This course will be an introduction to homotopy theory and characteristic classes, focusing on those aspects of the subject most needed to for work in geometry and topology (of all sorts), geometric analysis, and algebraic geometry. All of these subjects depend on this material to a greater or lesser extent, and it is not taught in our regular first-year courses.
The course will pretty much take off where MATH 734 finishes off (i.e., cohomology groups, cup products, and duality on manifolds). Thus MATH 734 is the basic prerequisite, though students who have done well in MATH 730 should be able to follow everything if they are willing to fill in a bit on cohomology on their own.
There will be no assigned textbook to purchase. Good free introductions to the subject are available in Peter May's A Concise Course in Algebraic Topology and Allen Hatcher's Algebraic Topology, Ch. 4 and his supplements on Spectral Sequences and Vector Bundles.
For the syllabus the last time the course was given, click here, though I will probably change the approach to some of the topics in light of more recent trends in homotopy theory.