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.

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.

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.

An update!

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.

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Review: LiveScribe Echo Digital Pen

I’m still here! Though posting will be a bit sporadic in the future, as I’m spending the majority of my time working on my thesis.  I’ll be back to blogging a bit more often once that’s been finished.  So, here’s a review I started writing back in November, but has sat in the draft box since then:

About six months ago I asked people for opinions about digital pens, and did my own research, so I decided to pick up a LiveScribe Echo based on what I’d heard, and the campus computer store has a 10 day return policy (so long as packaging is intact) so I decided to give it a try.  For those who are impatient, the short version of my review is: the digital pen is good, and I’m keeping mine, but there are a few improvements that would go a long way to mainstreaming, so I give it a 4/5.

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Prym Varieties

Let ${\pi:\tilde{C}\rightarrow C}$ be an unramified double cover, where ${C}$ is geneus ${g}$. Then ${\tilde{C}}$ has genus ${2g-1}$ by the Riemann-Hurwitz formula. Now, ${J(C)}$ encodes lots of information about the geometry of ${C}$, especially with the additional data of the theta divisor. It turns out that for double covers, there’s an abelian variety that contains a lot of this data.

Rational Normal Scrolls

Today, we’re going to talk about a very important class of rational varieties, that show up all the time, in quite a variety of different contexts, and at the end, we’ll talk about why.