Symplectic geometry grew out of classical mechanics and is now a thriving field of study. It has important connections with low dimensional topology and there are still some exciting foundational questions to which we don’t know the answer. This course will discuss the basics of symplectic geometry and then focus on constructions of symplectic manifolds, obstructions to the existence of symplectic structures on smooth manifolds, and a conjectural approach to when symplectic structures exist on manifolds. If time permits we will then discuss the same questions for contact structures (the odd dimensional analog of symplectic structures).
The prerequisites for the course are differential topology and algebraic topology. Specifically, I will be assuming you are familiar with smooth manifolds, bundles, differential forms, De Rham cohomology, fundamental groups, and homology and cohomology.
- Lecture notes, sections I-II
- Lecture notes, section III
- Lecture notes, section IV A-C
- Lecture notes, section IV D-F
- Lecture notes, section V A-C
- Lecture notes, section V D-E
- Lecture notes, section V F
- Lecture notes, section VIA-B
- Lecture notes, section VIC-D
- Lecture notes, section VIIA-B
As background here are some notes on Hodge theory.
Monday-Wednesday-Friday 9:05-9:55 in Skiles 246.
Office: Skiles 106
e-mail: etnyre "at" math .gatech.edu
Your grade will be based on class participation and some class assignments. For the class assignments, in the each of the months of September, October, and November, by the end of the month you need to either turn in 5 problems mentioned in class or in one of the text books (related to the material being covered in class) or alternatively give a talk in the Wednesday or Friday geometry/topology seminars. The written assignments must be turned in by the end of month. The assignments will be graded on a pass/fail basis. A reasonable attempt will receive a pass. By reasonable attempt, I mean that the attempts for solutions to 4 out of 5 of the problems essentially correct even if there might be some minor errors. If you give a talk, that will count as a completed assignments for that month.
For solving the written problems you are welcome to consult with me or work with other students. But the problems must be written up by you, in your own words.
To receive passing grade you must complete and pass the monthly assignments. For an A you must also attend at least 90% of the classes (or have an excused absence), for an B you must attend at least 80% of the classes (or have an excused absence), for a C, 70%, and for a D, 60%.
- Academic Integrity. All students are expected to comply with the Georgia Tech Honor Code.
- Students with Disabilities and/or in need of Special Accommodations. Georgia Tech complies with the regulations of the Americans with Disabilities Act of 1990 and offers accommodations to students with disabilities. If you are in need of classroom or testing accommodations, please make an appointment with the ADAPTS office to discuss the appropriate procedures.
- Intent for Inclusivity. As a member of the Georgia Tech community, I am committed to creating a learning environment in which all of my students feel safe and included. Because we are individuals with varying needs, I am reliant on your feedback to achieve this goal. To that end, I invite you to enter into dialogue with me about the things I can stop, start, and continue doing to make my classroom an environment in which every student feels valued and can engage actively in our learning community.
There are no official text book for this course, but the following books are quite useful for the material in the first part of the course:
Later in the course we will cover material related to the paper
- Symplectic submanifolds and almost-complex geometry
- Géométrie de contact: de la dimension trois vers les dimensions supérieures