Course details
Semester
 Spring 2021
 Monday, January 25th to Friday, March 19th; Monday, April 26th to Friday, May 7th
Hours
 Live lecture hours
 10
 Recorded lecture hours
 0
 Total advised study hours
 40
Timetable
 Wednesdays
 11:05  11:55
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Description
Symplectic geometry is a formalisation of the mathematics of classical mechanics. Therefore one might imagine it is only of historical interest, but in fact it is a central topic in modern mathematics research.
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
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
Exposure to the following:
 Linear algebra (abstract vector spaces; bases; dual spaces)
 Differential geometry (manifolds; tangent vectors; differential forms)
 Basic group theory (groups and homomorphisms)
 Linear algebra (abstract vector spaces; bases; dual spaces)
 Differential geometry (manifolds; tangent vectors; differential forms)
 Basic group theory (groups and homomorphisms)
but concepts will be reviewed as needed. Exposure to de Rham cohomology would be useful but is not necessary.
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

JG
Dr Joseph Grant
 University
 University of East Anglia
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