Presentation topics, Algebraic topology, Math 4152/9052, Fall 2010

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.

Possible Topics: These are suggestions, but you can also propose other topics. Topics need to be discussed with me and approved. I will encourage the graduate students to choose slightly more challenging topics.

Applications of homology, Hatcher 2.B. Jordan curve theorem, Alexander horned sphere, invariance of domain, division algebras, Borsuk-Ulam, ...
Introduction to cohomology, Hatcher Ch 3. Universal coefficient theorem, ring structure (cup product).
K(G,1) spaces, Hatcher 1.B.
Hurewicz Thm, Hatcher 2.A and maybe p. 366-?. Sketch proof of 2A.1, several examples, statement of 4.32, more examples.
Applications of homology, e.g. to division algebras. Hatcher 2.B.
Vector fields and Euler characteristic. See, for example, these notes.
Long exact sequence of homotopy groups for a fibration, Hatcher p.375-?. Define fibration, state result, sketch proof of one part, give examples.
Fundamental groupoid, generalized van Kampen thm. Higgins, "Categories and Groupoids". QA171.H57.
Classification of surfaces, Massey, GTM 127, Ch 1. (A lot of material.)
Brown representability for generalized cohomology. Hatcher 4.E, but there are probably better sources too. Connections to K(G,1)'s.
K-theory as a generalized cohomology theory? (Too much?)

Duration: 45-55 minutes for grad students, 40-50 for undergrads. The presentations are not long, so you will need to carefully select the appropriate amount of material to present.

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

knowledge of material
organization of material: what you choose to cover, and how you choose to organize it.
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
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.

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; students in the audience should feel free to ask questions.

Timeline:

In mid-October, look over topics and read about a couple of them.
Meet with me before the end of October to discuss topics and select a date.
Choose your topic ≥ 3 weeks ahead of your date.
Give me an 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. Decide which parts you will say and which parts you will write on the board.

M4152/M9052 home page.