## Math 215b Final project

The only official requirement for this course is to write an
expository paper, of about 10 pages, on some topic of interest in
algebraic topology or a closely related field; to "referee" (i.e. make
anonymous critical comments on) the paper of one of your classmates;
and then to revise your paper in light of the referee report you will
receive. I suggest the following schedule:

- by April 1: propose a topic to write about, either in
person or by email. I can help you find references for your topic.
- by April 30: finish a draft of your paper, which I will then
pass on to one of your classmates for refereeing.
- by May 10: write a referee report on one of your classmates'
papers.
- by May 20: revise your paper.

The topic should be fairly specific; it is better to understand a
small area in detail and be able to prove at least something. However
it is also good to indicate how the topic is connected to other
things. You can do pretty much anything you want. For example, here
are some topics which people already told me that they want to do
(although there is no reason why two people can't write about similar
things, and also your topic can be a little less advanced than some of
these):
- Why is it not always possible to represent an element of the
homology of a manifold by an embedded submanifold?
- The Grothendieck spectral sequence and how to use it.
- K-theory: what is it and why do we care?

There are many interesting topics in the sections of Milnor and
Stasheff, Hatcher, or Bott and Tu that we did not cover in class, so
you might want to learn about one of these (probably looking at other
sources as well in order to go into more depth). For topology up to
1960, a neat book is "A history of algebraic and differential topology
1900-1960" by Dieudonne, which summarizes an amazing amount of
material while still indicating the mathematical ideas. (However the
older papers referenced in this book might be hard to read so you may
want to find modern treatments.) It might also be fun to learn about
more recent developments. Low-dimensional topology (i.e. topology of
manifolds of dimension 4 or less) is currently extremely active, with
many fascinating topics you can start to learn about; I can help you
pick out something. (This is what i actually do research in.)

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