Math 595 – Poisson Geometry – Spring 2020

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 looking at the following slides. In this course I will be following a book that I am writing with Marius Crainic and Ioan Marcut. You can check the table of contents. Students taking this course are assumed to know differential geometry at the level of Math 518 – Differentiable Manifolds.

Email: ruiloja (at) illinois.edu
Office: 346 Illini Hall
Office Hours: MF 1:30-2.30 pm (or by appointment);
Class meets: 9:00-9:50 AM MWF 441 Altgeld Hall
Prerequisites: Math 518 or equivalent.

In this page:

  • Announcements
  • Syllabus
  • Textbooks
  • Grading Policy and Exams
  • On-line classes
  • Emergency information for students in Mathematics courses

Announcements:

  • Class will meet for the first time on Monday, January 27.
  • There will be no classes from Wednesday, March 11, to Friday, March 23.
  • According to the new rules, due to the SARS-CoV-2 pandemic, classes have moved on-line. I will be sharing videos in anticipation of each chapter of the lecture notes. See the section “on-line classes”.

Syllabus:

  • Basic Concepts. Poisson brackets; Poisson bivectors; The Darboux-Weinstein Theorem.
  • The Symplectic Foliation. Symplectic leaves and symplectic foliations; Poisson transversals; Symplectic realizations; Dirac geometry; Submanifolds in Poisson geometry.
  • Global Aspects. Lie groupoids, integrability, symplectic realizations, averaging and linearization, Van Est map.
  • Symplectic Groupoids. Complete symplectic realizations; Lie groupoids and Lie algebroids;Symplectic groupoids.
  • Special Topics. To be chosen from: Moduli space of flat connections; A-symplectic structures; Conn’s Linearization Theorem; Cluster algebras; Symplectic stacks; Symplectic foliations; Quantization deformation.

Textbooks:

I will provide to participants 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 encouraged to write (in LaTeX) and present a paper. This is not mandatory. Following the tradition of topics courses, there will be no homework and no written exams.

On-line classes

Chapters 1-7 of the Lecture Notes were covered in the face-to-face lectures.

(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, 2020.