Presentation topics, Representation Theory, Math 9140a

All students are expected to attend all presentations and to arrive on time.

Possible Topics: 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, which organizes information about direct sum and tensor product of representations. The wikipedia page gives some references, and sample computations. Working out some of these computations could be part of the presentation.
• 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 this mathoverflow question, e.g. Varadarajan's book (Thm 9.4, 2nd ed.). This article by Straumann might be helpful. There's also a long, inconclusive discussion at the n-category cafe.
• Representations of S_n (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. Many other sources cover this material as well.
• 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 S_n, 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, especially 11.4).
• 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 available here, 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; or many other sources).
• Representations of Lie algebras and su(2). (Sternberg 4.10 and 4.11 is one possible source, but there are many others.)
• To what extent does the representation theory of a finite group G determine the group? The answer depends on what you remember about the representations. Tannaka duality provides one answer, and there are other interesting things to discuss. Ask me for references.
• Group cohomology (e.g. Carlson's book, Weibel's book on homological algebra, this expository 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).
• Representations of quantum groups/Hopf algebras, or just SL_q(2). Connections between representation theory of quantum groups and knot theory.

Duration: 45-55 minutes. 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, motivation, necessary background, and history (e.g. attributions and years). You aren't expected to prove everything, but should give one or two short proofs. It should be regarded more like a seminar talk than a course lecture.

Grading: The presentations will be worth 1/3 of the overall mark in the course. They will be graded on:

• Outline and draft: Well-organized; appropriate choice of topics and amount of material; done on time.
• Knowledge of material. Be prepared to answer questions.
• 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.

Timeline:

• May 30-June 3: look over topics and read about a couple of them.
• Meet with me June 6-June 10 to discuss topics and select a date. Bring two possible choices of topic when we meet. Talks will take place roughly July 4 to July 15.
• Finalize choice of your topic ≥ 3 weeks ahead of your date.
• Give me a brief outline (1 to 2 pages) ≥ 2 weeks ahead of your date.
• 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.