Saturday, June 1, 2013

Summer break!

I apologize for completely deserting this blog for the last couple months. My semester was a bit too busy to get any writing done, but I seem to have learned quite a bit.

I took what turned out to be one of the best courses I've taken at Columbia so far: Professor Mu-Tao Wang's Intro. to Differentiable Manifolds course. We worked out of John Lee's Introduction to Smooth Manifolds (GTM218), which I thought to be a pretty good introductory text. Over the course of the semester we covered Chapters 1-6, 8, 10-16 (and bits of 9). The subject turned out to be much more interesting than I had initially assumed, especially since I took it in conjunction with Professor Peter Woit's course on quantum mechanics from the viewpoint of representation theory. Obviously I'm new to these ideas, but the fact that geometry and topology is so closely related to fundamental physics is pretty neat!

Still, I'm glad that the semester's over. I could definitely use a bit of a break. I'm free until mid-June, when I'll be flying over to Berkeley for a summer research program. Not quite sure what I'll be working on yet, but I'm sure I'll learn a lot. In the two weeks of totally-free-time that I have left, though, I think I'll try to start working through a couple of books on representation theory and Lie groups+algebras, commutative algebra, and Riemannian geometry. I've also really been itching to read Naber's two-part series on Topology, Geometry, and Gauge Fields, so hopefully I'll start digging into that soon. It may sound a little ambitious to try to learn five different things at once, but I think it might actually be better than concentrating on one topic for too long. It's all too easy to get stuck/frustrated on one little concept or exercise... perhaps it's better to be stuck on five different ones!

On a more serious note, since it's difficult to learn math without actually doing any, I will be typing up proofs to relevant propositions, worked exercises, and the like. (I'm already one chapter into Atiyah's Intro. to Commutative Algebra; I'll put up my notes at some point in the near future.) I'm currently debating whether I should take notes on this blog or instead just commit them to my GitHub notes repo. Perhaps I'll do both... somehow.

Well regardless, I hope to be blogging about a bit about math and physics over the next couple of months, hopefully at least once a week. Ciao.