Ph 229 Advanced Mathematical Methods of Physics

A more precise title for this course would be "Elements of Geometry and Topology for Theoretical Physicists." It is an introduction to areas of mathematics which find use in modern theoretical high-energy physics but are usually not taught to physics students. The topics to be covered in the course are The required books for the fall term are:

Theodore Frankel, "The Geometry of Physics" (2nd edition).

Shigeyuki Morita, "Geometry of Differential Forms."

The required book for the winter term are the same as for the fall term, plus

William S. Massey, "A Basic Course in Algebraic Topology."

In addition, I strongly recommend

Mikio Nakahara, "Geometry, Topology, and Physics" (it covers roughly the same ground as Frankel's book plus the Atiyah-Singer index theorem)

and, for the winter term:

Allen Hatcher, "Algebraic Topology" (it is more thorough and provides more motivation than Massey's book).

For the spring term, besides Frankel and Morita, the following is required:

R.O. Wells, Differential Analysis on Complex Manifolds.

Nakahara's book  will also be useful. Atiyah-Singer theorem is thoroughly covered in a book by John Roe, "Elliptic operators, topology, and asymptotic methods",
which I recommend; however our discussion will be rather superficial, and one can do without this book.

Other useful textbooks are listed here.


An outline of the course:

Fall Term: Basic Topology, Manifolds, Calculus on Manifolds.

Winter Term:  Vector Bundles and Connections, Basic Riemannian Geometry, Algebraic Topology.

Spring Term: Characteristic Classes, Spinors,  Atiyah-Singer Index Theorem, Basic Complex Geometry.

There will be homework problems, but no exams. Homework problems will be posted on this website each Wednesday.
The homework posted on week N is due by Friday of week N+1. You may leave them in TA's mailbox (see below), or
give them to him directly. Grading will be done on the basis of the turned in homework.
Each problem is worth 10 points, but if a problem has subproblems, then each subproblem is worth 10 points.

The TA for this course is Yi Li; he is located in 471 Lauritsen . Graded homeworks together with the solutions sets will be placed outside my office (451 Lauritsen).

I will have an office hour in 451 Lauritsen each Friday, from 4:30 p.m. to 5:30 p.m. Yi Li's office hour is on Thursday, 2:30 -3:50 p.m.

Homework problems (fall term)

Week 1 (due October 6)     Week 2 (due October 13)    Week 3 (due October 20)   Week 4 (due October 27) Week 5 (due November 3)

Week 6 (due November 10)   Week 7 (due November 17)  Week 8 (due November 24)  Week 9 (due December 1)  Week 10 (due December 8)

Homework problems (winter term)

Week 1 (due Jan. 13: note a later-than-usual date)   Week 2 (due Jan. 19)  Week 3 (due Jan. 26)  Week 4 (due Feb. 2)   Week 5 (due Feb. 9)

Week 6 (due Feb. 16)   Week 7 (due Feb. 23)  Week 8 (due March 2)  Week 9 (due March 9)  Week 10 (due March 16, sharp)

Homework problems (spring term)

Week 2 (due April 13)  Week 3 (due April 20)  Week 4 (due April 27)  Week 5 (due May 4)  Week 6 (due May 11)

Week 7 - no homework   Week 8 (due May 25)