Math 597B, Spring 2016

 


Course Title:  Symplectic Geometry


Instructor:  Nigel Higson


Class Meeting Times:  Mondays, Wednesdays and Fridays, 1:25-2:15, in 106 McAllister


Office Hours:  By appointment



Overview:  The major prerequisite for this course is some familiarity with smooth manifolds and differential forms (the course will include a review of these things, but the review will be very rapid).  The major goal of the course is to develop the theory of symplectic manifolds as it arises from efforts to formulate the mathematical foundations of classical mechanics.   Major topics will be normal forms, Lagrangian fibrations, symplectic reductions, and group actions, especially torus actions.



Class Work: I’ll provide a variety of homework problems that I hope will help deepen our understanding of the material.  There will be no exams.



Background Material on Manifolds and Differential Forms


    Spivak, Comprehensive Introduction to Differential Geometry, Vol. 1


    Warner, Foundations of Differentiable Manifolds and Lie Groups, Chapters 1,2,4



Three references that I particularly recommend


   Meinrenken - Symplectic geometry


   Duistermaat - Symplectic geometry


   Heckman - Symplectic geometry



A Survey of Symplectic Geometry


    Weinstein - 1981 - Symplectic geometry



Other Lecture Notes on Symplectic and Poisson Geometry


    Bates, Weinstein - Lectures on the geometry of quantization


    Bryant - Introduction to Lie groups and symplectic geometry


    Cannas da Silva - Lectures on symplectic geometry



Two classic and comprehensive texts


    Guillemin, Sternberg - Geometric asymptotics


    Guillemin, Sternberg - Symplectic techniques in physics



Academic Integrity: All Penn State policies regarding ethics and honorable behavior apply to this course (see links below for policy statements). Academic integrity is the pursuit of scholarly activity free from fraud and deception and is an educational objective of this institution. All University policies regarding academic integrity apply to this course. Academic dishonesty includes, but is not limited to, cheating, plagiarizing, fabricating of information or citations, facilitating acts of academic dishonesty by others, having unauthorized possession of examinations, submitting work of another person or work previously used without informing the instructor, or tampering with the academic work of other students. For any material or ideas obtained from other sources, such as the text or things you see on the web, in the library, etc., a source reference must be given. Direct quotes from any source must be identified as such. All exam answers must be your own, and you must not provide any assistance to other students during exams. Any instances of academic dishonesty WILL be pursued under the University and Eberly College of Science regulations concerning academic integrity. 


(For a more compelling account of what honesty and integrity should mean, at least for a scientist, consider these famous words of Richard Feynman.



Disability Statement: Penn State welcomes students with disabilities into the University's educational programs. Every Penn State campus has an office for students with disabilities. The Office for Disability Services (ODS) Web site provides contact information for every Penn State campus: http://equity.psu.edu/ods/dcl. For further information, please visit the Office for Disability Services Web site: http://equity.psu.edu/ods.   In order to receive consideration for reasonable accommodations, you must contact the appropriate disability services office at the campus where you are officially enrolled, participate in an intake interview, and provide documentation: http://equity.psu.edu/ods/doc-guidelines. If the documentation supports your request for reasonable accommodations, your campus’s disability services office will provide you with an accommodation letter. Please share this letter with your instructors and discuss the accommodations with them as early in your courses as possible. You must follow this process for every semester that you request accommodations.

Nigel Higson - 2016