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It’s My Turn August 29, 2010

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This is from a movie called It’s My Turn starring Jill Clayburgh. I’m pretty impressed. Notice that this is perfectly correct math that belongs in a college course. Also notice that Jill Clayburgh is a woman mathematician, she is not reclusive or schizophrenic, and if you watch the whole movie she’s actually a pretty interesting character. She has normal worries — problems in love, doubts about a new job that might require her to do mostly administration, the sort of thing that actual academics stress out about.

When was the last time you saw any of this in a movie?

I don’t just mean the math (which is really remarkable, given how bad a job Hollywood usually does.) You just don’t see that kind of a female lead in movies today: actually intelligent, confused about life, falls in love but isn’t defined by the men in her life.

There’s a period in the late 70’s-early 80’s where you see a lot of movies that have a similar quality. (Offhand list: Kramer vs. Kramer, Hannah and Her Sisters, The Paper Chase, 84 Charing Cross Road, Manhattan.) Movies about smart grown-ups, usually living in New York, having complicated modern problems with their love lives. Strong (if neurotic) female characters, sort of feminist. Montages with happy Baroque music in Central Park in the fall. It’s kind of a vanished type — maybe Aaron Sorkin is a little bit in that tradition, in that he writes intelligent adult characters and makes maturity look cool. But really: can you imagine a romantic comedy written today, with a heroine who’s a mathematician, that doesn’t make fun of her “geeky” job or completely mangle the details?

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Bulldog! Bulldog! Bow wow wow! August 29, 2010

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Just a few notes as the week goes by:

1. I’ve got classes picked out now: modern algebra, measure theory and integration, algebraic topology, and harmonic analysis. And the new requirement, “ethical conduct of research” (which I assume means “don’t plagiarize and don’t torture monkeys.”)

2. I love Ikea. I know everybody loves Ikea, but there’s no reason to depart from convention here. Everything is beautiful and cheap and quite a lot of things come in orange.

3. The international students, apparently, were taken aside to learn Yale football chants and associated paraphernalia (including the “Bulldog! Bulldog! Bow wow wow!” cheer.) We Americans were never told anything of the kind. I guess this goes along with the tradition of immigrants knowing the Constitution better than we do.

4. Thanks to a fellow grad student, I’m remembering how much I like geometry and topology. Here’s something I learned (at a pub night, no less!) A hyperbolic manifold is the hyperbolic plane modulo a discrete group of isometries. That means, if you look at the Poincare disc model, some of the points on the circular boundary are limit points of the orbit of some point in the interior under isometries in the group. Consider the set of such points. Apparently: if it’s not the whole circle, it is a Cantor set, and its Hausdorff dimension equals the first eigenvalue of the Laplacian on the manifold. WHOA.

Back to School August 22, 2010

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I just arrived at school, though classes won’t start until the first of September. I have a nice little room; little, but shipshape (at least right now.)

New Haven is far superior to Princeton in terms of having affordable retail within walking distance. You hear a lot about the downsides of going to school in a city, but right now (while setting up the room) I’m really experiencing the upside of not living in a suburban island.

I’ve started meeting the other students — visited that local landmark, the GPSCY (pronounced “the Gypsy”) which is where grad students have social events — so far, people seem friendly.

Spectrum of the Laplacian as a shape-matching tool August 19, 2010

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I like the Laplacian. It’s funny but I just realized that every time I’ve gotten suddenly passionate about an area of mathematics, the Laplacian has been involved. So I enjoyed this paper by a couple of computer graphics guys in Hannover, using the spectrum of the Laplace-Beltrami operator as a “fingerprint” for shape matching.

Measuring the similarity between shapes is obviously a big deal for computer graphics or machine vision, and there are many competing similarity measures — you can use moments or spherical harmonics or Gromov-Hausdorff distance, among many others. But the spectrum of the Laplacian has the particularly neat property that for some shapes (for example, the disc) you can prove that it uniquely determines the shape — any shape with the same spectrum as the disc must in fact be a disc. The spectrum of the Laplacian also determines the dimension of a manifold (determining dimension is a common difficulty, as many procedures for fitting a manifold to data assume the dimension is known.) There are enough handy properties here that the spectrum of the Laplacian seems — in my untutored opinion — to be a very natural way to characterize shapes. (A propos of the previous post, and applications of the Selberg Trace Formula.)

(Possibly related: from the Geomblog.)

What is the Selberg Trace Formula? August 19, 2010

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Ngo Bau Chau was one of the winners of the Fields Medal for 2010, so I thought I’d try to understand what it is he studied. The laudation gives a clear explanation of his contribution to the Langlands program. Ngo’s proof applies the Selberg trace formula, and since that’s accessible, I thought I’d try to understand that.

If G is a Lie group and \Gamma is a cofinite discrete group, we want to understand the operator

R(f) = \int_\Gamma f(y) R(y) dy
on L^2(G/\Gamma).

R(f) is an integral operator, with kernel
K(f) = \sum_{\gamma \in \Gamma} f(x^{-1} \gamma y)
And now we can express the trace of the operator as
\int_G \sum_{\gamma \in \Gamma} f(x^{-1} \gamma x) dx
The result is
\sum_{[\gamma]} \int_{\Gamma_\gamma / G} f(x^{-1} \gamma x) dx = \sum_{\pi} tr(\pi(f))
where [\gamma] are the conjugacy classes of \gamma, \Gamma_\gamma is the centralizer of \gamma in \Gamma, and \pi are the automorphic representations.

When \Gamma = \mathbb{Z} and G = \mathbb{R}, the Selberg Trace Formula is the Poisson Summation Formula

\sum_{n = -\infty}^\infty f(t + nT) = \frac{1}{T}\sum_{k = -\infty}^\infty \hat{f}(k/T) exp(2 \pi i k/T t).

The Poisson Summation formula can be seen to be a trace formula as follows:
The eigenvectors of the positive Laplacian on the unit circle are m^2 for integer m, and the eigenfunctions are \phi_m = (2\pi)^{1/2} e^{i mx}.
Consider the linear operator
(Lf)(x) = \int_0^{2\pi} k(x, y) f(y) dy
with kernel
k(x, y) = \sum h(m)\phi_m(x) \bar{\phi_m(y)}
Then
L \phi_m = h(m) \phi_m and so the Poisson Summation Formula says
Tr L = \sum_m h(m) = \sum \int h(\rho) e^{2 \pi i n \rho} d\rho

The Selberg Trace Formula is proven in a similar manner to the proof of the Poisson Summation Theorem. If R is the regular representation of G on L^2(\Gamma/G)
[R(g) \phi](x) = \phi(xg), g\in G, x \in \Gamma / G
then we can write R as
R(f)\phi(x) = \int_G f(x^{-1}g) \phi(g) dg
fixing a Haar measure on G. Then, by splitting the integral
R(f) \phi(x) = \int_{\Gamma / G} \sum_{\gamma \in \Gamma} f(x^{-1} \gamma g)\phi(g) dg =  \int_{\Gamma / G} K_f(x, y) \phi(y) dy.

We now take the trace of this operator in two different ways:
on one side,
tr R = \int_{\Gamma / G} K_f(x, x) dx
We break the sum over \gamma into conjugacy classes of \gamma. Each conjugacy class contributes
\int_{\Gamma / G} \sum_{\delta \in \Gamma_\gamma / \Gamma}f(x^{-1}\delta^{-1} \gamma \delta x) dx = \int_{\Gamma / G} f(x^{-1} \gamma x)  dx
On the other side, we can compute the trace alternatively by quoting a result that says L^2(\Gamma / G) decomposes into a direct sum of irreducible representations of G:
\sum_{\pi} tr(\pi(f)).

This completes the proof. Conjugacy classes and irreducible representations are dual — in the same way that integers (in the Fourier series) and functions e^{2 \pi i nx} are dual in the Poisson summation formula.

So what can you do with the trace formula? One thing is to let \Gamma be the fundamental group of a Riemann surface, and describe the spectrum of differential operators such as the Laplace-Beltrami operator using geometric data like the lengths of geodesics. (Way back in sophomore year I was trying to learn about this from Peter Buser’s book.) The Selberg Trace Formula is also useful in analyzing the Riemann zeta function.

See these notes for a bunch of background on the Selberg Trace Formula.

Meyer/Basarab paper August 18, 2010

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this is what I’m reading through right now: “A variant on the compressed sensing of Emmanuel Candes,” by Yves Meyer and Matei Basarab. I’m not done with it, but it involves reconstructing a function by sub-Nyquist sampling of its Fourier coefficients, using an \ell^1-minimization technique. It’s a generalization of a theorem by Terence Tao about functions on finite fields. Looks to be interesting.

Packing: it begins August 11, 2010

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I really like cartoonist Lucy Knisely, and one of her latest panels rings pretty true to me right now:

The past few days have been a flurry of inconsistencies. I’m gonna be poor in grad school! No, I’m going to be making more money than I ever have in my life! I’m all grown up! No, I’m an overgrown little kid! I’m a scientist now! But I feel woefully unprepared! The one certainty is that I don’t wanna put huge numbers of textbooks in shipping boxes this morning, but I have to.

Still here August 9, 2010

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Despite the light posting, I’m still alive. Just very boring.

I’ve just come back from a family vacation; now I’m home, planning for the start of school and trying to locate things in New Haven. (There is a Target on the bus route — huzzah!) Also picked out math courses.

My only math-related news is continuing to read through Atiyah-Macdonald and Hatcher’s Algebraic Topology. I’m liking Hatcher — it’s much more concrete and example-based than the book I used for algebraic topology in undergrad (From Calculus to Cohomology). Little pictures of triangles, to be completely honest, make the boundary operator a lot more intuitive.

In non-math news: Truman Capote’s Breakfast at Tiffany’s is really worth reading. The movie is pretty much fashionista froth, not that there’s anything wrong with that, but the book is profound.