Practical information:
Contacts:
Overview of the course:
The subject of symplectic geometry deals with objects called symplectic manifolds. A symplectic manifold is a smooth manifold endowed with a 2-form which is closed and of maximal rank. While it may not be apparent from the definition why these objects should occupy an entire field of mathematics, symplectic manifolds turn out to be extremely rich. Their study is also connected to many other areas of mathematics including algebraic geometry, representation theory, dynamics, etc.
This course aims to provide a first introduction to symplectic geometry, assuming relatively minimal background. We will cover a selection of topics, including: basics of symplectic geometry; constructions of symplectic manifolds; Lagrangians submanifolds; Hamiltonian dynamics.
Importantly, and in contrast to last year's (V5D3), we will not discuss J-holomorphic curves or Floer theory. Students wishing to learn more about Floer theory can consider participating in the seminar (S2D1/S4D1).
The course will be co-taught by Nate Bottman and Laurent Côté in English.
Pre-requisites:
Optional exercise sheets:
Summary of lectures:
Lecture 1 (references: McDuff-Salamon "Introduction to symplectic topology" ed. 1, sections 2.1-2.2, 3.1). Definition of symplectic vector space. Dimension of symplectic vector space is even. Definition of standard symplectic vector space, linear symplectomorphisms, and symplectic matrices, including the group Sp(2n). Definition of symplectic complement, and isotropic, coisotropic, symplectic, and Lagrangian subspaces. Basic properties of symplectic complement. Any symplectic vector space is isomorphic to the standard one. Definition of symplectic manifold, and of the symplectic structure on R^{2n}. Definition of symplectomorphisms, and the symplectomorphism group Symp(M,omega). Definition of symplectic vector fields, and proof that they integrate to symplectomorphisms. Construction of the canonical 1-form lambda_can on T^*L (both coordinate-dependently and -independently; equivalence of definitions is in the exercise sheet).
Lecture 2 (references: McDuff-Salamon "Introduction to symplectic topology ed. 1, section 3.2-3.3). Moser's trick. A technical lemma: given closed 2-forms that agree and are nondegenerate on a closed submanifold Q, there is a local diffeomorphism near Q that takes one form to the other. Darboux's theorem. Definition and existence of Darboux charts. The Moser stability theorem for symplectic structures. Definition of symplectic/(co)isotropic/Lagrangian submanifolds. Weinstein's Lagrangian neighborhood theorem (up to a claim that we will prove at the beginning of Lecture 3).
Lecture 3 (references: McDuff-Salamon "Introduction to symplectic topology ed. 3, sections 1.1 and 3.1 [the portion about Hamiltonian vector fields and symplectomorphisms, and Poisson structures]). Proved Weinstein's Lagrangian neighborhood theorem (left over from last time), up to an exercise that's in the exercises sheet. Setup for classical mechanics, starting from the Lagrangian formulation. Derived the Euler-Lagrange equations. Applied the Legendre transformation to move to the Hamiltonian formulation, and derived Hamilton's equations. Derivation of the Poisson bracket, and statement that a quantity is conserved if and only if its Poisson bracket with the Hamiltonian is zero. C^infty(R^{2n}) is a Lie algebra via {-,-} (without proof). Note that these constructions can be recast in terms of the standard symplectic form. Definition of Hamiltonian vector field on any symplectic manifold. Definition of Hamiltonian flow, and proof that it is a symplectomorphism. H is conserved under the flow of its Hamiltonian vector field. Definition of Poisson bracket on general M.
Lecture 4 (references: Evans "Lectures on Lagrangian torus fibrations, sections 1.3-1.6).