# Math 595 – Poisson Geometry – Spring 2014

This course is an introduction to Poisson geometry. Poisson geometry is the study of differentiable manifolds equipped with a Poisson bracket. Its roots lie in Classical Mechanics, but it became an independent field of study in the 70’s and in the 80’s, in parallel to its close cousin Symplectic geometry. If you have a basic knowledge of manifolds, vector fields and differential forms, you can get an idea of what Poisson geometry is by reading a brief introduction (under construction). Students taking this course are assumed to know differential geometry at the level of Math 518 – Differentiable Manifolds. Lectures for this topic course will be delivered by Rui Loja Fernandes and Ioan Marcut.

Main Lecturer:Rui Loja Fernandes
Other Lecturer: Ioan Marcut
Email: ruiloja (at) illinois.edu
Office: 346 Illini Hall
Office Hours: Wed 11:00 AM (or by appointment);
Class meets: MWF 12:00-12:50 PM, 443 Altgeld Hall;
Prerequisites: Math 518 or equivalent.

## Syllabus:

• Foundational aspects. Poisson brackets, Poisson maps, Hamiltonian vector fields,Poisson submanifolds, coisotropic submanifolds, Dirac manifolds. Quotients and Reduction.
• Infinitesimal aspects. Poisson transversals, Weinstein Splitting, couplings. Lie algebroids, connections, parallel transport, holonomy. Cohomology and deformations.
• Global Aspects. Lie groupoids, integrability, symplectic realizations, averaging and linearization, Van Est map.

## Textbooks:

I will provide some lecture notes as the course progresses, but the following two references should also be very helpful:

• A. Cannas da Silva and A. Weinstein, Geometric models for noncommutative algebras, Berkeley Mathematics Lecture Notes, 10. American Mathematical Society, Providence, RI, 1999.
• J.-P. Dufour and N.T. Zung, Poisson Structures and Their Normal Forms, Progress in Mathematics, Vol. 242, BirkhÃ¤user, Basel, 2005.

• Expository Paper Students will 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.
• Problem Sessions There will be problem sessions every other week. Students are not required to turn in homework, but homework problems will be discussed in these sessions.

## Homework Assignments and Sections covered so far:

• Homework #1: Read Chapter 1 of the Lecture Notes and solve the Homework set at the end of the Chpater.
• Homework #2: Read Chapter 2 of the Lecture Notes and solve the Homework set at the end of the Chapter.
• Homework #3: Read Chapter 3 of the Lecture Notes and solve Exercise 3.3 and the Homework set at the end of the Chapter.
• Homework #4: Read Chapter 4 of the Lecture Notes and solve the Homework set at the end of the Chapter.

Sections of the 1st Set of Lecture Notes covered: Chapters 1-4.
Sections of the 2nd Set of Lecture Notes covered: Chapters 1.