New Series: Moduli of Riemann Surfaces

Though I’m not quite ready to start (next week!) I feel that, in the spirit of Jim getting back to the blogging and my continued promises, I’d announce my series now.  I’m going to start a detailed series on the moduli of Riemann surfaces, including both topological and geometric aspects.  And I figured I’d start it out with a list of references for some of the topics that I’d be covering:

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The torsion on CM elliptic curves over prime degree number fields

It’s good to be back! This weekend I’m going to Paris to give a talk in the London-Paris Number Theory seminar so I’m going to give a preview of that talk, based on joint work with Pete Clark and Abbey Bourdon. We will post this onto the arxiv soon.

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I know I’ve said it before, but…

Rigorous Trivialities will be returning! Not immediately, but it will be.  I’m reorganizing a bit, and as I’m going to be contributing to a blog for the general public via the Kavli Foundation, I’m also going to try to revive this blog.  Oh, and I’m on twitter now as @SiegelMath, and we’ll see if I’m capable of microblogging.  Lots of experiments in communicating math for me over the next year, we’ll see how it goes, hope that some people are still out there and watching, but even if not, I’ll do my best to draw people back.

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The sound you hear is another conjecture in birational geometry dropping like a fly

These are interesting times to look over the algebraic geometry arxiv postings. Just over a week ago, there was a posting by Tanaka which claimed the minimal model program was false in characteristic two. Then yesterday at the top of the page was a paper by Castravet and Tevelev claiming that the Mori Dream Space conjecture for \overline{M_{0,n}} was false. Then today, there is a paper by Fontanari claiming instead that the Mori Dream Space conjecture is TRUE for the same space, but modded out by the finite group S_n.

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Posted in Algebraic Geometry, Uncategorized | 1 Comment

Oops and Yay

First, the oops.  I DID intend to blog from Berlin.  Didn’t happen, got caught up in giving talks and starting collaborations.  It happens.  I MAY be posting again in the next couple of months, but I’m only back home for a couple of weeks before I go off again travelling.  Mid-May is the next long-term stable period I’ll have, but I have half written posts that should be up before then.  Probably.  Maybe.

As for “Yay” (cue youtube), the biggest reason for the “Oops” is that my thesis is finally posted to the arXiv! The next project won’t take so long.

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The Gauss Map

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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Posted in AG From the Beginning, Algebraic Geometry, Complex Analysis, Differential Geometry | 2 Comments

Understanding Integration III: Jacobians

Now, we’re going to talk a bit about the geometry of the periods, which were completely analytic in nature.  As we mentioned, for a compact Riemann surface X, we have a period matrix \Omega that encodes the complex integration theory on the surface.

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Posted in Abelian Varieties, AG From the Beginning, Algebraic Geometry, Complex Analysis, Curves | 2 Comments