Archive for March, 2007

A number theory question related to the Feit-Thompson theorem

March 31, 2007

In the book by Gelbaum and Olmsted on counterexamples in mathematics, they quote the following as being an open question as of 1990.

The problem comes from a long series of papers by Feit and Thompson on the solvability of groups of odd order (I couldn’t find where exactly, but for example their lemma 34.2 of chapter 5 certainly looks related). Not that I know anything about it all of course. The question as stated by Gelbaum and Olmsted is:

    given two different primes p and q, are the integers N_p(q):=\sum_{k=0}^{q-1}p^k=\frac{p^q-1}{p-1} and N_q(p):=\sum_{i=0}^{p-1}q^i relatively prime ?

and they comment that solving it in the affirmative would make the Feit-Thompson proof “considerably shorter”.

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Lp norms of eigenfunctions and conformal mappings

March 30, 2007

In France we have a very useful saying “le ridicule ne tue pas” (being ridiculous does not kill), so I’ll attempt to clear my mind on a few topics here, and come what may… 🙂

The first thing that puzzles me is an answer of Terry Tao on his blog to a silly question of mine about eigenfunctions of the Dirichlet Laplacian on a general manifold \Omega. He said that L^p(\Omega) norms describe the distributions of the eigenfunctions and that their understanding is still far from satisfactory.

Let’s pick an eigenfunction \phi_k. I didn’t see at all how a countable sequence of positive numbers, namely the N_{k,p}:=\parallel \phi_k \parallel_p, were going to tell us anything about \phi_k(x) for any x\in\Omega. Could one reconstruct \phi_k from (N_{k,p})_{p\in\mathbb{N}} ? (I heard in algebraic geometry people define objects as the union over all rings of zeros of a given polynomial, maybe there’s a similarity here, we look at \phi_k in all the L^p(\Omega). But I digress…)

So I started digging the arXiv and found a few papers.

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Some azur (2)

March 29, 2007

Avignon is a very beautiful city to visit, especially in spring before the big summer heats and flocks of tourists.

Panoramic view of Avignon

(Click here for a zoom of this picture, which is under the GPL courtesy of Wikimedia Commons.)

Mathematics papers containing wrong results

March 25, 2007

When doing mathematics one constantly looks at the literature to find theorems on which to build some further results. So a crucial issue is: how do we know when a published paper in fact contains an error ? If we are interested in understanding the method used by the author to prove a given statement then of course one can check directly for oneself.

But in doing so one could overlook an error in the proof. Another situation is when one wants to use a theorem proved by somebody else as a black box, something one takes for granted without having the need to actually understand the proof first hand.

Well, when an error has been noticed, either the journal subsequently publishes an errata (this is very common in the case of typos which alter an otherwise valid result) or even a retraction when the paper cannot be salvaged. Or the author takes the responsability to warn potential readers of the flaw in one way or another (as for example the renowned Michael Harris duly does on his webpage).

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Some azur (1)

March 23, 2007

In order to justify at last the title of this blog here’s a nice picture of the village of Murs from images-provence-gratuites.com and which is under CC licence

Village of Murs (provence, France)

Colin de Verdière invariant and Four-Color theorem

March 22, 2007

This is a follow-up to a previous post on the Colin de verdière graph invariant. I’ve come across a book in french by Colin de Verdière (Spectres de Graphes, SMF, 1998) in which he writes the following.

Let G denote some graph. Then:

(a) if G is embeddable in either the real projective plane P\mathbb{R}^2 or the Klein Bottle then \mu(G) \leq 5

(b) if G is embeddable in the 2-torus \mathbb{T}^2 then \mu(G) \leq 6

(c) if G is embeddable in a surface S whose Euler characteristic \chi(S) is negative then \mu(G)\leq 4-2\chi(S)

where (a) and (b) are “optimal”.

Moreover he states the Colin de Verdière Conjecture: for any graph we have C(G)\leq \mu(G) +1, where C(G) is the chromatic number.

He adds that a proof of this would provide a non-computational proof of the Four-Color theorem. Moreover this conjecture is implied by Hadwiger’s conjecture, so it is weaker and thus perhaps easier to prove.

Changes to the arXiv next april

March 21, 2007

Watch out: some changes to the arXiv are due to take place on 1st april 2007 (not a joke), read this.

Basically this affects everybody since all identifiers will change: the new format will be arXiv:0703.1234 (i.e. arXiv:YearMonth.Number) and all things like arXiv:math.NT/0703999 will disappear…

Undergraduate mathematicians

March 16, 2007

The 2007 AMS prizes were announced back in january. In the undergraduate student category, the Morgan prize, the winner is yet again a student from MIT (for the third year in a row!), with yet again an amazing publication record, Daniel M. Kane.

It’s a very interesting phenomenom these undergrads with high quality research-level contributions. I heard this was common place in the former USSR, and still is the case in Russia today (perhaps to a lesser extent?).

On the other hand it certainly doesn’t happen in France, where even the best students from the ENS don’t start publishing before graduating, including those who won a gold medal at the Olympiads. There must be some cultural aspects which somehow refrain them from blossoming early, possibly a big lack of dialogue between university researchers and high school and prépa teachers…

The Colin de Verdière graph invariant

March 12, 2007

A very interesting open problem in topology and combinatorics is to find out, for some given k\in\mathbb{N}, what the inequality \mu(G)\leq k for a graph G means, and in fact if it does mean anything at all for large k. Here \mu is the Colin de Verdière graph invariant, a minor-monotone invariant of spectral origin.

Known results are:

  • \mu(G)\leq 1 iff G is a disjoint union of paths;
  • \mu(G)\leq 2 iff G is outerplanar;
  • \mu(G)\leq 3 iff G is planar;
  • \mu(G)\leq 4 iff G is linklessly embeddable.

That’s very neat indeed! Unfortunately even guessing what it means for k=5 seems pretty hard. What is the next step after being planar and having no links ? Having only one connected component ? Or maybe having no cycle ? (Update: actually these two last ideas obviously wouldn’t work since they both imply linkless embeddability. Sigh.)

Princeton’s guide to math generals

March 9, 2007

Stumbling upon a guide for Princeton graduate students about general examinations I found a few amusing bits.

First, in Brian Conrad’s exam:

Can you give a heuristic proof of Fermat’s Last Theorem?
(this last one was asked by Aizenman, the physicist!! Wiles and Faltings
seemed only slightly amused).

🙂

And also in Terence Tao’s exam, with a certain hindsight:

Talk about primes in arithmetic progressions.

🙂