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Bleg: book recommendations for an undergraduate July 3, 2009

Posted by Ben Webster in blegs.
22 comments

Following Emily’s advice, I recently signed up to be mentor in the AWM Mentor Network. It’s been pretty good thus far (I recommend it to any of you who would like to do some menting), but I got a request from my mentee that I thought some of our audience might have better ideas about than me.

What math books would you suggest for relatively casual summer reading for an undergraduate math major finishing their third year? This is not the sort of thing I think about a lot, but I know a reasonable number of readers have a lot more experience with young mathematicians than I do.

Three geometric constructions of the irreducible representations of GL_n July 3, 2009

Posted by Joel Kamnitzer in Uncategorized.
2 comments

The past few weeks there has been a summer school and conference on geometric representation theory and extended affine Lie algebras at University of Ottawa. As part of this event, I gave a week long lecture series entitled “three geometric constructions of the irreducible representations of GL_n “. Specifically I discussed the Borel-Weil theorem, Ginzburg’s construction using Springer fibres, and the geometric Satake correspondence. I focused on GL_n to keep the root system combinatorics and the geometry as elementary as possible.

The typed lecture notes from my talk are now available. If you do read them, please let me know if you have any comments/corrections. (You can also find videos of the talks.)

The other lectures at the summer school were given by Neher, Kang, Wang, Savage, and Chari. I recommend reading their notes/watching their videos if you want to learn more about geometric representation theory, crystals, and affine Lie algebras.

Continued Fractions and Hyperelliptic Curves July 2, 2009

Posted by David Speyer in Algebraic Geometry, Number theory.
4 comments

I recently read a charming little paper: Quasi-elliptic integrals and periodic continued fractions, by van der Poorten and Tran. Most of us who have taken a number theory course of some kind learned how to solve Pell’s equation: x^2 - D y^2 =1 where D is a nonsquare positive integer. The usual method is to compute the continued fraction
\displaystyle{\sqrt{D} = a_0 + \frac{1}{a_1 + \frac{1}{a_2 + \frac{1}{\cdots}}}}.
One then defines the convergents of \sqrt{D} by
\displaystyle{x_0/y_0 = a_0}
\displaystyle{x_1/y_1 = a_0 + \frac{1}{a_1}}
\displaystyle{x_2/y_2 = a_0 + \frac{1}{a_1+\frac{1}{a_2}}} etcetera.

Then x_i^2 - D y_i^2 tends to be very small and, if you compute long enough, for some i you will have x_i^2 - D y_i^2=1.

What van der Poorten and Tran do is to ask what happens if D is not an integer, but a polynomial D(t) = t^{2g+2} + d_{2g+1} t^{2g+1} + \cdots + d_1 t + d_0. Before I get into details, I want to tell you about something gorgeous that I won’t explain at all. Using the methods in their paper, van der Poorten and Trap can discover identities like
\displaystyle{ \int \frac{3 x dx}{\sqrt{x^4+2x}} = \log \left( x^3+1+x \sqrt{x^4+2x} \right)}.
Isn’t that pretty?

It turns out that the continued fraction algorithm for \sqrt{D(t)} is actually much prettier than for integers. Everything should be understood in terms of the curve C cut out by y^2 = D(t). This is a curve of genus g, with two points at infinity. (One of these points is the limit of (t, \sqrt{D(t)}) and the other is the limit of (t, -\sqrt{D(t)}).) I’ll call these two points \infty_{+} and \infty_{-}. The theory is controlled by the line bundles \mathcal{O}(k \infty_+ + \ell \infty_-). In particular, there are nontrivial solutions to x(t)^2 - D(t) y(t)^2 =1 if and only if the continued fraction is periodic, if and only if \mathcal{O}(k \infty_+) = \mathcal{O}(k \infty_-) for some a >0.

Below the fold, I’ll explain what is meant by the continued fraction algorithm for an algebraic function, and tell you some of the other nice results from the paper.

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Man and machine thinking about SPC4 June 29, 2009

Posted by Scott Morrison in crazy ideas, link homology, low-dimensional topology, papers.
6 comments

I’ve just uploaded a paper to the arXiv, Man and machine thinking about the smooth 4-dimensional Poincaré conjecture, joint with Michael Freedman, Robert Gompf, and Kevin Walker.

The smooth 4-dimensional Poincaré conjecture (SPC4) is the “last man standing in geometric topology”: the last open problem immediately recognizable to a topologist from the 1950s. It says, of course:

A smooth four dimensional manifold \Sigma homeomorphic to the 4-sphere S^4 is actually diffeomorphic to it, \Sigma = S^4.

We try to have it both ways in this paper, hoping to both prove and disprove the conjecture! Unsuprisingly we’re not particularly successful in either direction, but we think there are some interesting things to say regardless. When I say we “hope to prove the conjecture”, really I mean that we suggest a conjecture equivalent to SPC4, but perhaps friendlier looking to 3-manifold topologists. When I say we “hope to disprove the conjecture”, really I mean that we explain an potential computable obstruction, which might suffice to establish a counterexample. We also get to draw some amazingly complicated links:

SPC4 link

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New Journal: Quantum Topology June 26, 2009

Posted by Noah Snyder in good journals, hopf algebras, link homology, planar algebras, subfactors, tqft.
1 comment so far

The European Math Society Publishing House (a non-profit publishing company which also publishes the Journal of the EMS, CMH, and half a dozen other journals) just announced a new journal: Quantum Topology. I think this is very exciting as it fills a nice hole in the existing journal options. The list of main topics include knot polynomials, TQFT, fusion categories, categorification, and subfactors. So there should be lots of material of interest to people here.

Conference Networking June 22, 2009

Posted by Scott Carnahan in conferences, math life.
14 comments

Early in my graduate student career, I was told by several people that I should go to conferences and talk to professors. If you work in mathematics, you’ve probably heard this piece of advice before, and it’s hard to see how you could damage your career by following it (given reasonable assumptions on your behavior). I encountered two problems:

  1. What sort of talking am I supposed to do with a professor if I don’t know anything?
  2. How do I make my way into one of those small circles of people that inevitably form between talks?

I’ve heard that some advisors actually go to conferences with their students and introduce them to colleagues, and this pretty much solves both problems, but I’d like to focus on the case that this doesn’t happen, since I imagine it will be the norm for a while. This isn’t meant to be a definitive guide, and I’d really appreciate further suggestions and anecdotes.
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Job at King’s College June 17, 2009

Posted by David Speyer in Uncategorized.
2 comments

Konni Rietsch writes to me

The Department of Mathematics at King’s College London is advertising a permanent Lectureship in the area of geometry with application deadline the 27th of July.  

Further information can be found at:

http://www.kcl.ac.uk/depsta/pertra/vacancy/external/pers_detail.php?jobindex=7951

King’s College London, part of the University of London, is based in central London with its mathematics department located in the Strand campus, next to the river Thames and surrounded by theatres and art galleries.

More information about King’s College London can be found on http://www.kcl.ac.uk/about/

Information on the department and its research groups can be found on http://www.kcl.ac.uk/schools/pse/maths/

KingsCollege

For those not familiar with British academic ranks, my understanding is that a permanent lectureship is the first stage leading to a professorship; and that the term “professor” at a British University is far more prestigious than at an American one.

I haven’t been to King’s College myself, but I am a frequent user and admirer of Konni’s work on total positivity. Looking through their department, I see a lot of a lot of geometry and representation theory, including classical Lie Theory, mapping class groups, Langlands, quantum groups and higher category theory. I also see a lot of number theory/arithmetic geometry, including Diamond, and a lot of mathematical physics. If you like the Secret Blogging Seminar, you might fit in very well there.

Local Systems: The connection perspective June 15, 2009

Posted by David Speyer in Algebraic Geometry, D-modules.
5 comments

Welcome to the next installation of my series on local systems. In this post I’ll be talking about connections. This post should require less sophistication than the last few — no schemes, no functors — I’ll almost be coming at the subject afresh. There will be another post later, explaining how you might get to connections if you started out thinking about the infinitesimal site.

To start out with, let’s talk about derivatives; ordinary, single variable calculus derivatives. We have a function f of a variable x. Then the derivative of f is the function f'(x) := \lim_{h \to 0} (f(x+h) - f(x))/h. There are two directions in which we might want to generalize this idea. The first is to work with functions on a manifold, on a space which has no inherent coordinate system. This is the subject of your standard Calculus on Manifolds course, and I am going to assume that my readers are at least vaguely familiar with it. The second is to work, not with functions, but with sections of vector bundles. That’s our subject in this post.

So, let’s think about a vector bundle V on the line \mathbb{R}, and let \sigma be a section of V. If we want to define \sigma', we need to subtract \sigma(x+h) and \sigma(x), two vectors which live in different fibers. To think of it another way, we need to distinguish between f(x), the point in the fiber over x, and f(x), the constant function which assigns the same value at every point. Suppose that, for any v \in V_x, we had a local section c_v of V with c_v(x)=v; we think of c_v as a constant function. Then we could define \sigma'(x) = \lim_{h \to 0} \left( \sigma(x+h) - c_{\sigma(x)}(x+h) \right)/h.

A local system gives us the constant functions c_v. (Indeed, in definitions A.2 and B.6, we took a local system to be the constant functions, along with the data of certain maps between them.) Today, we will take the fundamental object to be the operation of derivation, and see how to build everything else from it.

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Benford’s law and the Iranian election June 15, 2009

Posted by Scott Morrison in Uncategorized.
3 comments

I’m obsessing over the aftermath of the Iranian election today, and thought our readership might be interested in this analysis by Walter Mebane, from UMich, of Iranian election results.

He’s taken the Ministry of Interior’s posted results (or a copy in Google docs), and done a quick check against Benford’s law, which predicts the statistical distribution of initial digits of numbers. With the available data (returns for ~350 districts), he reports that nothing looks particularly wrong, but would love to have more detailed (polling station level) data, as you shouldn’t expect to see much anyway.

Nate Silver also has a critique up of some earlier statistical complaints made about the announced election results. If anyone sees something else along this lines, I’d love to hear.

Have someone else write your bibliography June 14, 2009

Posted by Scott Morrison in papers, the arXiv, websites.
6 comments

Whenever I’m finishing off a paper, at some point I have to sit down and clean up all the references, which generally look something like \cite{Popa?} or \cite{that paper by Marco and co}. Wouldn’t it be nice if someone else could do the rest?

If you don’t already know about it, one great resource is mathscinet, which will produce nicely formatted BIBTEX entries for you (example). If you want to be even more efficient, you can wander around mathscinet, saving articles to your “clipboard”, and then ask mathscinet to give you the BIBTEX entries for everything at once. (After you have articles on the clipboard, follow the “clipboard” link in the top right of the page, then select BIBTEX from the drop-down box and click “SaveClip”.)

If you’re even lazier, you could use the two command-line scripts that I use (download find-missing-bibitems and get-mathscinet-bibtex and put them on your path; you’ll need linux/OSX/cygwin to run). Now, when you cite items in LaTeX, cite them via their mathscinet identifiers, e.g. \cite{MR1278111} instead of \cite{Popa?}. Now, if you usually type latex article to compile, and bibtex article to generate the bibliography, you can also type find-missing-bibitems article, and all the missing BIBTEX entries will appear! For example, after adding \cite{MR1278111} somewhere in my text, the output of find-missing-bibitems article is

@article {MR1278111,
    AUTHOR = {Popa, Sorin},
     TITLE = {Classification of amenable subfactors of type {II}},
   JOURNAL = {Acta Math.},
  FJOURNAL = {Acta Mathematica},
    VOLUME = {172},
      YEAR = {1994},
    NUMBER = {2},
     PAGES = {163--255},
      ISSN = {0001-5962},
     CODEN = {ACMAA8},
   MRCLASS = {46L37 (46L10 46L40)},
  MRNUMBER = {MR1278111 (95f:46105)},
MRREVIEWER = {V. S. Sunder},
}

If you’re brave, you could run something like

find-missing-bibitems article >> bibliography.bib

to automatically append any missing entries to your BIBTEX file. The really enthusiastic could incorporate this script into the standard latex-latex-bibtex-latex cycle.

Really, I like to have more in my BIBTEX file: I generally use the note field to include a link to the mathscinet review, and a link to the DOI for the paper on the publisher’s webpage. If available, I want a link to the arxiv version of the paper too, for people without institutional access to the published version. Currently, the scripts can’t do this automatically, but it’s might not be much more work. Maybe next time.