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As many readers may already know, my good friend and fellow mathematical blogger Tim Gowers, having wrapped up work on the Princeton Companion to Mathematics (which I believe is now in press), has begun another mathematical initiative, namely a “Tricks Wiki” to act as a repository for mathematical tricks and techniques. Tim has already started the ball rolling with several seed articles on his own blog, and asked me to also contribute some articles. (As I understand it, these articles will be migrated to the Wiki in a few months, once it is fully set up, and then they will evolve with edits and contributions by anyone who wishes to pitch in, in the spirit of Wikipedia; in particular, articles are not intended to be permanently authored or signed by any single contributor.)
So today I’d like to start by extracting some material from an old post of mine on “Amplification, arbitrage, and the tensor power trick” (as well as from some of the comments), and converting it to the Tricks Wiki format, while also taking the opportunity to add a few more examples.
Title: The tensor power trick
Quick description: If one wants to prove an inequality for some non-negative quantities X, Y, but can only see how to prove a quasi-inequality that loses a multiplicative constant C, then try to replace all objects involved in the problem by “tensor powers” of themselves and apply the quasi-inequality to those powers. If all goes well, one can show that for all , with a constant C which is independent of M, which implies that as desired by taking roots and then taking limits as .
The first Distinguished Lecture Series at UCLA of this academic year is being given this week by my good friend and fellow Medalist Charlie Fefferman, who also happens to be my “older brother” (we were both students of Elias Stein). The theme of Charlie’s lectures is “Interpolation of functions on “, in the spirit of the classical Whitney extension theorem, except that now one is considering much more quantitative and computational extension problems (in particular, viewing the problem from a theoretical computer science perspective). Today Charlie introduced the basic problems in this subject, and stated some of the results of his joint work with Bo’az Klartag; he will continue the lectures on Thursday and Friday.
The general topic of extracting quantitative bounds from classical qualitative theorems is a subject that I am personally very fond of, and Charlie gave a wonderfully accessible presentation of the main results, though the actual details of the proofs were left to the next two lectures.
As usual, all errors and omissions here are my responsibility, and are not due to Charlie.