August 2008


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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Ok, this might be a silly question, but I’m not sure if something I want to do is legal or illegal.

So, I’ve got some books in pdf format that are obtained legally (ie, EGA as downloaded from Numdam) which are rather out of print. I’d like to have nice copies of them to keep on my shelf and use as references…I don’t have an ebook reader, so I still prefer dead tree format. Now, there are companies like Lulu which will essentially take a PDF and convert it into an actual book, and are able to do print runs of 1, or 5, or however small. (At least, that’s my understanding)

So my question is: is it legal to have a company like Lulu print a copy of EGA (or similar out of print book) for personal use, with it not being set up to sell it to anyone else or to make a profit for me in any way?
If anyone out there actually knows what the laws on matters like this are, please help out and let me know if I’m allowed to do this.

We continue our quest to understand when quotients of schemes by actions of group schemes exist. Last time we defined group schemes, group actions, geometric quotients, and gave some examples. In this post, I’ll define what it means to be a reductive group scheme.  To give a full treatment of the theory of reductive group schemes here is impossible, so we’ll pick one way to define the notion of a reductive group scheme, state some equivalences, and give some examples.  I apologize for the long time between the posts, I’m on vacation and have been have been spending more time outside than at a computer!

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So, I don’t really have a full length post in me at the moment. However, here’s a nice trick that I’ve learned recently, plus some motivation. With this, it’s easy to write down explicitly a curve of arbitrary degree in the plane, which has specified nodes. (So long as it’s possible to have that many nodes.)

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So, we’ve got a new theme. This is for one simple reason: the old one didn’t list the author of posts. This wasn’t an issue before, but today, just before this, is the first post from one of the new cobloggers, Matt DeLand, from Columbia. The two themes seem close enough…comment here if there’s any trouble reading it or the like.

Hi! My name is Matt DeLand, I’m a graduate student at Columbia and I’m responding to Charles’ call for cobloggers. I also study Algebraic Geometry, and have been enjoying Charles’ posts; hopefully I can help out and make some positive contributions. I apologize in advance for the quality of my first post….

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We talked before about elimination theory, doing it entirely in the affine case. The question was asked about how to do it projectively. There are a couple of subtleties to it, but the idea is simple: we eliminate in each affine chart and then glue together. The problems that arise most naturally here actually involve working on \mathbb{P}^n\times\mathbb{A}^m, and projecting down to the affine space.

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Ok, well, I’m going to be posting less, and hopefully settling into a regular schedule at some point, but the rest of the month is going to be erratic at best, with my orals being (in theory) the first week of September.  I had been intending to write up some stuff on Jacobians of Curves and Theta Functions, but davidspeyer at the Secret Blogging Seminar is doing a far better job than I could, with these two posts.

So, a few things for the moment:

  1. I’m looking for some new cobloggers.  I’ve talked to a couple of people here at Penn, about it, but if anyone out there on the wider internet is interested, get in touch with me (my email is siegelch [at] math [dot] upenn [dot] edu, or leave a comment).  I had originally wanted this blog to be a collaborative thing, and would prefer to get it back there.
  2. I’m TAing an abstract algebra course this Fall (woo! delaying teaching Calc a bit) and was curious if anyone had any ideas about how I could use this blog to help with that.  I figure that I’ve got it, so I might as well try to use it.
  3. I have a static web page! It’s here and will eventually contain all sorts of notes that I’ve taken over the years in my courses.  Other than that, it’s pretty much a link here, so far.

I believe that’s all for the moment.  Will post something more mathy in the next couple of days.

Ok, this post is going a little bit out of my field. If anyone can fill in some of the gaps in my understanding of phylogenetics and in how to get from there to the math, please do so in the comments. So, anyway, phylogenetics is the study of how various organisms are related, and is very closely related to taxonomy, the classification of organisms. It doesn’t look, at first, like it’s the sort of thing that would grab techniques from algebraic geometry, the study of solutions to polynomial equations. My thought is that it just looks too hard on the surface to hope to have anything nice popping out mathematically that’s that elementary. Well…that’s wrong. Without realizing it, algebraic geometers have been looking at problems relevant to phylogenetics for quite some time.

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Now, the way that the Riemann-Roch theorem was phrased before, the geometry wasn’t obvious. We had to extract it in terms of rational functions with poles given by a divisor. Now that we’ve talked about canonical curves, we can use them to give a more obviously geometric form of the Riemann-Roch Theorem.

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