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:

  1. Category theory
  2. Group theory
  3. Vector spaces (in detail, including infinite dimensional spaces, tensors, exterior algebra etc.)
  4. A bit more category theory
  5. Various algebras very briefly including associative, Lie, Clifford, C* and quaternion algebras
  6. Topology in detail
  7. 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

  1. The Lorentz group
  2. Minkowski Space
  3. Dynamical Systems
  4. Numerical PDEs
  5. 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