Written: Oct. 10, 2002

It is all too familiar. You go to a colloquium talk, supposedly meant for a "general mathematical audience" and "graduate students", and you don't understand a word. OK, being clueless for one hour per week is not the end of the world, but what if you go to a National Meeting?, or worse, THE BIG HAPPENING of mathematics, ICM, and try to go to ALL the plenary talks?

I recently came back from ICM 2002, in Beijing, and once again was struck by the fact that the Tower of Babel hit us mathematicians particularly hard. In addition to the intrinsic compartimization, over-specialization and splintering of math, most mathematicians (even, or perhaps especially, the most prominent) have no idea how to give a general talk.

Luckily, there are exceptions. Noga Alon's talk "Discrete Mathematics: Methods and Challenges" was by far the best talk. In addition to the lucid and very accessible presentation, the transparencies were very readable, even from far away, and he spoke slowly and clearly, so everybody, regardless of specialty or native language, could easily follow (and be fascinated!).

The second-best talk (in the first week, I couldn't stay for the second week, and I am sure that at least Ed Witten's talk was excellent, as usual) was also in discrete math, by Shafi Goldwasser. Both the content and presentation were fascinating and very accessible (in spite of the few power-point glitches, that made her exclaim: "Computers hate me, and I am a computer scientist"). The only minor flaw was that she talked too fast.

If you think that it is not surprising that talks in Discrete Math are accessible, since it is a young field, and hence requires less background, then you should go and hear Michael Hopkins. His talk was first-rate, and even I, who has "topology anxiety", understood almost everything. So it is possible to give lucid and accessible talks, even in topology. Besides the math, Michael also had great one-liners, that could serve us all well, two of them being "It is very important in algebraic geometry not to divide by anything you don't have to divide by" (probably good advice in all areas), and "The reason it was so hard to prove it before was that we were asking for too little".

The next-in-line in my ratings (of the first 11 plenary talks) is a tie between David Mumford and Douglas Arnold. Mumford's talk was insightful and entertaining, and had many good quotes, my favorite being: ``Images of the real world are Renormalization-Group fixed-points''. Douglas Arnold showed very convincingly that Numerical Analysis has gotten to be very sophisticated (differential complexes and de Rahm cohomology!), and that good a priori error analysis can save lives and avoid catastrophes.

Neither good nor bad (as far as quality of exposition goes, I am sure that all the talks presented excellent mathematics) was Victor Kac's talk. It was a bit too technical, and dwelt too much on the speaker's own research, rather than giving an overview of the field. But it had many good moments (like some beautiful q-identities) and the grande finale that claimed that, notwithstanding Genesis, Light was created on the 4th day of Creation.

Bad, but not terrible (once again, from the point of view of exposition) were the talks by Luis Caffarelli and Uffe Haagerup. I have to admit that by then I got a little burnt-out, and sneaked out, to the exhibits, for about half of each of these talks.

The third-worst (among the 11 talks of the first week)
was the first, given Wed. morning by the brand-new
Fields medalist Laurent Lafforgue, who talked
about "Chtoucas des Drinfeld" and the Langlands Programme.
He started by lamenting that French is no longer the
lingua franca of math, and hypocritically cited "diversity"
(if I had to name a culture that abhors diversity it is the French,
they want everyone to Only speak French). As a
"compromise" he talked in (excellent) English, but
the transparencies were in French and Chinese.
Frankly, Laurent, I don't think that it matters what
natural language you choose for your talks,
since no one understands them anyway,
except perhaps ten experts (out of more than
4000 potential listeners). The only interesting part
was the human factoid that the initial manuscript of
600 pages contained a gap, that luckily got fixed
by doing away with it, like in
Wiles's FLT proof, but in this case Lafforgue did not
need Richard Taylor (I was told later that Lafforgue behaved
very honorably when he found the mistake in the earlier draft
and offered to withdraw his acceptance
of the IHES offer, unless he
fixed it, which he did). On second thought, there was at least
one mathematical statement that I *did* understand.
Sometime during the talk, he had the equation 2=1+1
(when he discussed the previous work of Drinfeld for rank 2,
where there are only two partitions: 2 itself and 1+1).

By the way, the second Fields medalist, Vladimir Voevodsky, gave an excellent "informal seminar", that was much more accessible, and it is clear that he is not just a brilliant specialist, but posseses great mathematical culture.

The second-worst plenary talk was by Yum-Tong Siu. I am sure that he made a great effort to only require, as he claimed at least four times during the talk, "minimal background", and he also used (not intentionally, it was clear that he is a very nice and warm human being) the intimidating phrase "everybody knows...", e.g. about the Kodaira vanishing theorem. Sorry, Prof. Siu, but at least I never heard of it. But you can still get something from even such an incomprehensible talk, because of the "marginalia". These can start a sequence of amusing day-dreams. Siu mentioned (at least twice) that Andre Bloch did his great work in a "mental asylum". That lead me to ponder about other mathematical murderers I have known (or heard of, like the Unabomber). Also he kept quoting familiar names, of people I know personally, like Harold Boas (and his collaborator Emil Staube), and Harold Boas lead, in my mind, to his father, Ralph Boas, whom I remember fondly as the editor who accepted my first single-authored paper.

However, the name that pleased me the most amongst those mentioned by Yum-Tong Siu was that of Marc Levine, who was my colleague at Penn, back in 1982-1983, and was coordinating the Calculus sections, when he was tenure-track, and I was visiting. It was nice to hear his name so often, also at the opening ceremony, for work connected to the work of one of the Fields-medalists. He also gave an invited talk. So Marc has certainly "made it to the Big Time". In addition for being happy for Marc, I was also gleeful that once again Penn is probably kicking itself, for not giving Marc tenure in the early eighties. If there is one math department that I really don't like, it is Penn's (as a whole(!), some of my best friends teach there, e.g. Herb Wilf), whose QP (Quality/Pretension) quotient is very low (in my opinion). Penn-Math also initially denied Vaughan Jones tenure, and once he came up with the Jones invariant, he didn't need them (and went to UCB).

The absolute worst talk (once again, expositorily speaking) was by Gang Tian. Here there was no room at all for productive day-dreaming. Tian's talk was extremely technical, very dry, and the transparencies were typed with small font, that probably made it hard to follow even for the ten or whatever experts in the audience.

I also went to many of the invited talks. Once again, the expository quality was variable. All the talks in Theoretical Computer Science (sec. 15) were really excellent (Ran Raz's talk, in particular, was (probably) the absolute best in the whole ICM, at least it was the best among the talks I went too, counting the plenary). The Combinatorics talks were also uniformly good and lucid. I also enjoyed very much the 3 talks on Probability (sec. 10) that I went to, by Gerard Ben Arous, Ofer Zeitouni and Kurt Johansson. Johansson's talk was particularly attractive to me because of the beautiful interplay of combinatorics and probability. It was very lucid and the transparencies were gorgeous. I was sorry that I had to miss Yuval Peres's interesting talk, but I read the paper, and it is really exciting, so I am sure that he also gave a great talk.

I tried to go to a random sample of talks in other fields. Eitan Tadmor's talk, in numerical analysis (16), was, if not exciting, at least very well-presented and lucid. Quite disappointing were the two talks in the Mathematical Physics section (13) that looked promising from the abstracts, by Jean-Pierre Eckmann (whose book on chaos I really love), and the brilliant wunderkind, Nikita Nekrasov, whom I heard speak, very clearly, in the Gelfand seminar, but whose talk was way too technical and boring.

Hugh Woodin's talk (in section 1 (Logic)) was out of this world. Superficially it was dry and boring, yet I wasn't bored for a second, because of the greater-than-life personality of the speaker. First, he was dressed-to-kill (which was particularly noticeable in 2002, where even plenary speakers are dressed like slobs). He reminded me of old-time movie heroes like Clark Gable. He also mentioned, with great glee, "Woodin cardinals" (to the point that he had to apologize for mentioning them so often). So even though I hardly had a clue of what he was talking about, I lasted almost to the end.

In the algebra section (02) I went to Zlil Sela's talk and the last part of Cheryl Praeger's. Zlil's content was fascinating. He proved long-standing conjectures of Alfred Tarski, but his presentation and lecture-style were only average. On the other hand Cheryl Praeger's talk about Permutation groups, was really tops! I would rate it second-best only slightly below Ran Raz's. Cheryl's talk also illustrated the power of patience, since the main result took ten years to prove! So, like Michael Hopkins, here is another proof that you don't have to be a lousy speaker just because you are a pure mathematician.

I also enjoyed the ad-hoc talk by Madhu Sudan on list-decoding, that was very lucid. The public talk by John Nash was, of course, a happening, and quite accessible, even though it was not originally intended for a general public. Mary Poovey's public talk (Thur. evening) was extremely stimulating and very depressing. She very eloquently made the case that the American Capitalist "free market" society is far from perfect, and not only statistics, but also math, can be abused in the hands of greedy crooks.

The organization was almost perfect. It would have been perfect if they had a crossing-guard to help cross the very busy street between the hotel and the conference.

Opinion 48