Differential Geometry


Posting is slowing down a bit, I’ve got a paper I’m trying to get out, and a couple of projects that are hitting some preliminary results, plus, I’m getting ready for holiday travel, and then two months at Humboldt.  Trying out an experiment with more rigid personal scheduling, and hopefully I’ll post more often.  Also, I’m reviewing Atiyah-Macdonald, Eisenbud, and Schenck so that perhaps in March I can begin a “Commutative Algebra from the Beginning” series, or perhaps just a series on geometric interpretation of commutative algebra theorems.

However, for today, we’re going to take something most of us first saw in differential geometry (I first met this map in do Carmo‘s book) and translate it into algebraic geometry.

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So, today I discovered something rather nice, that I think could easily get more time in books and the like, but doesn’t. We’ll first have need of a theorem that I don’t want to prove:

Theorem: Let X be a projective variety over \mathbb{C} and let \mathscr{L} be an invertible sheaf on X. Then \mathscr{L} is ample if and only if for all \mathscr{F} coherent sheaves, there is a number n>>0, such that we have H^i(X,\mathscr{F}\otimes\mathscr{L}^{\otimes n})=0 for all i>0.

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As part of my project to not let orals actually degrade my sanity beyond recovery, today we’re going to do some physics! Well, kind of. We’ll be citing a lot of physics, at the least, and using it to motivate things. After all, a lot of cool stuff is coming into algebraic geometry via physics these days, and the best way to understand how these things fit is to understand at least the basics of the physics.

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I’ve been a bit busy lately, and so I missed last week posting. So what I’ve decided to do is to take the talks I’ve given in various graduate student seminars over the last year or so and convert them into posts. This one is a particularly tough prospect, as the talk didn’t go very well. I’m following a paper of Hamilton‘s titled “Ricci Flow on Surfaces” and only present the high genus case. Comments and (especially!) corrections are encouraged.

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Today I’m going to talk about parallel parking. We’ve all had to do it at some point (at least, those who drive) and certainly we’ve all noticed how much of a pain it is to get into a small space. Well, as it happens, if your car has length L, then for any \epsilon>0, it is possible to parallel park, assuming some things like that the driver can make arbitrarily small movements.

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