Opinion 104: The Shocking State of Contemporary "Mathematics", and the Meta-Shocking Fact that Very Few People Are Shocked

By Doron Zeilberger

Written: Oct. 28, 2009

I just came back from attending the 1052nd AMS (sectional) meeting at Penn State, last weekend, and realized that the Kingdom of Mathematics is dead. Instead we have a disjoint union of narrow specialties, and people who know everything about nothing, and nothing about anything (except their very narrow acre). Not only do they know nothing besides their narrow expertise, they don't care!

The "meeting" was not really a meeting. It was many mini-meetings! 22 of them, running in parallel and in complete oblivion of each other. All that they shared was the coffee, tea, and donuts. That's a little reassuring that an algebraic combinatorialist has at least one thing in common with an algebraic geometer, a q-serieser, and a Heat-kernel group theorist: they all drink coffee (or tea), and eat donuts! But that's about it.

The highlight of such a conference used to be, and still should be, the invited hour address, where there are no conflicts, and that every participant is expected to attend. I believe that I was one of the few that went to all of them (another one is my Rutgers colleague Rodi Tumulka). Here is an exact count of the attendance (out of 380 registered participants):

• Sat. 11:00-11:50am: Laurent Saloff-Coste's talk " Subelliptic heat kernel measures and holomorphic functions on complex Lie groups": 65 people.
• Sat. 2:00-2:50pm: Kevin Payne's talk "PDE of mixed type: The twin challenges of globalization and diversity": 66 people.
• Sun. 11:00-11:50am: Robert Vaughan's "Diophantine approximation to curves and surfaces": 76 people, (the relatively high attendance is easily explained by the fact that Vaughan was playing a "home game", and possibly some of his colleagues who were not participants came to cheer him on.)
• Sun. 2:00-2:50pm: Michael Kiessling's talk "N-body problems in relativity" (34 people, ironically, Kiessling's talk was the most accessible to non-experts, and by far the best of the four).

Not that you have to feel bad for the speakers! They are just as narrow-minded specialists as the rest of the participants. The 4 by 4 0-1 matrix

M[i   ,   j]=1 if Speaker i attended Speaker j's talk,
M[i   ,   j]=0 if Speaker i did NOT attended Speaker j's talk,

is a very sparse matrix, very close to the identity matrix. (I know that M[4,2]=1, since I saw my colleague Michael Kiessling at Kevin Payne's talk, but I am almost sure that the other non-diagonal entries are all 0).

You can't really blame the audience for not showing up, since they were probably burnt out from countless previous invited talks where they didn't understand a word, or from reading the very technical abstracts of the current talks. Most speakers have no clue how to give a general talk. They start out, very nicely, with ancient history, and motivation, for the first five minutes, but then they start racing into technical lingo that I doubt even the experts can fully follow.

Please! Expand these first five minutes into fifty minutes, tell us about the history, background, motivation, and you don't have to even mention your own results. For example, Robert Vaughan could have gone into full details about Diophantus, and terminated with Khinchin's classical result, only mentioning that his work continues in the same vein. Kevin Payne could have reminded us, very slowly, with a simple example, what is a (i) PDE (ii) what are elliptic, parabolic, and hyperbolic types, etc. and never mention his own results. There is no way that in fifty minutes I can learn what took him twenty (or whatever) years to master.

One culprit is the pernicious laptop, it should be outlawed! It encourages the speaker to pass the cognitive speed-limit by orders of magnitude. Sure enough, the best invited talk was Michael Kiessling's talk that used the ancient technology of overhead projector, and it would have been even better if he only used the blackboard, and it would have been better still if he didn't use anything, just told us a story.

Here are some suggestions on how the American Mathematical Society can help the general mathematical education of the mathematical masses, and restore the mathematical culture, and save it from the dangerous fragmentation and over-specialization. (And ditto for the International Mathematical Union for the forthcoming ICM, in which once again, 20% of the plenary and invited talks would be excellent, but the rest incomprehensible). Here are some suggestions.

• Every invitation to give an invited talk should be conditional, and the talk should be rehearsed in front of a trial audience (preferably of graduate students), as well as a person known to be an excellent speaker (e.g. Ron Graham, Andrew Granville, Joe Gallian, Carl Pomerance, Avi Wigderson, and quite a few others). They should get feedback, and not allowed to speak unless pre-approved by the coach and the trial audience.
• One of the current reasons for invited talks (both AMS and ICM) is to bestow an honor. I used to be against the AMS Fellows proposal, but if the AMS would agree that it would only have good speakers speak, and bestow honors otherwise (by naming the greatest Lie-Group-Categorical-Sheaves-Heat-Kernelist in the world an "AMS fellow", rather than letting him bore the audience who never heard of any of those words with his latest results) I would support the AMS Fellows program.
• The AMS shouldn't be so greedy! Like many "non-profit" organizations, it still has the capitalist mentality of maximizing revenues rather than knowledge. That's why they try to pack as many talks and as many special sessions as possible, so that they can brag "record attendance expected". In fact, unlike the Sectional meetings, where the invited talks still do not conflict with the Special Sessions, during the Annual meetings, in order to maximize profit, and have "record attendance", all invited talks, except the Gibbs Lecture (at night) and the First Colloquium talk, and the Prize Session, conflict with lots of Special Sessions. So, please, AMS, don't be so greedy, and keep the invited talks, and especially ALL three Colloquium talks without conflict. Now you may say, "no one will show up at any rate, since these talks are so technical". Sure, but, if you follow the above suggestions you can make them good and accessible.

But, not only the AMS (and the IMU) should work hard to revive Mathematics as a culture, every department should! Analogous to the poor attendance at AMS invited talks is the poor attendance at the Mathematics Department's Colloquium. In the good old days, everyone went, it was part of the ritual. Now only a handful of specialists who are close to the speaker's narrow specialty show up (plus, in my own department, five or six old-timers). Even I stopped going regularly, since so many talks are over my head. But if people will make an effort, and be trained to give accessible and engaging talks, things would change. We can even make "general mathematical knowledge outside one's narrow specialty" a criterion for tenure and promotion, and every candidate should have to pass an oral exam. We can also make attendance at colloquium talks (once they get better) mandatory, like faculty meetings.

For the good of future mathematics we need generalists and strategians who can see the big picture. Narrow specialists and tacticians would soon be superseded by computers.

So let's get to work, and try to become mathematicians rather than topological algebraic Lie theorists, algebraic analytic number theorists, pseudo-spectral graph theorists etc.

Added Nov. 10, 2009: Read Edmund Harriss (alias Maxwell's Demon) eloquent defence of the laptop