Math 542, Fall 2012
Topics in advanced topology:
Differential forms and de Rham cohomology

Brendan Hassett
Herman Brown 402
hassett@rice.edu
http://www.math.rice.edu/~hassett
Office hours: online calendar and by appointment

Course objectives: This course will illustrate how cohomology on manifolds may be understood via calculus; our goal is for you to be proficient in using these techniques in a wide range of situations. We will assume knowledge of homology/cohomology at the level of MATH 445/540.

Topics:

Manifolds: tangent spaces, inverse/implicit function theorems, vector fields, distributions

Tensors: exterior algebra, differential forms

Integration: orientation, closed and exact forms

de Rham theory: sheaves and axiomatic sheaf cohomology, `classical' cohomology theories, proof of the de Rham theorem

Textbook: F. Warner, Foundations of Differentiable Manifolds and Lie Groups, Springer Verlag 1983
available at the bookstore

Assessment: The grade will be based on problem sets due Thursdays starting August 30. They should be turned in during class. These assignments are not pledged. You are strongly encouraged to work together, though each student should write up her/his own submission. The quality and clarity of the exposition will be reflected in the grades.

Attendance: Regular attendance is expected; classes may cover techniques not presented in the textbook.

Accommodations: Any student with a documented disability seeking academic adjustments or accommodations should speak with me during the first two weeks of class. Students with disabilities will need to also contact Disability Support Services in Allen Center.