E. Kowalski's blog






         Comments on mathematics, mostly.

24.05.2015

Latest adventures

Filed under: ETH,Language,Mathematics,Travel @ 14:29

The last two weeks were quite eventful…

First I spent four days in China for the conference in honor of N. Katz’s 71st birthday. I was lucky with jetlag and was able to really enjoy this trip, despite its short length. The talks themselves were quite interesting, even if most of them were rather far from my areas of expertise. I talked about my work with W. Sawin on Kloosterman paths; the slides are now online.

I only had time to participate in one of the excursions, to the Forbidden City,

Forbidden City

Forbidden City

were I took many pictures of Chinese Dragons…

Chinese Dragon

Chinese Dragon

That same evening, with F. Rodriguez Villegas and C. Hall, I explored a small part of the Beijing subway,

Subway map

Subway map

trying to interpret and recognize various Chinese characters, before spending a fair amount of time in a huge bookstore

Bookstore

Bookstore

(where I got some comic books in Chinese for fun).

Upon coming back on Thursday, I first found in my office the two volumes of the letters between Serre and Tate that the SMF has just published, and which I had ordered a few days before taking the plane. Reading the beginning of the first volume was very enjoyable in the train on Friday morning from Zürich to Lausanne, where the traditional Number Theory Days were organized this year. All talks were excellent again — we’re now looking forward to next year’s edition, which will be back in Zürich! And I’ll write later some more comments about the Serre-Tate letters…

And then, from last Monday to Friday, we had in Zürich the conference “Analytic Aspects of Number Theory”, organized by H. Iwaniec, Ph. Michel and myself with the help of FIM. It was great fun, and there were really superb and impressive talks. One interesting experience was the talk by J. Bellaïche : for health reasons, he couldn’t travel to Zürich, but we organized his talk by video (using a software called Scopia), watching it from a teleconference room at ETH. This went rather well.

07.05.2015

New versions, new bugs

Filed under: Computers,Mathematics @ 9:22

Ph. Michel found the first bug in the new version of Kloostermania: when entering a modulus for a sum from the menu, there was no check that it is prime (or adjustment to make it so). This is now corrected in version 1.01, to be found in the usual place.

As another bonus feature, a double tap on the screen will cycle between the types of sums presented: Kloosterman sum to Birch sum to both to Kloosterman…

04.05.2015

Kloostermania turns 1.0

Filed under: Computers,Mathematics @ 18:21

Since the Pocket Kloostermania has remained unchanged for almost five years, an update is probably welcome! This is now available, in time to honor N. Katz on the occasion of this 71st birthday (which I will help celebrate in China next week).

Important changes in the new version:

(1) It was recompiled — hence a new modern look instead of a style reminiscent of the dark ages –, and the resulting binary should work on any Android system with version 4.0.3 or later;

(2) To keep up with recent progress, the program now displays Kloosterman sums and/or Birch sums, instead of Kloosterman and/or Salié sums. On the other hand, the moduli used are now restricted to primes, to simplify things a bit, and only one parameter is used: the sums displayed are
\sum_{x\in\mathbf{F}_p^{\times}}e\Bigl(\frac{ax+\bar{x}}{p}\Bigr)
and
\sum_{x\in\mathbf{F}_p}e\Bigl(\frac{ax+x^3}{p}\Bigr)

(3) In addition to being able to change the modulus by swiping horizontally, a vertical swipe on the screen scrolls among the values of the parameter a. As was the case in the previous version (and even more because phones are faster now), the scrolling is usually too fast for a single step, so tapping once close to one of the edges of the window displaying the sum will perform just one step of the corresponding move (e.g., tapping close to the right of the screen goes to the next prime modulus). The value of the sum and its parameters and then displayed quickly.

(4) The plots are presented in orthonormal configurations, to give a more faithful representation of the paths in the plane;

Kloostermania

Kloostermania

(5) In the “About” dialog, a single click will display (if a PDF viewer is installed) the paper of W. Sawin and myself that explains the limiting distribution of Kloosterman (and Birch) paths…

(6) It is possible to save (or rather to “share” in the usual Android way) a picture of the sums currently displayed, as a PNG file.

(6) And the launcher icon is better.

Icon

Icon

The installation file is available right now on the updated Kloostermania page!

11.04.2015

Worms, brains and “graphes expanseurs”

Filed under: Computers,France,Mathematics,Science @ 13:03

Two days ago, I was in Paris and visited the École Polytechnique there (for the first time, as it turns out), in order to give the traditional introductory lecture presenting various aspects of mathematics that is given each year to the incoming class of students. (Many people would wonder how a new class can be incoming in April; this is because the students of École Polytechnique, which is a military school, begin their studies with a military/civil service from September to April).

I had selected expander graphs as the topic of my talk, as being simultaneously a very modern subject, that can be presented with very basic definitions (most students, coming from the French “Classes préparatoires” have had two very solid years of mathematical studies, but of a rather traditional kind), and that has links to many different areas, including more applied ones.

The slides of my presentation can be found here, in French, and I prepared an English translation there (up to the handwritten bits of information on certain pictures, which will remain in French…) Actually, the slides by themselves are not particularly interesting, since there are many remarks and details that I only described during the talk.

The talk begins with a discussion of graphs, continues with basic notions of expansion, and then defines expander graphs, with a bit of the history (especially the paper of Barzdin and Kolmogorov that I like very much), and concludes with two applications (among many…): Apollonian circle packings, and the Gromov-Guth knot-distorsion theorem. If this looks like a lot of ground to cover, this is because the talk was 1h30 long…

While preparing the first part of the talk, I researched some examples of graphs with more care than I had done before. Some of the things I found are extremely interesting, and since they might not be so well-known among my readers, I will present them quickly.

(1) The worm is Caenorhabditis elegans, more chummily known as C. elegans, one of the stars of neurosciences, as being the only animal whose entire nervous system has been mapped at the level of all individual neurons and connections between them. This was done in 1986 by White, Southgate, Thomson and Brenner, with minor corrections since then, and an important update and representation in 2011 by Varshney, Chen, Paniagua and Chklovskii. The resulting graph has 302 neurons (this number is apparently constant over all individuals), and about 8000 edges. (Interestingly, it is naturally a mix of directed and un-directed edges, depending on the type of biological connection). The paper of Varshney, Chen, Paniagua and Chklovskii is quite fascinating, as it investigates many mathematical invariants of the graph, and for instance compares the number of small subgraphs of various types with the expected number for random graphs with comparable parameters (certain subgraphs arise with higher frequency…)

Much more about C. elegans is found on dedicated websites, including Openworm, which seems to have as a goal to recreate the animal virtually…

(2) The brain is the humain brain; it can’t be mapped at the level of C. elegans (there are about 10^{11} neurons and 10^{15} synapses, from what I’ve seen), but I read a very interesting survey by Valiant about attempts to understand the computational model underpinning the reasoning capacities of the brain. He presents four basic tasks that must be among those that the brain performs, and explains how he succeeded in earlier work (from 1994 to 2005) in finding realistic algorithms and models for these problems. He comments:

In [5,14] it is shown that algorithms for the four random
access tasks described above can be performed on the
neuroidal model with realistic values of the numerical
parameters. The algorithms used are all of the vicinal
style. Their basic steps are all local in that they only
change synaptic strengths between pairs of neurons that
are directly connected. Yet they need to achieve the more
global objectives of random access. In order that they be
able to do this certain graph theoretic connectivity prop-
erties are required of the network. The property of
expansion [15], that any set of a certain number of
neurons have between them substantially more neighbors
than their own number, is an archetypal such property.
(This property, widely studied in computer science, was
apparently first discussed in a neuroscience setting [16].)
The vicinal algorithms for the four tasks considered here
need some such connectivity properties. In each case
random graphs with appropriate realistic parameters have
it, but pure randomness is not necessarily essential.

Here reference [16] is to the Barzdin-Kolmogorov paper.

(3) Lastly, I was wondering what is the size (number of vertices) of the largest graphs for which the first non-zero Laplace eigenvalue has been computed, approximately. The best I found is a paper of Kang, Breed, Papalexakis, and Faloutsos, which describes an algorithm that they have applied successfully to real-world graphs with about a billion vertices. This is rather impressive…

26.03.2015

Kummer extensions, Hilbert’s Theorem 90 and judicious expansion

Filed under: ETH,Exercise,Mathematics @ 17:45

This semester, I am teaching “Algebra II” for the first time. After “Algebra I” which covers standard “Groups, rings and fields”, this follow-up is largely Galois theory. In particular, I have to classify cyclic extensions.

In the simplest case where L/K is a cyclic extension of degree n\geq 1 and K contains all n-th roots of unity (and n is coprime to the characteristic of K), this essentially means proving that if L/K has cyclic Galois group of order n, then there is some b\in L with L=K(b) and b^n=a belongs to K^{\times}.

Indeed, the converse is relatively simple (in the technical sense that I can do it on paper or on the blackboard without having to think about it in advance, by just following the general principles that I remember).

I had however the memory that the second step is trickier, and didn’t remember exactly how it was done. The texts I use (the notes of M. Reid, Lang’s “Algebra” and Chambert-Loir’s delightful “Algèbre corporelle”, or rather its English translation) all give “the formula” for the element b but they do not really motivate it. This is certainly rather quick, but since I can’t remember it, and yet I would like to motivate as much as possible all steps in this construction, I looked at the question a bit more carefully.

As it turns out, a judicious expansion and lengthening of the argument makes it (to me) more memorable and understandable.

The first step (which is standard and motivated by the converse) is to recognize that it is enough to find some element x in L^{\times} such that \sigma(x)=\xi x, where \sigma is a generator of the Galois group G=\mathrm{Gal}(L/K) and \xi is a primitive n-th root of unity in L. This is a statement about the K-linear action of G on L, or in other words about the representation of G on the K-vector space L. So, as usual, the first question is to see what we know about this representation.

And we know quite a bit! Indeed, the normal basis theorem states that L is isomorphic to the left-regular representation of G on the vector space V of K-valued functions \varphi\,:\, G\longrightarrow K, which is given by
(\sigma\cdot \varphi)(\tau)=\varphi(\sigma^{-1}\tau).
(It is more usual to use the group algebra K[G], but both are isomorphic).

The desired equation implies (because G is generated by \sigma) that Kx is a sub-representation of L. In V, we have an explicit decomposition in direct sum
V=\bigoplus_{\chi} K\chi,
where \chi runs over all characters \chi\,:\, G\longrightarrow K (these really run over all characters of G over an algebraic closure of K, because K contains all n-th roots of unity and G has exponent n). So x (if it is to exist) must correspond to some character. The only thing to check now is whether we can find one with the right \sigma eigenvalue.

So we just see what happens (or we remember that it works). For a character \chi\in V such that \chi(\sigma) = \omega, and x\in L^{\times} the element corresponding to \chi under the G-isomorphism L\simeq V, we obtain \sigma(x)=\omega^{-1}x. But by easy character theory (recall that G is cyclic of order n) we can find \chi with \chi(\sigma)=\xi^{-1}, and we are done.

I noticed that Lang hides the formula in Hilbert’s Theorem 90: an element of norm 1 in a cyclic extension, with \sigma a generator of the Galois group, is of the form \sigma(x)/x for some non-zero x; this is applied to the n-th root of unity in L. The proof of Hilbert’s Theorem 90 uses something with the same flavor as the representation theory argument: Artin’s Lemma to the effect that the elements of G are linearly independent as linear maps on L. I haven’t completely elucidated the parallel however.

(P.S. Chambert-Loir’s blog has some recent very interesting posts on elementary Galois theory, which are highly recommended.)

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