Presentation topics, Algebraic topology II, Math 9152, Winter 2011

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. When you meet with me, I can give more information about the topics and can suggest further references.

Simplicial sets and geometric realization.
The classifying space of a topological group and G-bundles. Milnor's paper is good.
The classifying space of a category. Segal's original paper is good.
Bott periodicity.
K-theory. As a cohomology theory. Atiyah-Hirzebruch spectral sequence and the K-theory of projective spaces.
Cobordism. E.g. the h-cobordism theorem, which has a nice description in The wild world of 4-manifolds, by Alexandru Scorpan. See also Milnor's notes.
Brown Representability. Look at the original paper 1962 paper by Brown. Hatcher 4.E seems good too. This topic is also covered in Spanier and many other sources.
Spanier-Whitehead duality. Adams' blue book on Stable Homotopy Theory has a section on this.
Spectra. Adams' blue book on Stable Homotopy Theory. Or Hatcher 4.F.
Steenrod operations. E.g. book by Steenrod and Epstein. Or Hatcher 4.L.
Characteristic classes, e.g. Chern and Steifel-Whitney classes. Milnor and Stasheff's Characteristic classes is an excellent reference, using Steenrod operations to define SW classes. Steenrod's The Topology of Fibre Bundles, sections 29, 32, etc, uses obstruction theory to define characteristic classes. Husemoller's Fibre Bundles defines SW classes by computing the cohomology of Grassmanians.
Morse theory.
Model categories. Paper by Dwyer and Spalinski is good, or book by Hovey.
The fundamental groupoid. E.g. 1971 book Higgins, Categories and groupoids and 2006 book by Brown, Topology and groupoids. See also online notes by Baez.

Duration: 45-55 minutes. 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 35% 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 and not to me; students in the audience should feel free to ask questions.

Timeline:

During reading week, look over topics and read about a couple of them.
Meet with me the following week to discuss topics and select a date.
Finalize choice of 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.

M9152 home page.