Symplectic Geometry (MAGIC105)
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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
Spring 2021 (Monday, January 25 to Friday, March 19; Monday, April 26 to Friday, May 7)
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) but concepts will be reviewed as needed. Exposure to de Rham cohomology would be useful but is not necessary.
1. Introduction; symplectic vector spaces; Lagrangian subspaces 2. Differential geometry revision; definition of symplectic manifold 3. Examples of symplectic manifolds; Liouville form on the cotangent bundle 4. Lagrangian submanifolds; examples 5. Tubular neighbourhood theorem; homotopy formula 6. Moser trick; Darboux theorem 7. Hamiltonian vector fields; integrable systems 8. Symplectic actions; coadjoint representations 9. Hamiltonian actions and moment maps 10. Noether principle; symplectic reduction
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