Lynn S. Bennethum
Office: CU 638, Phone (303) 556-4810
Office hours: Mon. and Wed. 5:30-6:30pm in CU 638, or by appointment.
e-mail: Lynn.Bennethum@cudenver.edu
home page: http://www-math.cudenver.edu/~bennethm
fax: (303) 556-8550
home phone: 683-6983 (Please call after 9am and before 9:30pm - I generally work at home on Fridays).
 
 

This is a 1 credit-hour course covering the basics of lie group theory.  With John Starrett's help, we'll cover some pre-requisite differential geometry, and then go into Lie group methods which can be used to solve ODEs and help with determining the form of solutions of PDEs.  In the introduction of the textbook,  Olver states:

When beginning students first encounter ordinary differential
equations they are, more often than not, presented with a bewildering
variety of special techniques designed to solve certain particular,
seemingly unrelated types of equations, such as separable, homogeneous
or exact equations.  Indeed, this was the state of the art around
the middle of the nineteenth century, when Sophus Lie made the
profound and far-reaching discovery that these special
methods were, in fact, all special cases of a general integration
procedure based on the invariance of the differential equation
under a continuous group of symmetries.  This observation at once
unified and significantly extended the available integration
techniques, and inspired Lie to devote the remainder of his
mathematical career to the development and application of his
monumental theory of continuous groups...

Textbooks:
Applications of Lie Groups to Differential Equations, by Peter J. Olver, 2nd edition, Springer-Verlag (available at Amazon.com)
Geometrical Methods of Mathematical Physics, by Bernard Schutz, Cambridge
 Applications of Lie's Theory of Ordinary and Partial Differenctial Equations, by Lawrence Dresner, ISBN 0-7503-0531-2.  Available from Fat Brain for $31 (includes shipping).  Suggested by Martin Golubinsky, it is geared towards 4th year undergrads/ 1st year graduate students.  Clearly written, with lots of example problems and detailed solutions.  Definitely recommended!

Grading:  It is expected that each person will participate in discussion and will take his/her turn in leading the discussion.  This is a cooperative learning experience, so it is not expected that you understand all before leading a discussion.
 

Syllabus:  We will try to cover chapters 2 and 3 of Geometrical Methods of Mathematical Physics, and then chapters 1 and 2 of Applications of Lie Groups to Differential Equations.

Aug. 22:  Schutz, Sections 2.1-2.4;  definition of a manifold.  John Sterrett
Aug. 29:  Schutz, Sections 2.5-2.11;  functions, curves, vector fields on a manifold;  Fiber bundles.  John Starrett
Sept. 5:   Schutz, Sections 2.12,2.13, introduction to 2.14  Vector fields, integral curves, intro to Lie brackets.  Lynn Bennethum
Sept. 12:  Schutz,  Sections 2.14-2.17    Lie bracket and one-forms.       Lynn Bennethum.
Sept. 19:  Schutz,  Sections 2.18  One-forms.  Randy Chase.
Sept. 26:  Rico Argentati  2.19, 2.20, 3.1  More on one forms, introduction to Lie Groups
Oct.  3:  Rico Argentati  3.1, 3.2, 3.3.   Lie Derivatives.
Oct. 10:  Saulo Oliveira:  3.4 (Lie Derivatives), 3.5, 3.6, 3.9, 3.14 from Schutz, Section 1.2 from Olver
Oct. 17:  Saulo Oliveira:  Continued
Oct. 31:  (Halloween!!!)  Barry Ashworth - Examples of Lie Groups (Olver)  Schutz Section 3.15 and Section 1.2 from Olver
Nov 7:    Barry Ashworth and Dave Brown ?  - Examples of Lie Groups (Olver) continued
Nov 14:  Lynn Bennethum:   Brief discussion on first-order nonlinear PDE's.