Opinion 111: Congratulations Elon Lindenstrauss, Ngo Bao Chau, Stas Smirnov, and Cedric Villani for winning a 2010 Fields Medal, but PLEASE, don't spend the next 40 years of your life continuing to do the same old (of course very deep but SO) BORING stuff, but instead use your great talents to do research in INTERESTING mathematics (that not only Terry Tao can understand, but ALSO the rest of us).

By Doron Zeilberger

Written: Aug. 25, 2010

The ICM has just promoted four mathematicians from the rank of "excellent mathematician" to that of "truly great mathematician". Being a good mathematical citizen, I tried to understand, at least superficially, what they did. Except for Stas Smirnov, with whose work I was already familiar (and that I find the most exciting, being kind-of-combinatorics), I had no clue. So I turned to Terry Tao's blog entry and got some very general idea.

(BTW, Terry Tao is the greatest mathematician alive today, in spite of, (not because!) the fact that he is a Fields medalist. He understands (and can explain so well) more mathematics than anyone else (80% of the Fields medalists (and 99.99% of the rest of human-mathematics-kind) are not at all mathematicians but rather narrow-minded specialists who happen to be very good in that tiny part (at most 2 percent of mathematics) that they work on). A close second to Terry Tao is Tim Gowers, who also knows a lot!)

I then went to MathSciNet trying to delve more deeply into the details of the work of Elon, Chao, Stas, and Cedric, and barely understood a word (with the exception of Smirnov's beautiful work that I was already familiar with), and felt really dumb.

In order to cheer myself up, I bought last Friday's Israeli newspaper Ha'aretz, and surprise! mathematics was in the front page, something that never happened before! There was a big photo of Elon Lindenstrauss standing in front of a large blackboard (covered, with arcane formulas, and even some Hebrew words (tavnit ribuit)). The article stated that Elon was the first Israeli to ever win a Fields medal, and this is like a Nobel prize, and it is really a big deal, and even the prime minister took a few minutes off his busy schedule usually spent with hanging out with soccer players and celebrity entertainers, to congratulate Elon. The article also mentioned that Elon belongs to an illustrious family (that includes the state comptroller) and that his father is the eminent Israeli functional analysisist Joram Linenstrauss. The latter was quoted saying:

"To be honest, I don't really understand Elon's mathematics. There are probably only twenty mathematicians in the whole world who can really understand his work."

This made me feel much better. Maybe I am not so dumb after all. If one of the greatest functional analysts in the world, whose research area is much closer to Elon's area of Dynamical Systems than to mine, has no clue, what would you except from a simple-minded discretian like myself?

And that is the problem. "Mainstream" Mathematics has gotten so fragmented, so specialized, so out-of-reach-with reality and so boring. While I have (hardly) any clue what the new Fields medalists achieved, I know very well what they did not achieve:

Now here at least I understand the problems. The problems that Elon and company solved (with the possible exception of Stas S.) are so boring. Who cares about the so-called "Fundamental Lemma"? (Ans.: at most 30 people in the whole world). Who cares about a "rigorous" proof of Boltzmann's H-theorem? (Ans.: ditto).

(Speaking of "rigor", this is a soon-to-be-obsolete hang-up of 19th and 20th-century mathematics that did some good, but much more harm by hindering its progress. We often brag about how "useful" mathematics is in science and engineering, true, but usually the scientists and engineers discover it all by themselves (e.g. physicists Seiberg and Witten in topology and engineers Hamming and Golay in coding theory), and Einstein would have easily developed Riemannian Geometry, his way if Riemann didn't do it before).

I was particularly unimpressed by the "almost" solution of the Littlewood conjecture. "Almost" does not count!. The whole concept of "Lebesgue measure" gives me the creeps. It is so artificial and in fact an artifact of mathematicians' superstitious belief in the infinity and their fanatic insistence on (the appearance) of "rigor". "Almost" proving something is often a piece of cake (for example that almost all x wind up at 1 after iterating the 3x+1 map). I am sure that in the case of the Littlewood conjecture it was a major technical feat, so I am not saying that what Elon et. al. did is trivial. Quite the contrary, it is extremely deep, in fact too deep for my taste. It is so deep that I (and 5 billion people take away 20) couldn't care less about.

But, the point of this opinion is not to put you down or discourage you. All the four of you are so brilliant. Such great minds are terrible things to waste on current "main-stream", over-specialized, mathematics, that you have been doing so far.

First and foremost, learn how to program! well!, by yourself, and not just let students do it for you. Once you will learn how to think algorithmically you would be much better off. If you would have taken my Experimental Math class either in 2010 or 2009, or 2008, or 2007, or 2006, and learned how to program Maple (or Mathematica, but not with me), you would have been able do do so much more, than what you accomplished by mere paper-and-pencil.

You would also realize that not all mathematical results could be proven with full rigor, and sometimes one has to settle with semi-rigorous, and even non-rigorous proofs. Look at the great mathematics done by physicists Ken Wilson, Leo Kadanoff, and others when they developed the "renormalization group", and the attempts by some mathematicians to make it rigorous is "who cares?". I admit that sometimes attempt to make things "rigorous" leads to beautiful new insights, and sometimes "non-constructive" approaches (like Furstenberg's ergodic approach to Ramsey theory that lead to insights into purely combinatorial proofs) are worthwhile. But other times attempts to "rigorize" a piece of mathematical physics while technically very challenging, is a futile and pointless exercise.

Please diversify!, expand!, and try to prove (in whatever level of rigor you can master) results that are (i) interesting (ii) that I and my fellow mortals can understand and appreciate.

We have to thank Professor Fields for the 40-year-old upper bound. You still have (at least!) forty years of productive mathematical life. I agree with G.H. Hardy that mathematics is a young man's game (except for the "man" part), but "young" does not mean "under thirty", and not even "under forty". "Young" means, "young at heart", and willing to learn new things and new methodologies and master new technology. If you will follow my advice, I am sure that you would achieve much more than your already very impressive (albeit mostly boring) feats, and your future achievements will not only be technically challending, but also exciting, not just to you and to your thirty cronies, but to all of us common folks.

Opinions of Doron Zeilberger