Math 257A

Introduction to Symplectic Geometry

(MWF 9:30-10:20 @ 200-303)

Office hours: W, 4-5pm

Grades and homework:

Bi-weekly homeworks will be posted here. Grade is entirely based on homework.

Problem set 1 (due Oct 4)  Solution to q4 (by Dylan)

Problem set 2 (due Oct 18)

Problem set 3 (due Nov 1)

Problem set 4 (due Nov 15)

Problem set 5 (due Dec 6)

Plan for the course:

- The first 8 lectures of "Lectures on Symplectic Manifolds", by Alan Weinstein

- Gromov non-squeezing theorem (possibly from Gromov's original paper, cited below)

- Symplectic rigidity of Lagrangian submanifolds (from Audin-Lalonde-Polterovich article cited below)

Schedule

Week 1: Three levels of symplectic objects, cotangent bundle and its symplectomorphisms

Week 2: Symplectic linear algebra, Lagrangians, almost complex structures

Week 3: Complex manifolds, Stein manifolds, Kahler manifolds, Fubini-Study form on CP^n

Week 4: Constructions of closed symplectic manifolds, symplectic reduction, Hamiltonian vector fields

Week 5: Hamiltonian diffeomorphisms, symplectic and Hamiltonian isotopies, flux

Week 6: Coadjoint orbits, Hamiltonian group actions, moment map 

Week 7: Moser's argument, Darboux's theorem, normal neighborhoods of isotropic submanifolds

Week 8: Gromov non-squeezing

Week 9: Reduction of Lagrangians, Lagrangian correspondances, generating functions

Week 10: Symplectic rigidity of Lagrangian submanifolds

Sources 

Weinstein, Lectures on Symplectic Manifolds (book)

Gromov, Pseudo-holomorphic curves in symplectic manifolds

Audin-Lalonde-Polterovich, Symplectic rigidity: Lagrangian submanifolds

Wendl, Lectures on holomorphic curves

Sandon, Generating functions in symplectic topology

---Some other books on basic symplectic geometry

McDuff-Salamon, Introduction to Symplectic Topology 

Da Silva, Lectures on Symplectic Geometry

Arnol'd, Mathematical Methods of Classical Mechanics

 

 

Last modified Tue, 10 Dec, 2019 at 17:29