Physics 250
Geometry and Topology for Physicists
or
Mathematics of Gauge Theories and Gravitation
Fall, 2008
University of California, Berkeley
 Instructor: Robert Littlejohn
 Office: 449 Birge
 Office Hours: TBA
 Telephone: 6421229
 Email: physics250@wigner.berkeley.edu
Reader: Hal Haggard hal@wigner.berkeley.edu

 Lecture: 402 LeConte
 Time: TuTh 23:30
 Text(s): M. Nakahara, Geometry, Topology and Physics
(IOP Pubishing, Bristol,
2003), 2nd edition, and Theodore Frankel, The Geometry of Physics
(Cambridge, 1997), recommended.

 The textbook by Nakahara hasn't
changed too much from the first (1990) edition, and you can probably
get by
with it
if you don't want to buy the new one.
Organization and Logistics
The email address for this course is physics250@wigner.berkeley.edu.
If you wish to be included on the mailing list for course
announcements, homework notices, etc., send an email to this address
with your name.
(You don't have to be enrolled.) If you drop the course or don't
want to receive any more announcements, send an email to this address
with a request to be dropped.
The course web site
(this site) will be used to post lecture notes, special notes and
homework assignments, and homework solutions.
This is a variable unit
course (24 units). Fewer units
means less work, and may be useful if you are busy in
research etc.
The grade will be
based on weekly homework. Those taking the course for 2, 3 or
4 units will be expected to make a good attempt on 50%, 75% or
100%, respectively, of the homework. Actually, since some
problems are easier than others, you can usually do 50% of them with
less than 50% of the work. For some problems it
will be possible to get full credit simply by writing "trivial" or
"done this before". I don't intend to grade very hard, the point
of the course is to present some material that may be useful to you.
This is by nature a survey course. The general plan will be
to
convey the key ideas, a working familiarity with the formalism and
terminology, and an intuitive feel for each of the main areas, without
going too far into technicalities or difficult proofs.
Certain key results will be stated without proof.
There are no specific prerequisites
beyond graduate standing in Physics.
No particular mathematics background is required beyond that
usually expected of graduate students in physics (linear algebra,
complex analysis, etc), but it will help if you have some familiarity
with mathematical notation and ways of thinking.
Lecture notes are
my handwritten notes that I prepare for
lecture. I don't guarantee that they will be identical to what is
presented in class, but they should be pretty close usually, so you
can probably do without taking notes in class if you want to download
these. These notes are available in pdf format only.
Lecture Notes.
 Notes 1: Manifolds, Tangent
Vectors and Covectors, in ps
or pdf format, with extra figures.
 Notes 2: Differential Geometry
of Lie Groups, in ps
or pdf format, with extra figures.
Homework assignments
will normally (I hope) be made available on this web site by Friday of
each week, and will be due one week later, Friday at 5pm, in the
envelope outside my office (449 Birge).
 Homework 1, due Friday, September 5 at 5pm in postscript or pdf
format.
 Homework 2, due Friday, September 12 at 5pm in postscript or pdf
format.
 Homework 3, due Friday, September 19 at 5pm in postscript or pdf
format.
 Homework 4, due Friday, September 26 at 5pm in postscript or pdf
format.
 Homework 5, due Friday, October 3 at 5pm in postscript or pdf
format.
 Homework 6, due Friday, October 10 at 5pm in postscript or pdf
format.
 Homework 7, due Friday, October 17 at 5pm in postscript or pdf
format.
 Homework 8, due Friday, October 24 at 5pm in postscript or pdf
format.
 Homework 9, due Friday, October 31 at 5pm in postscript or pdf
format.
 Homework 10, due Friday, November 7 at 5pm in postscript or pdf
format.
 Homework 11, due Friday, November 14 at 5pm in postscript or pdf
format.
 Homework 12, due Friday, November 21 at 5pm in postscript or pdf
format.
 Homework 13, due Friday, December 5 at 5pm in postscript or pdf
format.
 Unhomework 14, nothing due, in postscript or pdf
format.
 Homework 15, due Friday, December 12 at 5pm in postscript or pdf
format.
Solutions
 Solutions for hw1.
 Solutions for hw2.
 Solutions for hw3.
 Solutions for hw4.
 Solutions for hw5.
 Solutions for hw6.
Course Outline
1. Physical context. Gauge theories, monopoles, instantons, order
parameters, general relativity, Berry's phase, classical mechanics, etc.
2. Mathematical background. Maps, functions, hierarchy of spaces and
manifolds, vector spaces, metrics, pullbacks and adjoints, raising
and lowering indices, quotient spaces, group actions etc.
3. Homology and homotopy. Abelian groups, simplexes, homology
groups, fundamental group, higher homotopy groups, defects in liquid
crystals, textures in superfluid ${\rm He}_3$.
4. Differentiable manifolds, tangent, cotangent spaces and bundles,
flows and Lie derivatives, differential forms, Stokes' theorem,
differential geometry of group manifolds, Lie algebras, de Rham
cohomology, Poincare' lemma, Frobenius theorem.
5. Riemannian manifolds. Parallel transport, connection, curvature,
holonomy, isometries and Killing vectors, vielbeins, Hodge theory,
Cartan structure equations.
6. Fiber bundles, vector bundles, principal bundles, spin and
Clifford bundles, connections, holonmy, curvature, gauge field
theories, general relativity as a gauge theory, geometrical phases.
7. Characteristic classes, topology of fiber bundles, Chern classes
and characters, ChernSimons forms, StiefelWhitney classes.
This is a little more than I covered the last time I taught the
course. I may leave out some material on general relativity, which is
covered in Physics 231 anyway, to make time to finish the list of
topics. I may also skip some of the introductory material.