Homepage of M392C, Introduction to Lie Groups and Symplectic Geometry; Spring 2003
Motto
"We may as well cut out the group theory. That is a subject that will never be of any use in physics."
[James Jeans discussing a syllabus in 1910]
Course
Syllabus online, printable version here.
Professor: Tamás Hausel, Office Hours: MWF 10-11am, Office: RLM 11.168, Phone: 471-7169, Email:
hausel@math.utexas.edu
Time: MWF 9-10am, Room: RLM 10.167
Description of course
This course gives an introduction to Lie groups. The emphasis will
be on applications and examples, i.e. Lie groups appearing as transformation groups and symmetry groups, rather than abstract theory. The applications will include Lie's original theory of differential equations of Lie type and Noether's theory of symmetries
and conservation laws. This latter subject will lead us to symplectic geometry (the geometry of Hamiltonian mechanics) and Hamiltonian group actions.
I will discuss symplectic reduction, with many examples, like toric varieties and moduli spaces of flat connections on Riemann surfaces.
If time permits I will discuss kähler and hyperkähler reductions and/or the Borel-Weil-Bott theory,
which gives a
complete description of the representation theory of compact Lie groups
using the symplectic geometry of coadjoint orbits in the dual
of the Lie algebra.
Prerequisites
The course will be self-contained though basic
familiarity with any of ODE's, classical mechanics,
differentiable manifolds and (finite) group theory is an advantage; physics students are welcomed! Contact me
if you are uncertain if this course is for you!
Text books
We will roughly follow
R. Bryant: An introduction to Lie Groups and Symplectic Geometry (in
Geometry and Quantum Field Theory, eds: Daniel Freed and Karen Uhlenbeck
, AMS-IAS, 1995) (for an updated version check out this link)
Other relevant text books:
V. Arnold: Mathematical Methods of Classical Mechanics,
Graduate Texts in Mathematics, 60. Springer-Verlag, New York
for symplectic geometry; and
F. Warner: Foundations of differentiable manifolds and Lie groups.
Corrected reprint of the 1971 edition. Graduate Texts in Mathematics, 94.
Springer-Verlag, New York-Berlin, 1983.
for differentiable manifolds and Lie Groups.
Links
A quick introduction to point set topology (including the Lebesgue
covering lemma) is available here.
A nice down to earth introduction to the tangent spaces of matrix Lie
groups could be found in 24.2 of this book. (This book is part of a series of three books which I can warmly
recommend as general background books for any subject in geometry.)
A leisurely account of Cohen's proof of the immersion conjecture is available here.
An immersion of the projective plane into 3-space: Boy surface
For quick introduction to some basic notions in point-set topology, group theory, Galois theory, differentiable manifolds, Lie groups and the like check out the handouts of
the Warmup Programme of the University of Chicago.
Cubic formula.
Classification of finite simple groups.
Analyzing Rubik's cube with GAP.
Check out a paper by Golomb: Rubik's cube and a model of quark confinement.
Links to the biographies of Group theory, Galois, Lie, Klein, Wigner, the Standard Model and Superstring theory.
Watch in streaming video Bryant's lecture "On the geometry of differential equations", MSRI, 2000.
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