Math @ Duke

Publications [#318275] of Robert Bryant
search www.ams.org.Papers Published
 Bryant, R, An introduction to Lie groups and symplectic geometry,
in Geometry and quantum field theory (Park City, UT, 1991), IAS/Park City Mathematics, edited by Freed, D; Uhlenbeck, K, vol. 1
(1995),
pp. 5181, American Mathematical Society, ISBN 0821804006 [MR96i:58002]
(last updated on 2019/08/25)
Author's Comments: This is a series of nine elementary lectures on Lie
groups and symplectic
geometry that were the basis for a short course in the
subject that I gave
in Park City, Utah in 1990 as part of the Regional
Geometry Institute Summer
School. These notes cover the usual introductory
material in Lie groups
(with some extra material on Lie's method of solving
differential equations
with symmetry), Lagrangian mechanics, Noether's
Theorem, symplectic manifolds,
the moment map and reduction, and concludes with a
brief look at the elliptic
methods that have become so important in symplectic
geometry in the last
ten years, largely due to the influence of M. Gromov.
As a result of my using these lectures again as a
resource for a graduate
course in Spring 2003 and also as a result of
Eugene Lerman pointing out some problems with
Lecture 8 (wherein I
discuss hyperKähler reduction), I have found
some serious flaws in
Lecture 8. In particular, the purported Theorems
2 and 5 (Kähler
and hyperKähler reduction) are not true in the
generality in which
I stated them in the published version. I
apologize for these mistakes
and thank Eugene for bringing them to my attention.
I have corrected these
mistakes, and in doing
so, have found it to be a good idea to modify both
Lectures 7 and 8.
I have posted the corrected
version here.
Please be aware, though, that these lectures are
essentially the same as
the original lectures, i.e.,
they are meant to
be a very quick and cursory introduction to the subject in
the title, not
an exhaustive treatment (of
which there are now
several very good ones by experts in symplectic
geometry).
Abstract: A series of lectures on Lie groups and symplectic
geometry, aimed at the beginning graduate student level.


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