• Basic Topology: General Topology, CW spaces, Homotopy, Homology.
• Smooth Manifolds: Calculus on Manifolds, De Rham theory, Fiber Bundles, Connection and Curvature.
• Basic Riemannian Geometry: Parallel Transport, Levi-Civita Connection, Riemann Tensor.
• Characteristic Classes, Index Theorems.
• Complex Geometry: Calculus on Complex Manifolds, Holomorphic Vector Bundles, Coherent Sheaves, , Kahler Manifolds.

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.

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.
 

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)