## Mathematical Physics

Class Notes #1: Categories and Sets

Class Notes #2: Groups

Class Notes #3: Vector Spaces

Class Notes #4: Tensors, Inner Product Spaces, Operators

Class Notes #5: Operators, Numerical PDEs, Functors, a bit of Topology

*errata: on page 13, three-space with a cross product is a Lie algebra rather than an associative algebra.*

Class Notes #6: Topology

Class Notes #7: Differential Topology I

Homework #0 Solutions #0

Homework #1 Solutions #1

Homework #2 Solutions #2

Homework #3 Solutions #3

Homework #4 Solutions #4

## Mathematical Physics

###
Topics:

- Category theory
- Group theory
- Vector spaces (in detail, including infinite dimensional spaces, tensors, exterior algebra etc.)
- A bit more category theory
- Various algebras very briefly including associative, Lie, Clifford, C* and quaternion algebras
- Topology in detail
- Differential topology (manifolds, differential forms, vector fields, index theory, integration, cohomology, Lie groups).

The course is organized around the mathematical topics with examples taken from Physics, including

- The Lorentz group
- Minkowski Space
- Dynamical Systems
- Numerical PDEs
- Maxwell's equations.

For this course, we will begin at the very beginning, so there are no prerequities as such. However,
this is a graduate level course - the pace will be rapid and the level of the course will be rigorous
with theorems and proofs and there will be lots of homework.
**Lectures:** One 1.5 hour lecture per week, June-August 2004.

**Books:** Required: *Mathematical Physics*, Robert Geroch, University of Chicago Press 1985.
Nice: *Topology from the Differentiable Viewpoint*, John W. Milnor, The University Press of Virginia 1990; *Differential Topology*, Victor
Guillemin and Alan Pollack, Prentice-Hall, 1974.

Saul Youssef