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 book by M. Nakahara ( "Geometry, Topology, and Physics," Adam Hilger (1990)) covers much of the
same material and therefore is the recommended textbook. I will use other sources as well, and students
are advised to consult them.

A rough outline of the course:

Fall Term: Basic Topology, Vector Bundles, Characteristic Classes.

Winter Term: Smooth Manifolds, Basic Riemannian Geometry.

Spring Term: Index Theorems, Complex Geometry.

There will be homework problems, but no exams. Grading will be done on the basis of the turned in homework.
The homework posted on week N is due by Friday of week N+1. The TA for this course is Takuya Okuda;
he is located in 422 Downs. 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.

Fall term

A more detailed outline of the fall term is here . Besides Nakahara, the most useful books for the fall term are
Massey and Vassiliev.  A copy of Massey's book is on reserve in Millikan Library.

Homework problems (fall term).

Week 1     Week 2      Week 3  Week 4     Week 5     Week 6    Week7     Week 8     Week 9

Winter term

A detailed outline of the term can be found here.  Nakahara contains almost all the material for this term,
but those wishing for a more pedagogical exposition may want to consult Warner.  In fact, during the first half
of this term I will follow Warner pretty closely. Warner does not cover Riemannian geometry,  the theory of
connections on vector bundles, or symplectic geometry. For the first two topics, one may consult Bishop and
Crittenden (it can also serve as an alternative to Warner), or, if one wants an exposition geared towards
physics applications, Curtis and Miller. These topics are also discussed in detail in a classic monograph by
Kobayashi and Nomizu.  As for symplectic geometry, I recommend Arnold, Guillemin and Sternberg, or
Cannas da Silva and Weinstein. I particularly like the last of these: it is up-to-date, and contains lots of
fascinating mathematics  which could be relevant for string theory.

Homework problems (winter term).

Week 1 (due Jan. 17)   Week 2 (due Jan. 24)   Week 3 (due Jan. 31)  Week 4 (due Feb. 7)  Week 5 (due Feb. 14)

Week 6 (due Feb. 21)  Week 7 (due Feb. 28)  Week 8 (due March 7) Week 9 (due March 14)

Spring term

I will discuss characteristic classes of vector bundles, index theorems for elliptic operators, and their
applications to physics. The rest of the spring term will be devoted to complex manifolds. I plan to
discuss calculus on complex manifolds, Dolbeault cohomology, holomorphic vector bundles, coherent
sheaves and sheaf cohomology, Riemann surfaces, Kahler manifolds, and in particular Calabi-Yau
manifolds.   A useful summary of characteristic classes and index theorems can be found in
a review by Eguchi, Gilkey, and Hanson, Physics Reports 66, p. 213.  The original papers by Atiyah, Singer,
and collaborators are also quite readable. A good textbook on complex manifolds is Wells.  Chapter 0 of
Griffiths and Harris is a compressed account of the first few chapters of Wells.  A good new book on
complex manifolds is Zheng.  For Riemann surfaces, you may consult a book by Miranda, or a review
by Shokurov.  A good textbook on complex algebraic geometry (I will barely mention this subject in this course)
is Shafarevich.  There is also an excellent review of modern algebraic geometry by Danilov.

Homework problems (spring term).

Week 1 (due April 11) Week 2  (due April 18)  Week 3 (due April 25)   Week 4 (due May 2)  Week 5 (due May 9)

Week 6 (due May 16)  Week 7 (due May 23)  Week 8  (due May 30)