Topics. The Fall 2008 Math 623 course is a continuation of the Spring 2008 Math 622. Several topics on the Math 623 syllabus (including curvature of Riemannian manifolds, vector bundles, connections, the Maurer-Cartan form) will be covered in Math 622. We will continue with geodesics and the Hodge decomposition theorem (cohomology). The bulk of the term will then be devoted to vector bundles, curvature and characteristic classes.
References.
Good sources for geodesics are
[Sp1] Spivak, volume 1, and
[dC] do Carmo's Riemannian Geometry.
The lectures on the Hodge Decomposition Theorem will follow
[W] Warner's Foundations of Differentiable Manifolds and Lie Groups.
We will spend the greater part of the semester working out of
[MT] From Calculus to
Cohomology: de Rham Cohomology and Characteristic Classes by Madsen and Tornehave.
We will begin around Chapter 11 and work though to the Gauss-Bonnet Theorem.
I recommend that you obtain a copy of [MT]. The texts [dC], [Sp1] and [W] will be on reserve at the library.
Homework. Due in class on Thursdays.
| Assignment | Due |
| [dC], Ch. 3: 1, 2 (a,b,e), 8, 11, 14. | Sep 4. |
| [Sp1], Ch. 9: 40, 41 (a-e), 42. | Sep 11. |
Midterms. Will be 24 hour take-homes. (See syllabus for tentative schedule.)
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