MATH 9601A Symplectic Geometry

Instructor: Martin Pinsonnault

Office: MC 132

E-mail: mpinson [at] uwo.ca

Course website: The MATH 9601A OWL website will be available later in August.

Course Outline: The aim of this course is to provide an introduction to symplectic geometry with a view to some recent results. We will cover the foundations: symplectic and complex vector spaces, linear symplectic geometry, symplectic manifolds, key examples and problems, connections to classical mechanics, normal forms, connections to complex geometry and contact geometry, symplectic transformations, Hamiltonian actions and moment maps, symplectic reduction, classical invariants, brief introduction to $J$-holomorphic curves and the Nonsqueezing theorem.

This course should prepare students for more advanced topics like classifications of Hamiltonian actions, full-fledged $J$-holomorphic curves theory, Gromov invariants, quantum homology, Floer theory, or Mirror Symmetry.

What is symplectic geometry ? At the linear level, an inner product $g$ on a real vector space $V$ is a non-degenerate symmetric bilinear pairing that defines an isomorphism between $V$ and its dual $V^*$. This structure induces a geometry based on the usual notions of lengths and angles. A symplectic structure $\omega$ is a non-degenerate skew-symmetric bilinear pairing that defines an isomorphism between $V$ and $V^*$. It induces a geometry based on 2-dimensional signed areas. Over the complex numbers this structure is just as natural as an inner product since any hermitian product $h$ on a complex vector space $W$ is made of an inner product $g$ (the real part) and a symplectic form $\omega$ (the imaginary part).

At the non-linear level, symplectic structures on manifolds appear naturally when one wants to write the equations of classical mechanics on arbitrary curved spaces using the minimal action principle (the symplectic form measures the action) as a guideline. Less well known is the fact that symplectic structures also appear in quantum mechanics (together with inner products and complex structures) as the space of states of a quantum system is the projective space of a Hilbert space.

To know more about symplectic geometry, take a look at the nLab entry at:

https://ncatlab.org/nlab/show/symplectic+geometry

From a physics perspective, there is this nice discussion at:

https://cohn.mit.edu/symplectic/

Of course, the Wikipedia page is also a good entry point:

https://en.wikipedia.org/wiki/Symplectic_geometry

Prerequisites: Familiarity with differential geometry (in particular, differential forms and vector fields on manifolds as covered in Math 9055A/4155A - Calculus on Manifolds) and with algebraic topology (in particular, de Rham cohomology as covered in Math 9052B/4152B - Algebraic Topology) will be assumed. Note that we will emphasize the geometrical aspects of the theory. In particular, we will try to stay away from purely topological questions and we will minimize the use of algebraic topology machinery.

References: I will mostly follow the textbook

• A. Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics, vol. 1764, Springer.

which is freely available at

https://people.math.ethz.ch/~acannas/Papers/lsg.pdf

or through the UWO Library online textbook program. An errata may be consulted at

https://people.math.ethz.ch/~acannas/Papers/lsg_errata_website.pdf

We will also draw from other well-known references as listed below. In orther to develop a deeper understanding of the subject matters, we encourage students to use more than one reference. When using online references, students must keep in mind that they may contain typos and other inaccuracies.

• Introduction to Symplectic Topology by D. McDuff and D. Salamon
• An Introduction to Symplectic Geometry by R. Berndt
• Mathematical Methods of Classical Mechanics by V. Arnold
• Symplectic Techniques in Physics by V. Guillemin and S. Sternberg
• J-holomorphic Curves and Symplectic Topology by D. McDuff and D. Salamon

Evaluation Scheme: The evaluation will consist in

• 6 short homeworks (every other week): $\sim$ 30%
• Final exam: $\sim$ 40%
• Presentation: $\sim$ 15%
• Written report: $\sim$ 15%.

The homeworks are an integral part of the course and special attention must be paid to the presentation. They will be evaluated based on both correctness and clarity. Students are allowed to discuss assignment problems, but each student should write their solutions individually, with proper credit given to those who helped you and to any references used.

At the end of the semester, each student will give a presentation ( $\sim$ 50 minutes) on a subject related to the course material. The topic should be selected as early as possible. We will maintain a list of possible topics that students can choose from. Alternatively, students may decide on another topic of particular interest/relevance for them after discussion with the intructor.

Course Learning Outcomes

Upon successful completion of the course, students will:

• Be able to define terms and restate theorems related to symplectic geometry and topology.
• Be able to give examples of symplectic manifolds with specific properties.
• Be able to apply the theory from the course to solve problems and prove theorems within symplectic geometry.
• Be able to describe a few problems from other fields that admit symplectic interpretations and/or solutions.
• Be able to write a thorough, carefully written proof, involving key concepts from symplectic geometry.
• Be able to present clear proofs to the class.
• Be able to converse with peers on the course topics.

Academic Policies and Regulations

UWO e-mail: In accordance with this policy, the centrally administered e-mail account provided to students will be considered the individual’s official university e-mail address. It is the responsibility of the account holder to ensure that e-mail received from the University at his/her official university address is attended to in a timely manner.

Scholastic offences: Scholastic offences are taken seriously and students are directed to read the appropriate policy, specifically, the definition of what constitutes a Scholastic Offence, at the following Web site: https://www.uwo.ca/univsec/pdf/academic_policies/appeals/scholastic_discipline_grad.pdf

The penalty for an academic offence will be a grade of 0 on that component of the final grade (referring to the percentages listed above).

In some cases, the penalty can include expulsion from the program. All academic offences are added to your student record.

Medical Accommodation: Most issues involving medical accommodation can be handled by contacting the intructor.

In other situations, a student requiring academic accommodation due to illness should bring a Student Medical Certificate with them when visiting an off-campus medical facility and use a Record Release Form for visits to Student Health Services.

Support Services: Learning-skills counsellors at Learning Development & Success are ready to help you improve your learning skills. Students who are in emotional/mental distress should refer to Mental Health at Western for a complete list of options about how to obtain help. Additional student-run support services are offered by the USC. The website for Registrarial Services is https://www.registrar.uwo.ca.

Student Accessibility Services: Please contact the course instructor if you require lecture or printed material in an alternate format or if any other arrangements can make this course more accessible to you. You may also wish to contact Accessible Education at http://academicsupport.uwo.ca/accessible_education/index.html if you have any questions regarding accommodations.

Western is committed to achieving barrier-free accessibility for all its members, including graduate students. As part of this commitment, Western provides a variety of services devoted to promoting, advocating, and accommodating persons with disabilities in their respective graduate program.

Graduate students with disabilities (for example, chronic illnesses, mental health conditions, mobility impairments) are encouraged to register with Student Accessibility Services, a confidential service designed to support graduate and undergraduate students through their academic program. With the appropriate documentation, the student will work with both SAS and their graduate programs (normally their Graduate Chair and/or Course instructor) to ensure that appropriate academic accommodations to program requirements are arranged. These accommodations include individual counselling, alternative formatted literature, accessible campus transportation, learning strategy instruction, writing exams and assistive technology instruction.

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Additional student-run support services are offered by the USC.

If students need assistance with the course OWL Brightspace site, they can seek support on the OWL Brightspace Help page. Alternatively, they can contact the Western Technology Services Helpdesk. The Helpdesk can also be contacted by phone at 519-661-3800 or ext. 83800.

Key Sessional Dates:

• September 5, 2024: Classes begin.
• September 13, 2024: Last day to add / drop a first-term half course (without showing WDN).
• September 30, 2024: National Day for Truth and Reconciliation. No classes.
• October 12 – 20, 2024: Fall Reading Week.
• December 2, 2024: Last day to withdraw from a first-term half course without academic penalty.
• December 6, 2024: Classes End.
• December 7 - 8, 2024: Study Days.
• December 9 – 22, 2024: Exam period.