Updated Outline

I was thinking about how to put this blog to use, and especially how to make it worthwhile for myself. In 2010, I ran a learning seminar on math/physics which in my opinion was extremely successful. I made detailed notes for nearly all of the talks, and I had several people ask me at the end of the seminar to give them copies of the notes I had made. We managed to do something very special in the seminar, and I think it's worth writing up notes from the best talks (and reorganizing/elaborating/etc. when necessary) and putting them online for all to see. It gives me a chance to practice my math writing, and if I ever run another seminar of this flavor (or teach a course, or write a book...) then a lot of the work will already be done for me.

Here is a rough outline of the topics we covered (or that need to be covered as background):

1. Newton's Laws (review of high school physics).
2. The Euler-Lagrange equations derived from Newton's 2nd law.
3. Hamilton's action principle.
4. Legendre transform, Hamilton's equations, basic symplectic geometry.
5. Symmetry, Noether's theorem, and moment maps.
6. Completely integrable systems.
7. Special relativity.
8. Classical field theory.
9. Classical electrodynamics.
10. General relativity.
11. Canonical quantization
12. Deformation quantization
13. Geometric quantization.
14. Path integrals in quantum mechanics.
15. Quantization of free fields.
16. Perturbation theory and Feynman diagrams.
17. Feynman rules for QED.
18. Berezin integration and quantization of fermionic fields.
19. Gauge fields.
20. Faddeev-Popov method for gauge fields.
21. BRST.
22. Path integral proof of the index theorem.

As you can see, it was an ambitious seminar! I will update the outline as I go through my old notes and start posting them.

Is it worth the trouble?

I've realized talking to fellow students and occasionally postdocs that I frequently have to explain the same few topics over and over. Since I have my own personal notes on these things, I might as well write them up for people to read, since then I can just point people towards the notes. And if I'm going through the trouble of writing things up, maybe I should post them on an indexed, google-searchable blog? Why not? Maybe I will.

First day of 2010

It's the first (academic) day of 2010. I figured that the best use of this blog is as a log--that is, to log and plan my work for the semester/year/life. If you want to accomplish goals, the fist thing to do is to write them down! So here we go, crude outline for the next semester:

1. Work through Milnor's Morse theory book, cover to cover. This should be easy since I'm taking a class in morse theory anyway.

2. Work through Kirwan's thesis cover to cover. I've already been through quite a bit of it, and the only things that caused me any trouble last summer have since been cleared up.

3. Work though Gulliemin and Sternberg's Equivariant cohomology book. Again, quite a bit of the material I already know, so this should be doable.

4. Finish working through HKLR. Really the only remaining part is supersymmetric nonlinear sigma models.

5. The details of the ADHM construction, once and for all. I should know this already.

6. Hilbert schemes of points on a surface. Really, the hyperkahler metric for the scheme of points on \(\mathbb{C}^2\). Again, I should know this already.

We'll see how these go--this is probably ambitious, and many of these will get extended into the summer.