Comments for E. Kowalski's blog http://blogs.ethz.ch/kowalski Comments on mathematics, mostly. Wed, 25 Jan 2012 12:23:22 +0000 hourly 1 http://wordpress.org/?v=3.3.1 Comment on Random CIRM happenings by valuevar http://blogs.ethz.ch/kowalski/2012/01/23/random-cirm-happenings/#comment-114306 valuevar Wed, 25 Jan 2012 12:23:22 +0000 http://blogs.ethz.ch/kowalski/?p=3062#comment-114306 A native Spanish speaker suggests (based on five minutes spent with the book in Luminy) that it is a very pretentious thriller by a young writer (or a shallow novel with pretensions of being a thriller, by the same). A native Spanish speaker suggests (based on five minutes spent with the book in Luminy) that it is a very pretentious thriller by a young writer (or a shallow novel with pretensions of being a thriller, by the same).

]]> Comment on Explicit growth for generating subsets of SL_2 over finite fields by Kowalski http://blogs.ethz.ch/kowalski/2011/12/13/explicit-growth-for-generating-subsets-of-sl_2-over-finite-fields/#comment-107849 Kowalski Mon, 02 Jan 2012 17:25:32 +0000 http://blogs.ethz.ch/kowalski/?p=3001#comment-107849 Thanks! I guess the current bound (which is now the amazingly large number $latex 2^{-32}$, in the latest version) is not as funny, but the "how large is $latex p$ large enough" is still $latex 2^{2^{46}}$... Thanks!

I guess the current bound (which is now the amazingly large number 2^{-32}, in the latest version) is not as funny, but the “how large is p large enough” is still 2^{2^{46}}…

]]> Comment on Explicit growth for generating subsets of SL_2 over finite fields by Victor Miller http://blogs.ethz.ch/kowalski/2011/12/13/explicit-growth-for-generating-subsets-of-sl_2-over-finite-fields/#comment-107834 Victor Miller Mon, 02 Jan 2012 16:57:34 +0000 http://blogs.ethz.ch/kowalski/?p=3001#comment-107834 Emmanuel, Happy New Year, and thanks for your wonderful blog. Your lower bound for $latex \lambda_1$ caused me to giggle uncontrollably for about 15 seconds (and I'm sure that it wasn't easy to get the bound!). Victor Emmanuel, Happy New Year, and thanks for your wonderful blog. Your lower bound for \lambda_1 caused me to giggle uncontrollably for about 15 seconds (and I’m sure that it wasn’t easy to get the bound!).

Victor

]]> Comment on What’s special with commutators in the Weyl group of C5? by Kowalski http://blogs.ethz.ch/kowalski/2011/08/30/whats-special-with-commutators-in-the-weyl-group-of-c5/#comment-106218 Kowalski Tue, 27 Dec 2011 07:48:57 +0000 http://blogs.ethz.ch/kowalski/?p=2739#comment-106218 Thanks for the comments! I've already seen talks on this (including one by you...) and since I like any quantitative aspects of interesting discrete groups, I certainly hope to learn more of this one day. Thanks for the comments! I’ve already seen talks on this (including one by you…) and since I like any quantitative aspects of interesting discrete groups, I certainly hope to learn more of this one day.

]]> Comment on And then there was one in 744 by Kowalski http://blogs.ethz.ch/kowalski/2011/12/16/and-then-there-was-one-in-744/#comment-106216 Kowalski Tue, 27 Dec 2011 07:47:16 +0000 http://blogs.ethz.ch/kowalski/?p=3021#comment-106216 That's certainly the right approach indeed... I had already started implementing this idea, for the expansion bound à la Bourgain-Gamburd, since it's in fact the least efficient part of the argument when it comes to explicit bounds. I'll report soon on what I can get now... That’s certainly the right approach indeed… I had already started implementing this idea, for the expansion bound à la Bourgain-Gamburd, since it’s in fact the least efficient part of the argument when it comes to explicit bounds. I’ll report soon on what I can get now…

]]> Comment on What is the Cayley graph of the cyclic group of order 2? by Richard Kent http://blogs.ethz.ch/kowalski/2011/11/27/what-is-the-cayley-graph-of-the-cyclic-group-of-order-2/#comment-105284 Richard Kent Thu, 22 Dec 2011 21:31:11 +0000 http://blogs.ethz.ch/kowalski/?p=2980#comment-105284 As a topologist, I think the right definition is Serre's, as it allows you to avoid talking about symmetric generating sets and just take the Cayley graph to be the 1-skeleton of the universal cover of a presentation 2-complex (the complex with 1-skeleton a rose on your generating set and 2-cells for the relations). As a topologist, I think the right definition is Serre’s, as it allows you to avoid talking about symmetric generating sets and just take the Cayley graph to be the 1-skeleton of the universal cover of a presentation 2-complex (the complex with 1-skeleton a rose on your generating set and 2-cells for the relations).

]]> Comment on What’s special with commutators in the Weyl group of C5? by Danny Calegari http://blogs.ethz.ch/kowalski/2011/08/30/whats-special-with-commutators-in-the-weyl-group-of-c5/#comment-104993 Danny Calegari Wed, 21 Dec 2011 18:13:34 +0000 http://blogs.ethz.ch/kowalski/?p=2739#comment-104993 Maybe the moment has passed, but I thought it might be worth making a few comments. Obviously you are interested in finite groups above, but commutators are very important in the theory of infinite groups too. The *commutator length* of an element (of the commutator subgroup) is the least number of commutators whose product is the given element, and the *commutator width* is the supremum of this number over all elements. In the world of infinite groups, the "typical" phenomenon is that commutator width is infinite. In fact, one can define the *stable commutator length* of an element g in [G,G] to be the limit of cl(g^n)/n. In a (nonabelian) free group F (in fact, in a torsion-free hyperbolic group), the stable commutator length of *every* element of [F,F] is positive - in fact, it is at least 1/2. Stable commutator length is closely related to 2-dimensional bounded cohomology, and has many interesting connections to geometry, dynamics, etc.; see eg. http://www.its.caltech.edu/~dannyc/scl/toc.html Maybe the moment has passed, but I thought it might be worth making a few comments. Obviously you are interested in finite groups above, but commutators are very important in the theory of infinite groups too. The *commutator length* of an element (of the commutator subgroup) is the least number of commutators whose product is the given element, and the *commutator width* is the supremum of this number over all elements. In the world of infinite groups, the “typical” phenomenon is that commutator width is infinite. In fact, one can define the *stable commutator length* of an element g in [G,G] to be the limit of cl(g^n)/n. In a (nonabelian) free group F (in fact, in a torsion-free hyperbolic group), the stable commutator length of *every* element of [F,F] is positive – in fact, it is at least 1/2. Stable commutator length is closely related to 2-dimensional bounded cohomology, and has many interesting connections to geometry, dynamics, etc.; see eg. http://www.its.caltech.edu/~dannyc/scl/toc.html

]]> Comment on And then there was one in 744 by Harald http://blogs.ethz.ch/kowalski/2011/12/16/and-then-there-was-one-in-744/#comment-104467 Harald Mon, 19 Dec 2011 14:14:54 +0000 http://blogs.ethz.ch/kowalski/?p=3021#comment-104467 Emmanuel, >in order to avoid dealing with elements of trace 0, I >first had to replace the generating set H with H\cdot >H, alas... Surely, whether your goal is to get a bound on the diameter or a bound on the spectral gap, it is best to prove a theorem of the following form? Let A, B be sets of generators of G=SL_2(F_p). Then either |BBABB^{-1}BBABBBABB| > |A|^{1+\delta} or (same word) = G. I just made that word up - it's just an example. The point is that you could use a word with many occurences of B and few occurences of A. That way, if you are bounding the diameter of the Cayley graph Gamma(G,S) (say), you can initially set A = S, B = S, and then A = word(S,S), B = S, and then A = word(word(S,S),S), B = S... In other words, one of the sets is kept small and cheap - and is used to escape from special situations (such as zero trace). Also notice that the tripling Lemma (Ruzsa) does involve a certain cost, so, if you care about constants, you are better off not using it if at all possible. Emmanuel,

>in order to avoid dealing with elements of trace 0, I >first had to replace the generating set H with H\cdot >H, alas…

Surely, whether your goal is to get a bound on the diameter or a bound on the spectral gap, it is best to prove a theorem of the following form?

Let A, B be sets of generators of G=SL_2(F_p). Then
either

|BBABB^{-1}BBABBBABB| > |A|^{1+\delta}

or

(same word) = G.

I just made that word up – it’s just an example. The point is that you could use a word with many occurences of B and few occurences of A. That way, if you are bounding the diameter of the Cayley graph Gamma(G,S) (say), you can initially set A = S, B = S, and then A = word(S,S), B = S, and then
A = word(word(S,S),S), B = S…
In other words, one of the sets is kept small and cheap – and is used to escape from special situations (such as zero trace).

Also notice that the tripling Lemma (Ruzsa) does involve a certain cost, so, if you care about constants, you are better off not using it if at all possible.

]]> Comment on And then there was one in 744 by Kowalski http://blogs.ethz.ch/kowalski/2011/12/16/and-then-there-was-one-in-744/#comment-103925 Kowalski Fri, 16 Dec 2011 18:31:34 +0000 http://blogs.ethz.ch/kowalski/?p=3021#comment-103925 I am pretty certain this is just a coincidence... I am pretty certain this is just a coincidence…

]]> Comment on And then there was one in 744 by Owen Maresh http://blogs.ethz.ch/kowalski/2011/12/16/and-then-there-was-one-in-744/#comment-103922 Owen Maresh Fri, 16 Dec 2011 18:09:17 +0000 http://blogs.ethz.ch/kowalski/?p=3021#comment-103922 I can't help of thinking of the fourier coefficients of the j-invariant. (but this could just be a random coincidence too) I can’t help of thinking of the fourier coefficients of the j-invariant. (but this could just be a random coincidence too)

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