Presentation topics, Representation Theory, Math 9140a
Each student will give one presentation near the end of the course.
All presentations will be done using the blackboard.
All students are expected to attend all presentations and to arrive on time.
These are suggestions, but you can also propose other topics.
Topics need to be discussed with me and approved.
You should choose a topic that is not something you already know about.
When you meet with me, I can give more information about
the topics and can suggest further references. You should also
do some research about the topics.
Some of these may be covered in class, as the final choice of
course material is still being worked out.
Sternberg's book is available here.
- The representation ring of a group; not sure if there is enough
to say here to fill a talk.
- Representations of semidirect products of groups.
3.8 in Sternberg is one possible source, but I believe there are slicker ways to do it.
See Section 8.2 of Serre's book on representation theory and/or
Section 3.7 of Andy
Baker's notes for brief treatments.
- The representations of the Poincare group, using the previous item.
This gives Wigner's classification of particles by mass and spin.
Sternberg does this in 3.9, but some parts aren't clear.
Other references are given in
book (Thm 9.4, 2nd ed.).
This article by Straumann
might be helpful.
There's also a long, inconclusive discussion at the
- Representations of Sn (Steinberg, Chapter 10). This is too
much for one presentation, but it could be split into two, with
one focusing on the combinatorics of Young diagrams/tableaux/tabloids
and the other on the representation theory. Even then, there would
only be time for highlights.
Chapter 7 of the book Diaconis, 1988
is probably an even better source than Steinberg, as it is more efficient.
- An application of Fourier analysis to graph theory, as in
Section 5.4 of Steinberg. Remark 5.4.13 contains pointers to
further work, which might be good to incorporate.
(If you are interested in this, tell me today!)
- Voting and Fourier analysis on the symmetric group Sn,
e.g. via the references mentioned in Example 5.5.8 of Steinberg
(e.g. Diaconis, 1989).
- Probability and random walks on groups (Steinberg, Chapter 11,
- The fast Fourier transform; most treatments don't have much
representation theory, so this is a bit tangential.
(One exception is a survey paper by D. Maslen and D. Rockmore
but it's fairly intricate.)
- Representations of SU(2) (e.g. Sternberg 4.3).
- Representations of compact topological groups or Lie groups,
possibly covering Haar measure and the Peter-Weyl theorem
(e.g. Sternberg 4.1, 4.2 and Appendix E; or J.F. Adams, Lectures
on Lie Groups).
- Representations of Lie algebras and su(2).
(Sternberg 4.10 and 4.11 is one possible source, but there are many others.)
- Representations of quantum groups/Hopf algebras, or just SLq(2).
- Tannaka duality: to what extent does the category of
representations determine the group? Ask me for references.
- Connections between representation theory and knot theory.
- Group cohomology (e.g. Carlson's book,
Weibel's book on homological algebra,
article by Isaksen, etc.)
- Modular representation theory, i.e. representations over finite
fields; maybe also the stable module category of a group
(e.g. Carlson's book).
- Applications to particle physics (e.g. Sternberg 3.10, 3.12, 5.9 to 5.12, ...).
- Applications to crystalography (e.g. Sternberg 1.5, 1.8, 1.9, 1.10).
- Other applications (e.g Sternberg 4.5 to 4.9).
The presentations are not long, so you will need to carefully
select the appropriate amount of material to present.
You should focus on the key ideas, with illustrative examples.
You aren't expected to prove everything, but should give one or two
It should be regarded more like a seminar talk than a course lecture.
The presentations will be worth 1/3 of the overall mark in the course.
They will be graded on:
Note that knowledge of material is just a small part of the grade.
The presentation itself is much more important. Because of this, you
should practice the talk at least once or twice beforehand, on
a blackboard, with someone listening, and you should time how long
it takes. This is extremely important. You should also address
your presentation to your fellow students, not to me; students in the audience
are strongly encouraged to ask questions during and after the talk.
- Knowledge of material. Be prepared to answer questions.
- Organization of material: what you choose to cover, and how
you choose to organize it. Don't forget to include motivating
examples, history (e.g. attributions and years), and context.
- Clarity and style of presentation: speaking clearly, looking
at audience, giving clear explanations, etc.
- Blackboard use: use boards in order, don't erase what you've
just written, don't stand in front of what you've written,
use coloured chalk when appropriate, use the side board for
things you want to leave up, etc.
- Duration: if you end within the time span given, you get full
marks for this category; otherwise, you lose marks. You might
want to build some flexibility into the end of your presentation
so you can adjust on the fly. And take into account that there
may be questions during your talk.
- Right away: look over topics and read about a couple of them.
- Meet with me May 26-27 to discuss topics and select a date.
Bring two possible choices of topic when we meet.
Talks will take place roughly June 30 to July 4.
- Finalize choice of your topic ≥ 3 weeks ahead of your date.
- Give me an outline (1 to 2 pages) ≥ 2 weeks ahead of
- Give me a draft of the whole talk ≥ 1 week ahead of your date.
Indicate which parts you will say and which parts you will write on the board.
Course home page.