# Math 520 – Symplectic Geometry – Fall 2019

Symplectic structures originated from the geometric formulation of classical mechanics. Nowadays, symplectic geometry is a central field in Mathematics with many connections with other fields, both in and outside Mathematics. This course presents an introduction to the foundational tools, ideas, examples and theorems of symplectic geometry. It is intended for PhD students studying symplectic geometry, Poisson geometry, and symplectic topology, as well as students in related areas such as dynamical systems, algebraic geometry, complex geometry, low dimensional topology and mathematical physics. The course covers the local and global structure of symplectic manifolds, their submanifolds, the special automorphisms they support (Hamiltonian flows), their natural boundaries (contact manifolds), their special geometric features (almost complex structures), and their symmetries. Students taking this course are assumed to know differential geometry at the level of Math 518 – Differentiable Manifolds.

Lecturer:Rui Loja Fernandes
Email: ruiloja (at) illinois.edu
Office: 346 Illini Hall
Office Hours: Mondays and Fridays 1.30-2.30 pm (or by appointment);
Class meets: MWF 9:00-9:50 AM, 447 Altgeld Hall;
Prerequisites: Math 518, or equivalent.

## Syllabus:

• Linear Symplectic Geometry:symplectic forms, examples and constructions, Darboux’s Theorem, Lagrangian submanifolds, Weinstein’s tubular neighborhood theorem, blowing up and down, symplectic cuts and sums.
• Symplectic Manifolds: symplectic forms, examples and constructions, Darboux’s Theorem, Lagrangian submanifolds, Weinstein’s tubular neighborhood theorem, blowing up and down, symplectic cuts and sums.
• Symplectomorphisms: fixed point theorems, Hamiltonian flows, Poisson brackets, integrable systems, the group of symplectomorphisms.
• Contact Manifolds: Contact structures, contact forms, Gray’s theorem, Reeb flows.
• Almost Complex Structures: almost complex structures, integrability, complex manifolds, Kahler forms, compact Kahler manifolds.
• Hamiltonian Group Actions and Reduction: group actions, moment maps, Marsden-Weinstein-Meyer reduction, toric manifolds.
• Advanced Topics (to be determined by instructor and student interests)

## Textbooks:

• Homework and Expository Paper: I will assign homework every other week, from the first reference above. Students will also be required to write (in LaTeX) and present a paper. The papers will be due by the last week of classes. Presentations will be in class, during the last week of classes.

## Homework Assignments, Lecture Notes and Sections covered so far:

All homework sets are from the book by Cannas da Silva:

• Homework set #1: (due Friday, Sep. 6):
• HW1 Problems 7, 8, 9;
• HW2 Problem 2;
• HW3 Problems 1, 2, 3;
• Homework set #2: (due Monday, Sep. 20):
• HW5 Problems 2, 3;
• HW6 Problems 1, 2, 3;
• Homework set #3: (due Monday, Oct. 21):
• HW7 Problems 1, 2, 3, 4;
• Homework set #4: (due Monday, Nov. 18):
• HW10 Problem 1;
• HW12 Problems 1-7;

Lecture Notes (Disclaimer: this contains mistakes! Better read the book…)

Sections of the book covered so far: 1.1-1.4; 2.1-2.4; 3.1-3.4; 4.1.-4.3; 5.1, 7.1-7.3, 8.1-8.3, 9.1-9.4, 10.1-10.3, 11.1-11.3, 12.1-12.3, 13.1-13.3, 14.1-14.3, 15.1-15.3, 16.1-16.4, 17.1-17.4, 18.1-18.4, 21.1-21.5, 22.1-22.4, 23.1-23.3.