Tuesday, April 24, 2007

Edinburgh

Just got back from a workshop on Geometry and Algorithms in Edinburgh. Was there any Fourier analysis? But of course! What would geometry and algorithms be without it?

Unfortunately, though, it's hard to get it into one's talk in 50 minutes, since one has to cover the geometry and algorithms parts. Guy Kindler mentioned some Fourier analysis in connection with UGC-hardness for Max-Cut. Assaf Naor talked a bit about his upcoming magnum opus with Manor Mendel on "Metric Cotype" which will appear in the Annals of Math. That paper is pretty much all Fourier analysis for functions $f : {\mathbb{Z}_m}^n \to \mathbb{R}$, but with $m > 2$. And Robi Krauthgamer came closest to actually doing some Fourier analysis in his slides, while talking about his work on the communication complexity and embeddability of Edit Distance. Unfortunately, his paper (joint with Alexandr Andoni) doesn't seem to be available yet.

I did my best to get there while talking about the parallel repetition/foams stuff I talked about at the ACO Seminar in March, but no dice. I do think the Fourier analysis we used in that result was pretty cute though; basically: (1) Talagrand's Lemma from Homework 5 Problem 2; and (2) the reversed form of the Hypercontractive Inequality, namely:

If $f : \{-1,1\}^n \to \mathbb{R}^{\geq 0}$ and $-\infty < q \leq p \leq 1$, then $\|T_\rho f\|_q \geq \|f\|_p$ so long as $0 \leq \rho \leq \sqrt{(1-p)/(1-q)}$.


You know -- just like the usual inequality, but reversed.

1 comment:

myinvestorsplace said...

Viewing the Fourier transform as a limit process allows us to apply Fourier analysis in these cases and treat these cases as we would more conventional cases.

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