November 2008


Hey all, in December, Rigorous Trivialities is hosting the Giant’s Shoulders, a carnival of posts on the history of science.  So, please sent submissions through blog carnival.  Let’s see if we can get some math in this one.

We’ve previously discussed moduli spaces a bit. The point of this is that we’ll need to understand a certain moduli space, the moduli of stable maps, in order to continue. So we’ll be starting out with moduli of stable curves, and then continuing on.

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I also have an excuse for my absence… I’ve also been working on grant applications, job applications, my thesis, etc!  I’ll take a little time now to write more about rational varieties.  In general, it can be quite difficult to decide if a given variety is rational or not.  Charles mentioned in a comment that he would like to see the proof that cubic threefolds are not rational.  This is perhaps a good topic for the future, but maybe it requires more work to explain than I have time for right now.  In any case, in this post I’ll talk about one way to verify that a variety is not rational.  We won’t be distinguishing any unirational varieties from rational ones today though.

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Ok, this has been quite a long time in the making (did the research years ago) but thanks to my NSF application, I’ve finally managed to push this through to the point where it’s on the arxiv.  It’s numerical analysis (I wasn’t always an algebraist!) but still, I’m excited.  Here it is:

Improved Error Bounds for Dirchlet-to-Neumann Absorbing Boundaries

I don’t care if you’re Republican or Democrat, Socialist or Fascist (ok, I actually DO care if you’re a Fascist, but I’ve got a rhythm going here), black or white, urban or rural, whatever.  Go vote.  Today’s the day.  Don’t let intimidation tactics or inclement weather (just started raining here in Philly) stop you.  Just do it.  It’s important.

Yes, this is one of those stupid “why I haven’t been posting” posts.  Yes, I know it’s a cop-out.  Anyway, I’ve been writing fellowship applications this last month, and for most of them it’s my last chance to apply, so I’m rather focused on it.  The NSF is due on Wednesday, and probably after that I’ll resume a bit of sporadic posting.  However, I’ve lost my steam on the toric geometry series, so I’m going to sharply change things up and do a (mostly) complete solution to the following problem: how many rational nodal plane curves of degree d are there passing through 3d-1 points.  The short series will follow a series of actual lectures I’m giving, and will culminate with Kontsevich’s recursive formula.  I’ll even say in advance what step I’m skipping: constructing the moduli space of stable maps.  It’s long and technical anyway.  So that’s what’s coming from me, and then possibly more enumerative geometry and intersection theory.

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