Publications [#9565] of Robert Bryant

Papers Published

1. An introduction to Lie groups and symplectic geometry, in Geometry and quantum field theory (Park City, UT, 1991), IAS/Park City Mathematics, vol. 1 (1995), pp. 5–181, Amer. Math. Soc., Providence, RI [MR96i:58002]
   (last updated on 2004/04/06)

   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.