The past 20 years have been an exciting time for the interaction of mathematics and physics.
Remarkable new mathematical discoveries have been made using the methods of quantum field
theory. Conversely, powerful new mathematical techniques have been successfully applied to
gain nontrivial insights in sophisticated theories like supersymmetric gauge theories and string theory.

The purpose of this course is to provide some of the mathematical background which one needs
in order to learn about these modern developments. The course surveys some aspects of  topology and differential geometry of manifolds, with an emphasis on relations to modern mathematical physics. The course will also cover some aspects of supersymmetry.

This course is primarily intended for physics graduate students specializing in theoretical physics.

Some knowledge of manifolds, differential forms, and cohomology will be assumed, but we
will try to keep the course material self-contained.
 

An optimistic goal is to end with the proofs of the index theorems for elliptic operators using supersymmetric quantum mechanics. An approximate sequence of topics will be as follows:

 
1. Riemannian geometry. Orthonormal frames.

2. Clifford algebras and spinors.  Supersymmetry algebras.
 
3.  Hodge *,  Hodge theory, harmonic forms

4. Maxwell theory and generalized Maxwell theories. Electric-magnetic duality of  gauge theories of forms.  Self-dual equations of motion.

5. Connections and curvature on fiber bundles.
 
6. Supersymmetric quantum mechanics and the index.
 
7. Proofs of the index theorems using supersymmetric  quantum mechanics.