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Michi’s blog » archive for February, 2008

 Introduction to Algebraic Geometry (2 in a series)

  • February 21st, 2008
  • 10:43 pm

I want to lead this sequence to the point where I am having trouble understanding algebraic geometry. Hence, I won’t take the usual course such an introduction would take, but rather set the stage reasonably quickly to make the transit to the more abstract themes clear.

But that’s all a few posts away. For now, recall that we recognized already that any variety is defined by an ideal, and that intersections and unions of varieties are given by sums and intersections or products of ideals.

This is the first page of what is known as the Algebra-Geometry dictionary. The dictionary is made complete by a pair of reasonably famous theorems. I won’t bother proving them – the proofs are a good chunk of any decent commutative algebra course – but I’ll quote the theorems and discuss why they matter.

 Introduction to Algebraic Geometry (1 in a series)

  • February 21st, 2008
  • 12:33 pm

I’m growing embarrassed by my lack of understanding for the sheaf-theoretic approaches to algebraic (and differential) geometry. I’ve tried to deal with it several times before, and I’m currently reading up on Algebraic Geometry again to fill the void that the finished thesis, soon arriving travels and non-existent job application responses produce.

So, why not learn by teaching? It’s an approach that has been pretty darn good in the past. So I thought I’d write a sequence of posts on algebraic geometry, introducing what it’s supposed to be about and how the main viewpoints develop more or less naturally from the approaches taken.

Varieties

The basic objective of algebraic geometry is to study solution sets to systems of polynomial equations. That is, we take some set f_1,\dots,f_r of polynomials in some polynomial ring k[x_1,\dots,x_n] over some field k. And we write V(f_1,\dots,f_r) for the set of all simultaneous roots to all these polynomials:
V(f_1,\dots,f_r)=\{p\in k^n:f_1(p)=0, \dots, f_r(p)=0\}

 Scripting Games in Haskell

  • February 21st, 2008
  • 1:15 am

I saw the Cerebrate solve the first Scripting Games challenge: Pairing off. And immediately thought “I can do that in Haskell too”.

So, here it is.

import Data.List

cards = [(1,7),(0,5),(3,7),(2,7),(2,13)]

countpairs [] = 0
countpairs [a] = 0
countpairs (a:as) = length . filter (((snd a)==) . snd) $ as

pairingOff = sum . map countpairs . tails

And that’s that. Alas, the actual competition only takes Perl, VBScript and PowerShell, so I won’t be submitting this.

 PROPs and patches

  • February 15th, 2008
  • 7:59 pm

Brent Yorgey wrote a post on using category theory to formalize patch theory. In the middle of it, he talks about the need to commute a patch to the end of a patch series, in order to apply a patch undoing it. He suggests a necessary condition to do this is that, given patches P and Q, we need to be able to find patches Q’ and P’ such that PQ=Q’P’, and preferably such that Q’ and P’ capture some of the info in P and Q.

However, as such, this is not enough to solve the issue. For one thing, we can set Q’=P and P’=Q, and things are the way he asks for.

Now, I wonder whether we can solve this by using PROPs (or possibly di-operads or something like that). Let’s represent a document as a list of some sort of tokens. We’ll set D_n the set of all lists of length n, and we’ll set P_n^m to denote operations that take a list of length n and returns a list of length m.

 Thesis written

  • February 15th, 2008
  • 2:13 pm

In a mean push, these last two weeks my advisor has read three different drafts of my thesis. And I’ve worked on getting the corrections in quickly. The last push started yesterday, when I got a bunch of corrections in the morning, had the last draft ready at 4pm, and then sat reading it myself until 1am.

My advisor took it home with him, spent the evening on it, and had his batch of corrections in the morning.

Hence, today at 10-ish when I got myself in to the office, I had two batches of corrections in front of me, and a printer closing at 2pm. So I worked – and now, well, it’s done.

That’s it.

It’ll get printed.

Then read.

In May, we should get all the comments back from the external examiners.

 My topology students move into knot theory

  • February 1st, 2008
  • 2:27 pm

So, here’s the plan for my 10th grade topology students.

Today, we’ll abandon algebraic topology completely, and instead go into knot theory. I’ll want to discuss what we mean by a knot (embedding of S^1 in S^3), what we mean by a knot deformation (thus introducing isotopies while we’re at it) and the Reidemeister moves. Also we’ll discuss knot invariants – and their use analogous to topological invariants.

Later on, we’ll continue with other invariants; definitely including the Jones polynomial, and possibly even covering Khovanov homology. One possible end report would be to explain a bunch of knot invariants and show using examples how these have different coarseness.

Edited to add: I got myself some damn smart students. They figured out the Reidemeister moves on their own – as well as minimal crossing number in a projection being highly relevant – with basically no prompting from me. I’m impressed.