Course details
Semester
 Spring 2025
 Monday, January 27th to Friday, April 4th
Hours
 Live lecture hours
 10
 Recorded lecture hours
 0
 Total advised study hours
 40
Timetable
 Mondays
 13:05  13:55 (UK)
Description
In the first part of the course, we will introduce symplectic vector spaces and symplectic manifolds, revising concepts from differential geometry as they are needed. An important example will be the cotangent bundle of an arbitrary manifold. We will also study the key substructures of symplectic manifolds: Lagrangian submanifolds.
Symplectic vector spaces have a canonical form. In the second part of the course we will see that, locally, there is an analogous statement for symplectic manifolds. This is the Darboux theorem. It says that symplectic manifolds have no local invariants, unlike Riemannian geometry where we have a notion of curvature. For this reason, the subject is sometimes called symplectic topology.
The third part of the course uses ideas from classical mechanics to obtain precise results in pure mathematics. A symplectic manifold is like a curved phase space. In physics, the dynamics of phase space is controlled by conservation of energy. In symplectic geometry, this is formalised by the existence of a flow corresponding to a vector field constructed from some Hamiltonian function. This leads to the Noether principle, relating symmetries and integrals of motions on a symplectic manifold.
This would make a good second course on differential geometry, and is closely connected to subjects such as Lie groups, integrable systems, and Morse theory. It will provide a foundation for further study with connections to algebraic geometry, representation theory, and string theory
Prerequisites
 Linear algebra (abstract vector spaces; bases; dual spaces)
 Differential geometry (manifolds; tangent vectors; differential forms)
 Basic group theory (groups and homomorphisms)
Syllabus
 Introduction; symplectic vector spaces; Lagrangian subspaces
 Differential geometry revision; definition of symplectic manifold
 Examples of symplectic manifolds; Liouville form on the cotangent bundle
 Lagrangian submanifolds; examples
 Moser trick; Darboux theorem
 Hamiltonian vector fields; integrable systems
 Symplectic actions; coadjoint representations
 Hamiltonian actions and moment maps
 Noether principle; symplectic reduction
Lecturer

CM
Cheuk Yu Mak
 University
 University of Southampton
Bibliography
No bibliography has been specified for this course.
Assessment
The assessment for this course will be released on Tuesday 22nd April 2025 at 00:00 and is due in before Friday 2nd May 2025 at 11:00.
Assessment for all MAGIC courses is via takehome exam which will be made available at the release date (the start of the exam period).
You will need to upload a PDF file with your own attempted solutions by the due date (the end of the exam period).
If you have kept uptodate with the course, the expectation is it should take at most 3 hoursâ€™ work to attain the pass mark, which is 50%.
Please note that you are not registered for assessment on this course.
Files
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Lectures
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