## Differential Topology, M382D, Spring 2016

• Classes: MWF 10-11,  RLM 9.166
• Office: RLM 9.114
• Office Hours:  Tu2-3, Th10-11 and I keep an open door
• Phone: 471-7121
• TA: Laura Fredrickson,
• Office:  RLMXX.XXX
• Email: lfredrickson@math.utexas.edu

• Textbook: Differential Topology, by Guillemin and Pollack. Another useful reference is Spivak's Calculus on Manifolds. Of course, not everything in lecture will be from a text, and not everything in Guillemin and Pollack will appear in the lectures.

Homework: There will be weekly problem sets, due on Fridays. You are encouraged to discuss your homework with other students. (Not that anything I could say would stop you, anyway.) However, in the end, the paper you turn in should reflect your own individual understanding.

Exams: There will be two prelim-style in-class exams, one shortly before Spring Break and one on the last day of class.

Term paper: One of the course requirements is to write a term paper, either alone or in collaboration with up to three others. Pick a theorem of differential topology and discuss it. You can explain its history, the ideas behind it, applications of it, different approaches to it, an elementary way of viewing it, or whatever you like. Just say something interesting. The paper should be between 3+N and 4+2N TeX pages, where N is the number of people in the collaboration. The term paper is due one week before the end of classes. You should pick a topic and discuss it with me by spring break at the latest.

Grading: Your grade will be based 30% on homework, 20% on the term paper, 20% on the first exam and 30% on the second exam. Qualitatively, an A means you're probably ready to pass the prelim exam this summer, a B means that with some hard work you have a good shot to pass it, and a C means that passing the exam just isn't in the cards. You can do the obvious adjustments for plus and minus grades. My hope (and this is not a promise) is not to give any grades lower than a B-.

Syllabus: We'll be covering all of Guillemin and Pollack and a little bit more. At the beginning of the course we'll do the inverse function theorem and the implicit function theorem a bit more deeply than Guillemin and Pollack do, (reference: Spivak) and at the end of the term, time permitting, we'll do some de Rham cohomology theory. We will also consistently work with abstract manifolds, rather than just manifolds embedded in R^n.