**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.

**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.