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 | 1 Comment

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 | Leave a comment

Japanese for mathematics: Algebraic Variety

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 | 4 Comments

Grant Day!

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

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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Japanese for mathematics: Manifold

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

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