Topology ---Information on semester project
In the final weeks of the semester you will be given time to work on a project.
You may choose one of the following options:
1) The default option is algebraic topology (Introduction to homotopy,
simply connected spaces, the first fundamental group, covering spaces,
and
using these to compute some fundamental groups, etc). Munkres Sections
51-.... I will be lecturing on this project
in class and attendance is
optional (typed notes provided). There is a Homework assignment (Homework 9), and
there may be questions on this in the final exam. Also, students doing this project will
continue to do in-class presentations as before.
2) Filters, ultrafilters, and applications to the Stone-Cech compactification and real normal
spaces. Properties of the Stone-Cech compactification of the natural numbers. Ultrafilters can also be used to give a quick proof of Tychonoff's theorem (due to Chernoff).
3) State and prove a metrization theorem (and all necessary lemmas), eg Munkres section 39-40,
or paracompactness and Munkres section 41-42.
4) State and prove the Arzela, Ascoli, Arzela-Ascoli theorems (and all necessary lemmas), see
eg Munkres section 43, 45-47.
5) Dimension theory -- Munkres Chapter 8.
6) Another topic from Munkres of your choice (e.g. sections 55, 57, 58, 59, 60, topics from
Munkres Chapter 10, or Chapter 11, or Chapter 12,
or Chapter 13).
7) The theory of totally disconnected/zero dimensional spaces
and relations to the Cantor set.
8) Akemanns noncommutative topology in C*-algebra theory
9) Locally compact abelian topological groups and the Fourier transform
(Eg.\ Folland ``A course in abstract harmonic analysis", beginning in
Section 2.1 and some other sections of Chapter 2, going back to Chapter 1
where needed. Unitary representations are treated in Chapter 3, then topics like the Fourier transform and duality in Chapter 4).
10) Basic theory of Polish spaces (i.e. topological space homeomorphic to a complete
separable metric space).
One could prove all the assertions in the Remark before 2.4.3,
e.g. that a locally compact space $K$
is second countable iff
$K$ is a Polish space.
This is also equivalent to $C_0(K)$ being separable.
Note that any uncountable Polish space is Borel isomorphic to
(0,1) (Kuratowski's theorem, see e.g.\ Srivastava ``A course on Borel sets", Jesse Peterson's Notes on operator algebras Section 3.9, or Cohn's Measure Theory Chapter 8).
11) Boolean algebras, Stone spaces, and Stone's theorem (related to project 7).
12)
Another topic of your choice, perhaps related
to your PhD topic (but talk to me about it first).
Please ask if you have any questions...
Please tell me by November 10 which project you are doing.
If you do not do the default algebraic topology project: Remember
that your project needs to be substantial, the equivalent of two or
three weeks of lectures (what we did for algebraic topology).
You may do a longer project as a group, with some doing different parts of the project. The
project is worth 40 points. It should be
turned in at the final exam (with an extension to Decembeer 16 under extenuating circumstances). You may be asked to do a short presentation of your project
during the final exam three hour timeslot.
If you are doing the default project there may be questions on this on your final exam.