This syllabus is for Andrew Snowden’s Undergraduate Seminar in Topology at MIT.
This course is a seminar in topology. The main mathematical goal is to learn about the fundamental group, homology and cohomology. The main non-mathematical goal is to obtain experience giving math talks. Lectures will be delivered by the students, with two students speaking at each class. There are no exams. There will be some homework assignments and a final paper.
e-mail: asnowden at math dot mit dot edu
Office hours by appointment
Time and location
The seminar typically meets Monday, Wednesday and Friday from 12pm to 1pm in room 2-139. See the calendar for exceptions. Practice lectures will also take place in room 2-139.
We will mainly use Hatcher’s “Algebraic topology.” This book is available for free online at Hatcher’s webpage. (It is also available in print.) We may also make some use of Massey’s “A basic course in algebraic topology,” which is published by Springer in the Graduate Texts in Math series (GTM 127).
The final grade is determined as follows:
* 60% — Lectures and participation
* 30% — Final paper
* 10% — Problem sets
Attendance is mandatory. Every three missed classes will result in the drop of a letter grade; thus one can miss up to two classes with no effect on the grade. The classes on 2/2 and 2/4 will not count towards this. Classes missed for a valid medical excuse will also not count towards this. If you know you will miss a class for some reason, e-mail me a day or two in advance and we can try to work something out.
Lectures and participation
Each class two students will give lectures. Each lecture should be about 25 minutes long. Individual lectures will not be graded, but lectures make up a good portion of the final grade. In evaluating your lectures, I will look at their clarity, organization and preparedness. I will also consider how your lectures improve over the course of the semester.
You will give a practice lecture to a small audience (consisting of me, Susan Ruff and the other student lecturing in the same class as you) before your first lecture.
Each lecturer will give one or two exercises relevant to the material being presented. These exercises, and their solutions, should be e-mailed to me as a Latex file. The exercises can be stated during lecture, though this is not necessary. It’s ok if the exercises come from a book (although it’d be preferable if they did not, or at least if they were slightly modified), but be sure to give proper attribution.
As a member of the audience, I’d like you to write a few comments on each lecture you observe. I’m not asking for any kind of lengthy analysis; it would be enough to point out that the lecturer is writing too small. However, make sure the comments are useful — don’t just say “that proof was good,” say why. I will collect these comments at the end of class and e-mail them to the lecturer so that they can have some feedback. (The lecturer will not know who made which comments.)
There will be approximately four problem sets. These will count towards the final grade. Solutions are to be written in Latex. You may work together on the problem sets, but everyone must write up their own solutions.
There will also be exercise sets, mainly composed of exercises given by lecturers. These are optional and do not have to be turned in. If you are intested in learning the material, it is probably a good idea to do at least some of the exercises.
The final paper is an exposition of a topic in algebraic topology that we will not cover in the seminar. It must be at least 10 pages long and written in Latex. Topics will be selected for the papers in March. A first draft is due in April, and a final draft two weeks later. In the final six or so meetings of class, students will give talks on their final papers.