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:  642-1229
Email:  physics250@wigner.berkeley.edu
Reader:  Hal Haggard hal@wigner.berkeley.edu

Lecture:  402 LeConte
Time:  TuTh 2-3:30
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 (2-4 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.


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). 


Solutions



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, pull-backs 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, Chern-Simons forms, Stiefel-Whitney 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.