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.
In this page:
- Announcements
- Syllabus
- Textbooks
- Grading Policy and Exams
- Homework Assignments
- Emergency information for students in Mathematics courses
Announcements:
- Class will meet for the first time on Wednesday, January 22.
- The 4th Homework Assignment is available here. It will be discussed in class on Monday, March 31.
- We are now using a second set of lectures notes:
- M. Crainic and R.L. Fernandes, Lectures on Integrability of Lie Brackets, Geometry & Topology Monographs 17 (2011) 1-107.
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, Birkhäuser, Basel, 2005.
Grading Policy
- 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.
(PDF files can be viewed using Adobe Acrobat Reader which can be downloaded for free from Adobe Systems for all operating systems.)
Emergency information for students in Mathematics courses
For important emergency information related to fires, tornados or active threats, please look at the following leaflet.
Last updated April 20, 2014.