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algebraic topology is off the chain!

Off the chain complex, that is! :p My TA sabbatical is this quarter, so I'm taking classes instead of teaching. And furthermore, since my income isn't tied to research, I decided I'm not going to feel guilty if I spend nearly all my time learning neat shit and don't get much research done. This could change if the theory professors do a good enough job at talking some sense into me, but my intention is to stand my ground as much as possible. Especially since I doubt I'll have time to learn this stuff once I'm a postdoc and paid full time to do research.

So this quarter I'm taking General Relativity, Algebraic Topology, and String Theory. I intend to do all of the homework for GR and String Theory, and at least a good bit of the Algebraic Toplogy work depending on how much I care about the proofs they give us (so far they look incredibly fun, but I'm thinking by the end it might get esoteric enough that it's not worth my time). I also went to the first Differential Geometry class on Wednesday, but it was fairly boring/simple stuff and it looks like it will be pretty much a repeat of the (rather advanced) undergrad Differential Geometry class I took at Georgia Tech--it's even out of the same book. Nevertheless, I'll probably go to some of those lectures toward the end of the quarter. Actually, maybe I will go to more of them--there were a bunch of things that went over my head when I took it as an undergrad, even though I somehow miraculously ended up with an A. I just hope the pace picks up soon.

By far, I am the most excited about Algebraic Toplogy. Yeah, I should be psyched about string theory because it's actual physics as opposed to just math (heh... no matter what Peter Woit tells you) but it looks like the way this course is being taught is going to be very useful and relevant to physics. And I just really like his style of teaching. And the topic is even cooler than I'd imagined. Not to mention, I get to use mighty words like p-skeleton, surgery, sheaf cohomology, cobordism, homotopy, category fibered in groupoids, etc. (and stop feeling dumb when math people talk over my head).

It looks like the layout of the class will be particularly relevant to physics, and the professor tends to be very good about explaining the meaning of things before he gives you the boring rigorous mathy definition for stuff. Not to mention, it's his area of specialty so he's very excited about it himself.

Course Description (from http://count.ucsc.edu/~tamanoi/math211.html):

In addition to basic materials of algebraic topology (category theory, homology groups, cohomology groups, Poincare duality for manifolds, and homotopy groups), in this quarter we plan to include materials on the following topics as long as time permits. My aim is to expose students to various powerful mathematical ideas and methods without much technicalities which non-topology students may not have a chance to get exposed to during their career.

(i) (co)homology of fibre bundles, Lie groups, loop spaces, and Hopf algebras,

(ii) various cohomology theories such as de Rham cohomology and sheaf cohomology, and the relations among them,

(iii) spectral sequence (Serre, Eilenberg-Moore, etc)

(iv) rational homolopy theory

Good mathematical ideas have universal nature, and they can be applied to other areas of mathematics, often with success.

Fiber bundles, lie groups, loop space, and Hopf algebras are all particularly important in physics, so it's great for me that he's going to spend time on them. Hopf Algebras are tied into non-commutative geometry and "quantum groups" (something I didn't even know about, way back when limbsoup asked me about them and I was like "uh, I have no idea"). And non-commutative geometry is very relevant to Matrix Theory, which I've been really wanting to get back to at some point. Oh, and the other really nice thing is we're using Allen Hatcher's book which is downloadable online for free (albeit 550 pages).

All in all, I think I'm going to have boatloads of fun this quarter. Also, I recently registered to attend the Singularity Summit at Stanford on May 13th, which is going to rock. Nick Bostrom, Douglas Hofstadter, Ray Kurzweil, Eric Drexler, Eliezer Yudkowski, Max More, and many more speakers (check Wikipedia for more info on any of these people)... pretty much all the people who've coined the most terms on my livejournal interests list! Best part is, it's completely free (last thing they had like this was around $150 to get in).

Comments

( 4 comments — Leave a comment )
onhava
Apr. 8th, 2006 09:39 pm (UTC)
That's a huge amount of topology to cover in a quarter! Sounds like a lot of fun, though. I like problem 4 on the homework, about the fundamental group of a topological group; I remember puzzling over it for a while in the first algebraic topology class I took before having one of those "aha!" moments where I suddenly had a very nice picture of why it works.

By the way, Peter May's book is also free online now: http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf
spoonless
Apr. 8th, 2006 11:12 pm (UTC)

That's a huge amount of topology to cover in a quarter!

Yeah, when I first looked at the class, and the prereqs your supposed to have (graduate Manifolds I, II, and III) I was pretty intimidated, but after going the first day and understanding every word easily, and realizing he probably isn't aware of the prereqs (and there's enough other people skipping them too) I think it will be no problem. Unless it suddenly starts to accelerate.

I get the feeling what he's going to do is cover a lot of topics but not in much depth--which is perfect for a physicist, since that's all we care about.

By the way, Peter May's book is also free online now

Thanks, concise is good--I really doubt I'll have anywhere near enough time to read all 550 pages of the longer book, in the midst of working through Polchinski (and doing the GR homeworks, although I don't expect that to take much time). 250 is much more reasonable.
(Deleted comment)
spoonless
Apr. 10th, 2006 02:09 am (UTC)
Good to see you're still around! I figured by this time you'd long since stopped reading livejournal. :)

And thanks for the notes/advice. Don't know if they'll come in handy, but the more references I have the better.
spoonless
Apr. 12th, 2006 07:58 am (UTC)

I like problem 4 on the homework, about the fundamental group of a topological group; I remember puzzling over it for a while in the first algebraic topology class I took before having one of those "aha!" moments where I suddenly had a very nice picture of why it works.

I just had that "aha" moment! :) Definitely was the coolest problem... the only one I hadn't figured out yet; quite a powerful connection between two seemingly unrelated things! I also like how the abelianness comes right out of there too.
( 4 comments — Leave a comment )

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