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.

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.

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.

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.

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.

Let ${M=\mathbb{P}^{nm-1}}$ be the projectivization of ${\mathrm{Mat}_{n\times m}(\mathbb{C})}$. Then for all ${k}$, we have a variety ${M_k}$ given as the set of matrices of rank at most ${k}$, which is given by the vanishing of the determinents of ${(k+1)\times (k+1)}$ minors. We call these the generic determinantal varieties.

I know I still have a few readers out there, and some of you are probably technophiles, so here’s a question: I’ve been considering switching from live-texing of things to a digital pen of some sort.  Anyone have any experience with these things? Know which ones work well and which don’t? Which ones may work with linux (though I do have a windows partition if really necessary)?

I’ve looked at a couple of them, and I’m really not sure what their specific pros and cons may be, and I have no real way to try them out directly at the moment.  There’s the LiveScribe pens, which require their magic dot paper stuff, which is a drawback that I’m not sure how annoying it would be, plus I don’t know how useful sound recording would be, though I can see myself maybe using it at some point.  The other brand I’ve looked at is SolidTek’s DigiMemo, which is a bit bulkier being a clipboard, and amazon reviews suggest it’s finicky.  Is there another brand I should look at? I’m mostly looking for a way to nicely digitally archive all of my scratchwork (my collection of notebooks is expanding too fast!) and also maybe taking notes at seminars, conferences and the occasional advanced course.

So, anyone? I figure that any math person who has a digital pen or has at least considered them will have a lot of the same uses in mind as me, and so I’m very interested in opinions.