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

Posted on December 3, 2013 by Jim Stankewicz

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

Posted on February 26, 2013 by Charles Siegel

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.

Posted in Uncategorized | 1 Comment

The Gauss Map

Posted on December 9, 2012 by Charles Siegel

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

Posted on November 19, 2012 by Charles Siegel

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

Japanese for mathematics: Algebraic Variety

Posted on October 22, 2012 by Charles Siegel

Last time on this series, I talked about the word manifold.  Today, we’re going to add a modifier.

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Posted in Japanese for Mathematics | 5 Comments

Grant Day!

Posted on October 15, 2012 by Charles Siegel

No substantive post today, because my grant application is due.  New post next week!

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Understanding Integration II: 1-Forms and Periods

Posted on October 15, 2012 by Charles Siegel

Last time, we discussed integration theory of functions along paths on Riemann surfaces, and then we decided that we wanted to talk about compact Riemann surfaces.  Unfortunately, there aren’t any holomorphic functions on them, and meromorphic functions are the wrong choice about what to integrate along curves.  Today, we’ll talk about the correct things to integrate, and some of their properties.

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Posted in AG From the Beginning, Algebraic Geometry, Complex Analysis, Curves | Leave a comment

Japanese for mathematics: Manifold

Posted on October 8, 2012 by Charles Siegel

So, I’m trying to learn Japanese, being as I live in Japan, so I’ve decided to start this series.  I’m armed with a mathematical English-Japanese dictionary, a kanji look-up website, and a willingness to be corrected if I happen to have any Japanese readers.  So, this post may not appear correctly if you don’t have Japanese fonts installed, just a warning, and if I explain anything incorrectly, let me know in the comments and I’ll correct the post.

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Posted in Japanese for Mathematics | 4 Comments

Understanding Integration I: Riemann Surfaces

Posted on October 1, 2012 by Charles Siegel

I’m back! And now, posting from Kavli IPMU in Japan.  Now, I’m going to start a series on theta functions, Jacobians, Pryms, and abelian varieties more generally, hopefully with some applications, with my goal being at least one post a week, and eventually establishing a regular posting schedule again.  But today, we’ll start with basics, something that should be completely understandable to graduate students and advanced undergraduates.

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Posted in Abelian Varieties, AG From the Beginning, Algebraic Geometry, Complex Analysis, Curves | 5 Comments

An update!

Posted on April 3, 2012 by Charles Siegel

Hi everyone, it’s been a LONG time since I last posted to this blog, and I intend to do so a bit more now that a few things have been handled.  The first of the handled things: I just defended my thesis today! And it was successful! So, once I finish writing it up for publication, I’m going to put some version of exposition on it here.  The other thing is that I’m employed in the fall.  I’m going to be spending the next three years as a Postdoc at Kavli IPMU at the University of Tokyo.  So, I should be back to posting by the end of the month, and this blog may well start to include some things other than math, like involving moving to Japan and adapting, learning the language, and the like.

Posted in Uncategorized | 4 Comments