FALL2021: Student Seminar in Symplectic vs. Contact Geometry
Instructors:
Meetings:
Time: Wednesdays 14:00-16:00
Place: ETH HG G 19.2
Zoom Room: Click HERE (Contact instructors for the passcode.)
Live Streaming: Click HERE (Log-in required)
Videos: Click HERE (Log-in required)
Useful external links:
Please refer to ETH Coronavirus Task Force webpage for the up-to-date pandemic regulations and measures.
Course catalogue for this seminar class.
ETH Academic Calendar for important dates.
Objectives:
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.
Topics:
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
Prerequisites:
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:
Seminar on symplectic toric manifolds by Ana Cannas
Lectures on symplectic geometry by Ana Cannas
Introduction to symplectic topology by Dusa McDuff and Dietmar Salamon
Torus actions on symplectic manifolds, Michèle Audin
Lecture notes on contact gometry by Ko Honda
An introduction to contact topology by Hansjörg Geiges (c.f. brief version.)
Symplectic toric manifolds by Ana Cannas
Lecture notes on (contact) geometry of manifolds by Emmy Murphy
Contact toric manifolds by Eugene Lerman
Constructions of contact manifolds by Hansjörg Geiges
A convexity theorem for torus actions on contact manifolds by Eugene Lerman
A note on toric contact geometry by Charles P. Boyer and Krzysztof Galicki
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.
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Tentative Weekly Plan:
For the sake of consistent presentations and lecture notes, please follow the notations and conventions here:
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
Reto Kaufmann: Definition, examples and nonexamples of symplectic manifold
Daniel Rutschmann: Hamiltonian vector fields with examples
Week 4: October 13, 2021
Alessandro Imparato: Definition, examples and nonexamples of contact manifold
Bangxin Wang: Reeb/contact vector fields with examples
Week 5: October 20, 2021
Frederik Semmel: Darboux's Theorem and Moser's argument
Benno Wendland: Darboux's Theorem and Moser's argument, cont.
Week 6: October 27, 2021
Marius Henry: Pfaff's Theorem and Gray's Stability Theorem
Eudes Robert: A dichotomy: Tight vs. Overtwisted contact structures
Part II: Two Sister Submanifolds: Lagrangian vs. Legendrian
Week 7: November 3, 2021
Metehan Aksay: Lagrangian manifolds, definition, examples
Marek Kurczynski: Motivating their importance
Week 8: November 10, 2021
Alex Uhlmann: Legendrian manifolds, definition, examples
Elia Mazzucchelli: Motivating their importance
Part III: Two Toric Geometries: Symplectic Toric vs. Contact Toric Geometry
Week 9: November 17, 2021
Daniel Rutschmann: A review of Lie group actions; the notion of moment map
Alessandro Imparato: Hamiltonian torus actions; convexity of moment map images
Week 10: November 24, 2021
Marek Kurczynski: Group actions on contact manifolds and the contact moment map
Reto Kaufmann: Convexity of the moment map image
Week 11: December 1, 2021
Elia Mazzucchelli: Symplectic reduction
Frederik Semmel: Contact reduction
Week 12: Double Header Week (Notice the shrunk schedule!)
December 8, 2021, Wednesday
Eudes Robert: Symplectic toric manifolds
Alex Uhlmann: Delzant's classification theorem for symplectic toric manifolds
December 10, 2021, Friday from 12:00-14:00
Bangxin Wang: Equivariant Darboux
Marius Henry: Delzant's construction
Week 13: December 15, 2021
Week 14: December 22, 2021
No meeting