Written: Oct. 26, 1998
One of the best (on many-levels!) talks that I have ever heard was Dave Bayer's invited talk, delivered at the MSRI workshop on Symbolic Computation in Geometry and Analysis (Oct. 12-16, 1998, Oct. 13, 9:30-10:30, soon to be posted at the MSRI site www.msri.org ). In addition to the mathematics itself, that while beautiful, will probably not influence my own research very much, the meta-mathematics, the rich metaphors, and the general style of DOING and LOVING mathematics, certainly will.
One very wise suggestion was to take a break from hiking, and pause and enjoy the view. Most of us, myself included, are too busy trying to prove theorems, and leave too little time for retrospection.
But, perhaps the most useful lesson that I got out of Dave Bayer's outstanding talk was the advice that it sometimes could be useful to abandon the sacrosanct obsession, that we mathematicians have, with minimality. In other words, we are slaves of Occam's razor. We always want the shortest possible proof, the sharpest estimates with as few assumptions as possible, a canonical base that is minimal, the most succinct formula, the most efficient algorithm etc.
This was even quantified by Gregory Chaitin who defined a program to be `most elegant' if it is as short as possible.
In Dave Bayer's talk he mentioned, that he himself, as the pioneering co-developer (with Mike Stillman) of the Macaulay system, was always using Groebner bases because these are canonical and, in a certain sense, minimal (if they are reduced). While COMPUTATIONALLY, of course, Groebner bases (the amazing brainchild of Bruno Buchberger), are the bases of choice, for theoretical development, there may be non-minimal, and not-necessarily-canonical bases that may be better.
It so happened that Dominique Foata and I have spent the last summer trying, so far in vain, to prove Mark Haiman's notorious (n+1)^(n-1) conjecture. Our approach was to construct explicitly (recursively) a Groebner basis for the relevant ideal. But the Groebner basis seems to be a mess. Dave Bayer's advice gave us renewed hope, and now we are searching for other, not necessarily canonical, and not necessarily reduced, bases, that would, who knows?, prove the conjecture.
So Sometimes (mathematical) FAT is BEAUTIFUL, and LONG can be SWEET.
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