I’m done! Well, for now. I’m pretty sure I didn’t pass all three, so I’m still not done with these darned tests.

Now I have to decide what I’m going to talk about. I decided I was going to do no math for a week after these tests were done. But I don’t feel that way now. I actually feel sort of motivated to try to look at some things I don’t have time to look at when school is in session (or when I’m studying for prelims).

From my point of view, my options seem to be Morse theory (something I’ve been threatening to do for probably 6 months now). If I did this, I’d probably just try to get to a few of the results of the form: If M is a manifold that admits a smooth real valued function with precisely two critical points, then it is homeomorphic to a sphere. Or something like that, I haven’t looked at it for awhile, so it might not actually be that. But I think it is really awesome that you can somehow get at topological facts, based purely on what real-valued functions it can admit.

I could work through Topology from the Differentiable Viewpoint and learn some things about cobordism (a term I hear thrown around a lot, but only vaguely know that it is in reference to what manifolds are boundaries of other ones or something).

I could pick up where I left off on the algebraic geometry, although I’ll probably do that during school since I’ll be taking algebraic geometry, so it might not be the best choice for right now.

I could do some of Bott and Tu’s book. I’ve only actually read the first part, and am quit curious as to what is in the rest of it.

I could try for the third time to read Zwiebach’s A First Course in String Theory, because I’m darned determined to learn what string theory is. Although, I suspect it will go even worse this time than last time considering its been well over a year since I took quantum mechanics.

Or I could switch gears and do some posts on books I’ve read (which are quite a few since my last book post) and movies I’ve seen. It is still the case that every day my Lost in the Funhouse post has the most hits. Darn you “survey of modern american lit classes” for causing so much confusion.

Or you could suggest something, and I might ignore it or actually do it.

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September 18, 2009 at 11:30 am

I’d be especially interested in learning about Morse theory or anything related to differential or algebraic topology, in fact.

September 18, 2009 at 4:58 pm

I vote that you combine several of them. Something that might be worth doing is proving the Lefschetz Hyperplane Theorem, which is a theorem about complex projective varieties which is proved by Morse Theory. It’s what motivated me to learn what a Morse function is (though I by no means can claim to know Morse theory.)

September 19, 2009 at 11:49 am

That is an awesome idea. I found two papers on it which seem to be very similar in approach. It looks like I’m going to have to build quite a few things before actually getting to it, but that’s fine. Thanks.