M392C
SYLLABUS

INSTRUCTOR INFO

INSTRUCTOR: TAMAS HAUSEL

PHONE: 471-7169

OFFICE: RLM 11.168

OFFICE HOURS: MWF 10-11am; also by appt.

EMAIL: hausel@math.utexas.edu

COURSE WEB SITE http://www.math.utexas.edu/~hausel/m392c/

GRADES

Grade (A) will be given for either HOMEWORKS or ESSAY or TALK.

HOMEWORKS: Solve plenty (say at least 20) good homeworks from the Book, well distributed among the chapters. Deadline for submissions: May 2

ESSAY: Write an essay on a subject of your interest about an application of groups or/and symplectic geometry. Deadline for submission: May 2.

TALK: Give 25-50 minutes talk on a subject of your interest about an application of groups or/and symplectic geometry. The talks will be presented during the last two weeks of class.

BOOKS

During this course "Book" will stand for

  • R. Bryant: An introduction to Lie Groups and Symplectic Geometry (pages 7-181 in Geometry and Quantum Field Theory, eds: Daniel Freed and Karen Uhlenbeck , AMS-IAS, 1995)

    Other relevant text books:

  • 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,

  • M. Audin: The topology of torus actions on symplectic manifolds, for symplectic geometry

    for symplectic geometry,

  • V. Arnold: Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics, 60. Springer-Verlag, New York

    for classical mechanics,

  • I. Shafarevich: Basic notions of algebra Algebra. I, Encyclopaedia Math. Sci., 11, Springer-Verlag, Berlin, 1997

    for group theory.

    SYLLABUS

    We will roughly follow the following timeline (Lecture n means, Lecture n in the Book):

    WEEK 1: Basic notions (group theory, differentiable manifolds, ODE's and classical mechanics) -- Lecture 1

    WEEK 2-5: Introduction to Lie groups and Lie algebras --Lecture 2

    WEEK 6: Group actions on manifolds --Lecture 3

    WEEK 7: Symmetries and conservation laws -- Lecture 4

    WEEK 8-11: Symplectic manifolds -- Lecture 5,6

    WEEK 12: Classical reduction -- Lecture 7

    WEEK 13: Kähler and hyperkähler reductions -- Lecture 8

    WEEK 14-15: Student presentations

    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]