homology, Seiberg–Witten Floer homology), quantum topology, categorification, and Hamiltonian/Reeb dynamics. This course introduces contact geometry in dimension three, surveys developments in higher dimensions, and explores connections with cluster algebras.
| Instructor: | Ko Honda |
| Email: | honda at math dot ucla dot edu |
| Office Hours: | Friday 1pm or by appointment |
| Website: | math.ucla.edu/~honda |
| Class Meetings: | MWF 2:00–2:50pm, MS 5137 |
Topics
- Introductory notions: contact structures, symplectic geometry, Legendrian submanifolds
- In dimension three: Legendrian knots, tight vs. overtwisted dichotomy, convex surface theory, bypasses, open book decompositions, Giroux correspondence
- Weinstein and Liouville domains, Lefschetz fibrations
- Higher dimensions: h‑principles, loose Legendrian knots, flexible Weinstein manifolds, classification of overtwisted contact structures (Borman–Eliashberg–Murphy), convex hypersurface theory, bypasses, open book decompositions, Giroux correspondence
- Contact homology and Legendrian contact homology
- Lagrangian fillings of Legendrians and relation to cluster algebras
Prerequisites
- Math 225 sequence or equivalent (knowledge of differentiable manifolds and homology). Some symplectic geometry is helpful but not required.
Grading
- Students will form groups of 2–3 and each group will give a 50‑minute talk on a chosen topic.
References
Introductory Notions
- Aebischer et al., Symplectic Geometry, Birkhäuser, 1994.
- Etnyre, Introductory lectures on contact geometry, Proc. Sympos. Pure Math. 71, 2003.
- Honda, Contact geometry notes.
- Geiges, An Introduction to Contact Topology, Cambridge Univ. Press, 2008.
- McDuff–Salamon, Introduction to Symplectic Topology, 2nd ed., Oxford Univ. Press, 1998.
Convex Surfaces & Open Books
- Giroux, Convexité en topologie de contact, Comment. Math. Helv. 66 (1991).
- Honda, On the classification of tight contact structures I, Geom. Topol. 4 (2000).
- Giroux, Géométrie de contact…, Proc. ICM (2002).
- Etnyre, Lectures on open book decompositions and contact structures, Clay Math. Proc. 5 (2006).
Legendrian Knots, Fillings & Cluster Algebras
- Etnyre, Legendrian and transversal knots, Handbook of Knot Theory, 2005.
- Chekanov, Differential algebra of Legendrian links, Invent. Math. 150 (2002).
- Ekholm–Honda–Kálmán, Legendrian knots and exact Lagrangian cobordisms, JEMS (2016).
- Williams, Cluster algebras: an introduction, Bull. AMS; arXiv:1212.6263.
- Shende–Treumann–Williams–Zaslow, Cluster varieties from Legendrian knots, Duke Math. J. 168 (2019).
- Casals–Gao, A Lagrangian filling for every cluster seed, Invent. Math. (2024).
Higher-Dimensional Contact Geometry
- Eliashberg–Mishachev, Introduction to the h‑Principle.
- Murphy, Loose Legendrian embeddings in higher‑dimensional contact manifolds.
- Borman–Eliashberg–Murphy, Existence and classification of overtwisted contact structures, Acta Math. 215 (2015).
- Breen–Christian–Honda–Huang, Convex hypersurface theory in contact topology, arXiv:1907.0602.
- 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.
Last modified: March 12, 2026.