Student Seminar in Symplectic vs. Contact Geometry

Autumn 2021


Bahar Acu,

Ana Cannas,


Class Meetings:

Time: Wednesdays 14:00-16:00

Place: HG G 19.2


Zoom Room:  Click HERE (Contact instructors for the passcode.)

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Useful external links:



This student seminar aims to provide a glimpse of two sister geometries that have recently earned a central role in mathematics interacting with other areas. Side by side, we will discuss basics of symplectic and contact manifolds, some key submanifolds (Lagrangian and Legendrian) and the toric subclasses (symplectic and contact), which have gained prominence as testing grounds for other theories.


By giving two or three half-hour talks about each geometry, typing up notes for those talks, and participating in talks by others, each participant will have the opportunity to get acquainted with the landscape of symplectic and contact worlds, expand their command of geometry and topology, and develop presentation and collaboration skills.



  • Basics of symplectic and contact geometry
  • Lagrangian and Legendrian submanifolds
  • Hamiltonian actions, torus actions, moment maps, symplectic and contact reduction
  • Symplectic and contact toric manifolds
  • Delzant's classification theorem



Prior knowledge of differential geometry and algebraic topology is required. Details of the seminar organization will be discussed during the organizational meeting on September 22, 2021.


Suggested references and readings:

  1. Seminar on symplectic toric manifolds by Ana Cannas
  2. Lectures on symplectic geometry by Ana Cannas
  3. Introduction to symplectic topology by Dusa McDuff and Dietmar Salamon
  4. Torus actions on symplectic manifolds, Michèle Audin
  5. Lecture notes on contact gometry by Ko Honda
  6. An introduction to contact topology by Hansjörg Geiges (c.f. brief version.)
  7. Symplectic toric manifolds by Ana Cannas
  8. Lecture notes on (contact) geometry of manifolds by Emmy Murphy
  9. Contact toric manifolds by Eugene Lerman
  10. Constructions of contact manifolds by Hansjörg Geiges
  11. A convexity theorem for torus actions on contact manifolds by Eugene Lerman
  12. A note on toric contact geometry by Charles P. Boyer and Krzysztof Galicki
  13. The fundamental groups of contact toric manifolds by Hui Li

Tentative Plan:

  • The organizational meeting will take place on September 22, 2021.
  • The second week is reserved for two brief lectures overviewing the two geometries.
  • The remaining 12 semester weeks will alternate between two geometries. Each week, dedicated to one of the two geometries, will consist of two presentations given by two students.
  • Each presentation is planned to take 30min, leaving 15min for questions and/or discussion.
  • Students will be asked to closely follow the notations and conventions file shared below.
  • Prior to each presentation, presenters will share their LaTeX'd lecture notes to be uploaded to this website. Each presenter will then have a week following their presentation to submit their finalized notes.



Tentative Weekly Plan:


For the sake of consistent presentations and lecture notes, please follow the notations and conventions here:
Notations and Conventions



Prelude: Overview


Week 1: September 22, 2021

  • Organizational meeting


Week 2: September 29, 2021

        Brief overview of two geometries

  • Ana Cannas: A symplectic glimpse
  • Bahar Acu: A brief overview of fundamental (and relevant) structures in contact geometry


Part I: Two Sister Geometries: Symplectic vs. Contact


Week 3: October 6, 2021

Week 4: October 13, 2021

Week 5: October 20, 2021

Week 6: October 27, 2021


Part II: Two Sister Submanifolds: Lagrangian vs. Legendrian


Week 7: November 3, 2021

Week 8: November 10, 2021


Part III: Two Toric Geometries: Symplectic Toric vs. Contact Toric Geometry


Week 9: November 17, 2021


Week 10: November 24, 2021


Week 11: December 1, 2021


Week 12: Double Header Week (Notice the shrunk schedule!)

December 8, 2021, Wednesday


December 10, 2021, Friday from 12:00-14:00


Week 13: December 15, 2021


Week 14: December 22, 2021

  • No meeting



Disclaimer: The course syllabus provides a general plan and is, in principle, tentative and subject to change. Please check out this website regularly for updates.


Last modified: December 14, 2021.