Bill (1946-2012)

I still remember the aura of mystery that surrounded Bill Thurston’s work when I first encountered it as a graduate student in a past life: a late afternoon seminar on the Travaux de Thurston sur les surfaces, a reading course on the notes The Geometry and Topology of Three-manifolds. Laminations with transverse invariant measure? A hyperbolic structure on the figure-eight knot complement? How many hours spent poring over those diagrams and drawing new ones, until I convinced myself (rightly or wrongly) that I could see what he was talking about. It was unlike anything I had ever seen in mathematics.

Later, sometime in the early 1980s, for a while I attended his group’s research seminar at Princeton (called “graduate course” there). The topic was compactifying the moduli space of postcritically finite rational maps, and he was developing it during the course, using the audience as a sounding board.  Laminations of the disk were involved, and at some point I joined his students and postdocs in computing examples.  My notes on this have been lost; I just checked Bill’s Mathscinet list, and did not find a publication corresponding to this. Since, like myself, the other participants had other main interests, it could be that some beautiful mathematics has vanished.

 That’s the risk with doing things in “Thurston mode”: for it to become a permanent part of mathematics, somebody else has to understand it well enough to be able to record it for people who operate at a “normal level”, and that didn’t always happen. Implementing parts of a "Thurston program" has been a nontrivial creative activity in itself, inspiring the careers of many.

Bill was aware of this problem; he appears to have given a lot of thought to the gulf between understanding a mathematical situation intuitively/geometrically and the need to communicate  (and record) it in a necessarily downgraded symbolic version:

Mathematics is commonly explained and recorded in symbolic and concrete forms that are easy to communicate, rather than in conceptual forms that are easy to understand once communicated. Translation in the direction conceptual --> concrete and symbolic is much easier than translation in the reverse direction, and symbolic forms often replace the conceptual forms of understanding. (Mathoverflow)

Here is Bill’s comment on the real reason mathematics is important:

The product of mathematics is clarity and understanding. Not theorems, by themselves. Is there, for example, any real reason that even such famous results as Fermat's Last Theorem, or the Poincaré conjecture, really matter? Their real importance is not in their specific statements, but their role in challenging our understanding, presenting challenges that led to mathematical developments that increased our understanding.

Elsewhere he said “the goal of mathematics is to develop ways for humans to see and think about the world”.  But, in that case, it is not enough for one human (or a handful) to reach such “increased understanding”; the work is not finished—not really useful as a new way to “think about the world”—until it reaches a form in which (at least) scientifically educated non-mathematicians can make sense of it; but by that time it is no longer research, and mathematicians have moved on.

(Think, for instance, of the recent proof of Thurston’s Geometrization Conjecture in the style of Hamilton and Perelman. How long until the “understanding” implicit in this approach can be put into a form that would make sense to a student--or an expert-- in Geometric Topology, or to a physicist? Who is going to do the work?)

Bill was a unique mathematician, and his passing at a point when he was still contributing so much saddens geometers everywhere. The memorial page set up by the Cornell mathematics department has links to sources including some of his views on mathematics, like this one:

Mathematics only exists in a living community of mathematicians that spreads understanding and breathes life into ideas both old and new. The real satisfaction from mathematics is in learning from others and sharing with others.

Exactly.  You can do it (up to a point) while separated from your `tribe’, but it’s no fun. And while all mathematicians know this is true, I hadn't seen it stated so precisely and eloquently until now. We'll miss him.

First Day (Part II)

In the afternoon, it’s time to meet the graduate students. I walk into class…and count nine people there (all men). That’s good, since only six are enrolled in the course. My graduate student is among them, sitting in the front row. I recognize a couple of faces from last year’s graduate course; also good. This is a Ph.D-level course (second-year graduate), so nine is a very decent number.

The focus of the course is a “hot topic” in my field. Ten years ago, a program to solve a famous problem in one area of mathematics using techniques from another area was brought to fruition in unexpected and spectacular fashion, by a brilliant individual working alone (the kind of thing that only happens in math, I think.) The techniques involved in proving that theorem are very powerful, not fully developed, and still have the potential to prove interesting results in both fields; it’s the kind of topic where a beginner can still find a good problem and make a contribution.  “Regardless of what your main field of interest is, having a paper in this area would be a good thing”, I tell them. 

On the other hand…since the topic bridges two areas, having some knowledge of both at the beginning graduate level would be highly desirable, right? So I ask for a show of hands. How many have taken a basic course in area A? Three hands shoot up. A fourth student says `I took it in the Physics Department, does that count?’ Now, that could take us far afield; but I just smile and say, `sure’. How many have a basic course in area B behind them? Four hands (different ones). Okay, so my first mission here is not to scare them away. When I was a student, if something was new and difficult it was impossible so scare me off; mastering new things is what it’s all about, even if it’s hard work. But by now I know from experience that not everyone is like that.  I tell them “well, you’ll have to do some independent reading of background material.” And I promise I won’t get too technical regarding area B.

In fact, the requirement for a grade is minimal: I want them to give a talk on the topic of the course—either present a result found in one of several monographs in this area, or one from a recent paper. On the other hand (I tell them) the more ambitious students should be thinking in terms of finding a good research problem in this area by the end of the year. “You’re graduate students, and what do graduate students do? They ask questions” (hint, hint).  “Some of them will be good questions” (they smile). “And, since you have no experience, they’re not likely to be questions that would occur to me, or to someone already working in the area”.  That’s their edge: being able to ask new questions, out of sheer ignorance.

I start slowly, with a survey of the main results I would like to focus on during the course, including simple examples one can do “with bare hands” (that is, on the back of an envelope).  I get one good, creative question (from the guy who took it in Physics). “No, I don’t think this has been done”, is the answer. Then I move on to a very simple result that can be proved in the same spirit as more difficult theorems, and where the techniques appear in their most basic form. This is standard “area A” material. Unfortunately area A can get technical very quickly, and in the back of my mind is the fear of discouraging those who are just planning to sit in to get informed. I could do the “big ideas” thing, and never get technical; but that would be cheating them of any chance to work in this area. The right balance between ``technical” and “big ideas” (in lecture) is not a trivial one to find.

When the lecture is over, I chat with my student for a few minutes. He tells me he has a new job, as an instructor at a local community college. That’s good, but it means he won’t have time to take any graduate courses for a grade. He is at the thesis-writing stage, anyway (Master’s degree). He tells me he is “just about finished” with writing up the derivation I asked him to do.

First Day

It’s the perfect day.  Pleasantly warm, comfortably dry. The students are back, scurrying about campus looking for their classrooms; rested, happy, excited, expectant. Lots of skimpy shorts and tight-fitting t-shirts (they clearly had a fun summer), so the professors are in a good mood, too. Sure, it is mildly annoying to get to the campus at 9 and have to look for a spot on the staff lot (must be the graduate students), or to deal with long waits at the coffee shop. But it’s good to see them back, they’re our raison d’etre, after all. Not to mention that some of that sexual energy can’t help but radiate to the environment.

I love the first day of classes. I’ve written my syllabi, posted them on the web site. I checked that my textbooks are already at the bookstore (not always the case). I’ve written down notes for my first few lectures. Not too detailed—you have to leave room for spontaneity in the delivery; just enough to give the impression of flowing naturally, without embarrassing moments of hesitation. I rarely have to look at my notes, and students are strangely impressed by that. I’ve secured a set of brand new markers (hate the damn glass boards). About ten minutes before class starts, I close the laptop to collect my thoughts. This will be the term when I’ll be less professorial and more ``approachable”; when I’ll just talk to them, and they’ll respond as curious students of the Academy.  Shouldn’t be hard, it’s a small class.  Fortified by optimistic thoughts, I walk into the classroom two minutes early…

And sitting there are six students; another walks in a moment later. I wonder if my face betrays my feelings, and just in case, I casually move to the business at hand. I introduce myself, and using the online roster (they look like their pictures) try to learn their names.  I knew the enrollment was nine, so seven on the first day is not unreasonable. Still, it is hard not to be taken aback by the reality of it. Mind you, this is not a graduate seminar. It is “Introduction to Advanced Mathematics”, meant for sophomores and juniors who want to learn any math beyond Calculus; our “proofs course”.  I taught it last fall to a class of twenty-four, and it went well. There are two other sections of the course this fall, and both are full, with enrollments of thirty.

I can only conjecture as to the reason, and it goes like this. The science-motivated math majors take the honors section. The other sections are populated by teaching-oriented majors, computer science majors, and engineers for whom this class is a requirement. In practice, most are seniors, trying to clear this last, pesky obstacle to graduation. For somebody like that, the guy with a reputation like “you learn a lot, and he’s friendly; but he expects you to work really hard” would hardly be a first choice. Better take the section where you can reliably expect a B for going through the motions. Math isn’t their interest, after all; so one can’t blame them for optimizing their use of time. Now, the natural question is: why do these majors require students to take the “proofs course” designed for mathematicians? It makes no sense, but that will be for a future post. Today I’m teaching.

And it goes well. I’m not doing anything hard: logical connectives, truth tables, equivalent propositional forms, tautologies.  I tell them about Godel’s Incompleteness Theorem, twin primes, and the conjectures of Goldbach, Catalan, and Collatz. (All this to illustrate the difference between “truth” and “provability”).  To another audience this might seem mysterious, intriguing, or fascinating. Maybe to some of these students, too. But surely some of the engineers are thinking “and this is good for…?”  And they don’t ask, unfortunately.  But I did point out that the text includes discussions of Turing machines, P vs. NP and RSA cryptography. (Some of them have heard about those, so maybe I have a `hook’ there to draw them in.)  They all seem engaged (it’s a small room), sometimes nodding or smiling. I ask them to work on a problem on the spot, and they do it. I ask questions, and get intelligent answers. It’s a good start.

I always get a `high’ from teaching, and it can take half an hour or more to `come down’ enough to be able to think about anything else. Knowing this, I walk out of the building, into the beautiful sunny day and the scurrying half-dressed students, headed for the coffee line.