Letters to a Young Mathematician (13 page)

In British elementary schools, the educational establishment has managed to get this spectacularly wrong. We now have a highly prescriptive “national curriculum,” and teachers—quite literally—check hundreds of boxes to mark the student’s progress. Can they count to five? Check. Can they add five to three? Check. The assumption is that what matters is their ability to get the answer. But what really matters is
how
they get the answer. I’m old-fashioned enough to believe that either way they have to get the
right
answer; no easy grades for “method” from me. But I am absolutely certain that checking a series of boxes is not the way to teach anyone mathematics.

Dear Meg,

Now that you are on the verge of becoming a fully fledged member of the mathematical community, it’s a good idea to understand what that entails. Not just the professional aspects, which we’ve already discussed, but the people you will be working alongside, and how you will fit in.

There’s a saying in science fiction circles: “It is a proud and lonely thing to be a fan.” The rest of the world cannot appreciate your enthusiasm for what seems to them a bizarre and pointless activity. The word “nerd” comes to mind. But we are all nerds about something, unless we are couch potatoes who have no interests except what’s on TV. Mathematicians are passionate about their subject, and proud to belong to a mathematical community whose tentacles stretch far and wide. You will find that community to be a constant source of encouragement and support—not to mention criticism
and advice. Yes, there will be disagreements too, but generally speaking, mathematicians are friendly and relaxed, provided you avoid pushing the wrong buttons.

Pride is one thing, loneliness another. My experience is that today’s public is much more aware than it used to be that mathematicians do useful and interesting things. At parties, if you admit to being one, you are far more likely to be asked, “What do you think about chaos theory?” than be told, “I was never any good at math when I was at school.” In
Jurassic Park
, Michael Crichton says that today’s mathematicians no longer resemble accountants, and some are more like rock stars.

If so, this is very bad news for rock stars.

Even if people ask you about chaos theory at parties, it is still unwise to explain your latest theorem on semicontinuous pseudometrics on Kähler manifolds to the guy in a leather jacket. (Though nowadays he
might
turn out to be a mathematician. But don’t count on it.) So, despite the public’s newfound tolerance of math, there will be occasions when you want to be with people who understand where you’re coming from. Such as just after you’ve finally proved the semicontinuous case of the Roddick–Federer conjecture on the irregularity of Kähler manifold pseudometrics in dimensions greater than 34.

Science fiction fans go to conventions (“cons,” as they say) to talk to other science fiction fans. Whippet breeders go to whippet shows and compete with other
people who breed whippets. Mathematicians go to conferences to hang out with other mathematicians. Or they give seminars, or colloquia, or just visit.

Our first vice chancellor, Jack Butterworth, once said that no university was worth anything unless a quarter of its faculty was in the air. He intended this literally: air travel, not intellectual high flight. The best way to advance the cause of mathematics is to meet other mathematicians.

If you are lucky, they will come to you. The University of Warwick, founded in the 1960s, became a world-class center for mathematics because from day one it held symposia, year-long special programs in some area of math. (I was once told that “symposium” means “drinking together,” a theory that cannot be rejected out of hand.) But it’s a good idea, and more fun, if you go to
them
. Mathematics, like all the sciences, has always been international. Isaac Newton used to write to his counterparts in France and Germany, but today he could hop on a budget flight and meet them.

Mathematicians get together, usually over coffee; Erd~s said that a mathematician is a machine for turning coffee into theorems. They share jokes, gossip, theorems, and news.

The jokes are mathematical, of course. There is a lengthy compendium of classic mathematical jokes in the January 2005 issue of the
Notices of the American
Mathematical Society
, and its contents are a vital part of
your mathematical culture, Meg. There is, for instance, a Noah’s ark joke. (Actually, my favorite Noah’s ark joke is biological: a cartoon. The rain is coming down in sheets, the ark is loaded with two of every kind of animal, and Noah is on hands and knees grubbing around in the mud. Mrs. Noah is shouting from the ark, “Noah! Forget the other amoeba!”) Anyway, the mathematical Noah’s ark joke goes like this:

The Flood has receded and the ark is safely aground atop Mount Ararat; Noah tells all the animals to go forth and multiply. Soon the land is teeming with every kind of living creature in abundance, except for snakes. Noah wonders why. One morning two miserable snakes knock on the door of the ark with a complaint. “You haven’t cut down any trees.” Noah is puzzled, but does as they wish. Within a month, you can’t walk a step without treading on baby snakes. With difficulty, he tracks down the two parents. “What was all that with the trees?” “Ah,” says one of the snakes, “you didn’t notice which species we are.” Noah still looks blank. “We’re adders, and we can only multiply using logs.”

This joke is a multiple pun: you can multiply numbers by adding their logarithms. Other jokes parody the logic of proofs: “
Theorem
: A cat has nine tails.
Proof
: No cat has eight tails. A cat has one more tail than no cat.
QED.”

Mathematicians tell each other theorems. Quirky ones, like the “ham sandwich theorem”: if you have a
slice of ham and two slices of bread, arranged in space in any relative positions whatsoever, then there exists a plane dividing each of the three pieces exactly in half. Or the recently proved “bellows conjecture,” which says that if a polyhedron flexes (as, remarkably, some can), then its volume doesn’t change. But there is often a sting in the tail: “Proved that? OK, now do it with
n
objects in
n
dimensions.” Sometimes they tell each other conjectures, theorems not yet proved and that for all they know might be false. My favorite is the “sausage conjecture.” For starters, suppose you want to wrap a number of tennis balls in plastic film. What arrangement has the least surface area? (Assume that the film forms a convex surface: no dents.) The answer is that if you have fifty-six balls or fewer, they should be placed in a line to make a “sausage.” If you have fifty-seven or more, then they should be clumped together more like potatoes in a string bag.

In a four-dimensional analogue, the breakpoint is somewhere between fifty thousand and one hundred thousand. With fifty thousand balls they form a sausage. With one hundred thousand they clump. The exact breakpoint here is not known.

Here is the full conjecture: Three and four dimensions are misleading. Prove that in five or more dimensions, sausages are
always
the answer, no matter how large the number of balls may be.

The sausage conjecture has been proved in forty-two dimensions or more.

This is bizarre. I love it.

There will be gossip. Nowadays it may be about the topologist who ran off with her secretary, or the messy divorce of two well-known group theorists, but that’s a recent development that I trace to the bad influence of television. Traditionally, gossip is about who is in line for the Chair of Abstract Nonsense at Boondoggle University, or do you know anyone who has a postdoc position going for a young functional analyst like my student Kylie, or do you think Winkle and Whelk’s purported proof of the mass gap hypothesis has any chance of being right?

There will be serious news. As I write, a major topic of conversation is the latest information on Grisha Perelman’s alleged proof of the Poincaré conjecture. Has anyone found a hole in it yet? What do the experts think today? This is really exciting because the Poincaré conjecture is one of the great open questions in mathematics, second only to the Riemann hypothesis. It all went back to a mistake that Henri Poincaré made in 1900. He assumed without proof that any three-dimensional topological space (with some technical conditions) in which every loop can be continuously shrunk to a single point must be equivalent to a three-sphere, the three-dimensional analogue of the two-dimensional surface of an ordinary
sphere. Then he noticed the absence of any proof, tried to find one, and failed. He turned the failure into a question: is every such space a three-sphere? But everyone was so sure that the answer had to be yes that his question quietly turned into a conjecture. Its generalization to higher dimensions was then proved, for every dimension
except
3, which was disappointing, to say the least. The Poincaré conjecture became so notorious that it is now one of the seven millennium problems selected by the Clay Institute as the most important open questions in mathematics. Each problem carries a million-dollar reward for its solution.

In 2002 and 2003 Perelman, a rather diffident young Russian with a physics background, published two papers on the arXiv (“archive”), a website for mathematical preprints, with the offhand remark that they not only proved the Poincaré conjecture, they also proved the even more powerful Thurston geometrization conjecture, which holds the key to
all
three-dimensional topological spaces!

Usually this kind of claim turns out to be nonsense, but Perelman’s idea is clever and comes with a good pedigree. His trick is to use the so-called Ricci flow to deform the candidate space in a manner closely analogous to how space-time deforms under Einstein’s equations of general relativity. And that’s the snag. To understand the proof properly, you need to know three-dimensional topology, relativity, cosmology, and a dozen
other hitherto unconnected areas of pure math and mathematical physics. And it’s a long and difficult proof, with plenty of traps for the unwary. Moreover, Perelman followed the time-honored Russian tradition of not giving all the details. So the experts, who have been working through his ideas in seminars all over the world, are understandably wary of declaring the proof correct. But every time someone finds what might be a gap or a mistake, Perelman quietly explains that he’s already thought of that and why it isn’t a problem. And he’s right.

It’s gotten to the point where, even if the proof turns out to be wrong, the correct things achieved along the way are of major significance to mathematics. And as I write, the experts seem to be nudging ever closer to the view that the proof really does work. Keep your ears open over the coffee, Meg.

As your career develops, the worldwide mathematical community will be increasingly important to you. You will become part of it, and then you will have a home in every city on Earth.

Just arrived in Tokyo? Drop by the nearest university, find the math department, walk in. There will be at least one person you know, or who knows you by your work even if you’ve not met before. They will drop everything, call their baby-sitter, and take you out on the town for the evening. They may have a spare room, if you forgot to book a hotel. They will set up a seminar so that you can present your latest ideas to a sympathetic audience.
They may even be able to drum up a small financial contribution to your airfare.

You don’t get to fly business class, though. Or to sleep in a hotel suite. (Not
yours
, at least.) Math operates on the cheap and cheerful principle. I sometimes wish we didn’t undervalue ourselves in this manner, but it’s ingrained habit and it is far too late to change it.

It is of course more civilized and more organized to e-mail the University of Tokyo math department in advance. The result will be similar.

If you get on well with your host, they will invite you back. As you and they climb the career ladder, both of you will start being invited to conferences. Then you will find yourself organizing conferences, which means that you can invite everyone you want to talk to. There is some kind of “phase transition,” so that over a period of about a year, you will go from being invited to no conferences to being invited to far too many. Be selective; learn to say no. Learn sometimes to say yes.

There are big conferences and medium-sized conferences and small conferences. There are special conferences and general ones. The big, general ones are great for meeting people and trawling for jobs. Every four years, the International Congress of Mathematicians is held somewhere in the world. I last went when it was in Kyoto, and there were four thousand participants. I saw a lot of Kyoto, met lots of old friends and made some new ones, and learned a little bit about what people outside
my area were doing. The family came too, and they had a whale of a time exploring the city and its surroundings.

I much prefer smallish, specialized meetings with a specific research theme. You can learn a lot from those, because almost every talk is on something that interests you and is related to what you are working on at the moment. And once you’ve been in the business for a few years, you will know almost everyone else who is attending. Except for the youngest participants, who have only just joined the community.

Welcome, Meg.

Dear Meg,

Assistant Professor, indeed. I’m proud of you; we all are. At an excellent institution, too. You’re a professional mathematician now, with professional obligations. And it occurs to me that I’ve been so busy offering advice about what to do in various circumstances that I’ve left out the other side of the equation: what not to do. Now that you have a tenure-track position, you will be taking on more responsibility, so you will have more to lose if you foul up. There are plenty of ways for mathematicians to make complete idiots of themselves in public, and nearly all of us have managed it at some stage in our careers. People make mistakes; wise people learn from them. And the least painful way is to learn from mistakes made by others.

The longer you stay in the business of mathematics, the more blunders you will inevitably make; this is how experienced people gain their experience. I have witnessed,
and committed, plenty of mistakes myself. They can range from writing the wrong equation on the board to mortally insulting the president of your university at some significant public event. Be warned. You will no doubt invent some new mistakes of your own; occasional embarrassment is the natural human condition.

Most of my advice will be obvious. An assistant professor who wishes to be tenured at her university must find out what the requirements and expectations are, and then meet them. If you are expected to have published two papers beyond the subject of your dissertation and instead publish one paper, coach the math club, direct the study-abroad program in Budapest, obtain a major research grant, and win the teacher-of-the-decade prize, you may be denied tenure. Take care to be polite to your superiors, unless you have excellent reasons not to and want to change jobs. Be polite to everyone else, when they deserve it and even sometimes when they don’t. If you disagree with some decision or argument, make your point concisely, clearly, and without implying that the opposing view is insane, even when it is. Honor your commitments, whether they are tutorial sessions, office hours, examination grading, or plenary lectures at the International Congress of Mathematicians. If you agree to sit on a committee, turn up for its meetings. Listen to the discussion. Contribute, though not at length. Generally, remember that you are a professional, and behave like one.

Some mistakes, on the other hand, are obvious only after you’ve made them. There is a persistent story at Warwick University that I ended my first ever undergraduate lecture by walking into the broom cupboard. It is time to set the record straight. Yes, I admit that it
was
a broom cupboard, but it was also the emergency exit from the lecture hall. I had assumed, without finding out ahead of time, that when the students left the hall by the main doors, I would be able to leave by what looked like a side door. But when I tried it, I found myself surrounded by buckets and mops. Worse, I discovered that the only way to leave the building by that route was to push open an emergency exit, which would set off an alarm. I had noticed the EXIT sign over the door but had failed to spot the word “emergency” above it. So I was forced, rather sheepishly, to emerge from the so-called broom cupboard and join the students as they walked up the stairs to the back of the hall and out the main doors.

The message here is, don’t assume things. Check them beforehand. Not just the layout of the lecture hall or the location of the building where you are supposed to be giving a talk, or the city in which your meeting is due to take place, or the date of that meeting . . . Recall Murphy’s law: “Anything that can go wrong, will.” Above all, remember the mathematician’s corollary to Murphy’s law: “Anything that
can’t
go wrong will go wrong too.”

This was brought home very clearly to a good friend of mine, also a mathematician, on a trip to a country that it would be best not to name. He was attending a conference and he was making a flight from his first port of entry to another, fairly distant, city. As he sat in the airplane scribbling some calculations, he noticed the pilot emerge from the flight cabin, shutting the connecting door behind him. A few minutes later the copilot did the same thing. Soon after that, the pilot returned, and tried to open the door to return to the cockpit. He seemed to be encountering some difficulty.
The copilot tried to help, but neither of them could open the door. At this point my friend realized that the plane was flying on autopilot and no one could get to the controls. A female flight attendant joined the pilot and copilot, disappeared, and reappeared carrying a small hand ax. The pilot then attacked the door with the ax, made a hole in it, put his hand through, and opened the door. The flight crew then entered the cabin and closed the door behind them.

No announcement was made to explain these events to the bemused and distinctly frightened passengers.

My advice here really applies to the pilot and the copilot, not the passengers. If you go to conferences, you will sometimes have to fly on airlines that have been booked by the conference organizers. You can choose not to go, if you wish, but you can’t always choose not to travel on an airline with a dubious safety record or an aging fleet.
There really is no way for a passenger to anticipate such a problem, or to help solve it or avoid it.

Let me revert to the topic of lecturing. Another useful piece of advice is to make sure that you have plenty of time to get to the lecture room. Avoid taking on unpredictable commitments immediately beforehand. I still have vivid recollections of arriving late to give a lecture on algebra. I lived in a village at that time, and another member of the math department owned a small farm in the same village, so we carpooled. One day when it was his turn to drive, he decided to drop a pig off at the local abattoir on the way to work. The pig, perhaps sensing that the trip would not be to its advantage, had other ideas. It refused to climb the plank into the back of the truck. It is difficult to remain professorial when explaining that you were late because you could not get a pig into a pickup truck.

One of the main ways you will interact with other mathematicians is by attending, and giving, talks. These might be seminars, specialist talks for experts in your research area; they might be colloquia, more general talks for professional mathematicians but not specialists in the area concerned; or they might be public lectures open to anyone who wants to turn up. All lectures are fraught with potential disaster.

There was, for instance, a prominent professor of number theory who had the habit of turning up at the start of a visiting speaker’s seminar, falling asleep within
minutes of the talk starting, snoring loudly the whole way through, and then asking penetrating questions at the end when the audience’s applause woke him up. By all means, emulate the penetrating questions, but try to avoid the snoring if you possibly can; otherwise you will get a reputation for eccentricity.

If you are the person giving the talk, the opportunities for Murphy to strike are far more numerous, especially if you are using equipment. When I started lecturing, the only equipment we ever had was blackboard and chalk, and I confess to a continuing bias in favor of lo-tech visual aids, though I am capable of preparing the all-singing, all-dancing PowerPoint presentation with a video projector and moving graphics downloaded from the Internet if that is what is called for. I have used overhead projectors, whiteboards with those horrible pens that smell of solvent, even the businessperson’s ubiquitous flipchart.

Even chalk can go wrong. First, it may not be there. I developed a habit of taking my own box of chalk to lectures in case the previous lecturer had used it all, or the students had hidden it as a joke. Some chalk is very dusty and gets all over your clothes: I made sure I had the “antidust” kind with me. It still got on my clothes, but the cleaning bills were smaller. Many types of chalk can make horrible screeching sounds when you write, putting everyone’s teeth on edge. It takes practice to prevent that. And normal-sized chalk will not do if you are lecturing to
five hundred freshman calculus students in a cavernous lecture hall. You need supersize.

Other types of equipment can go wrong more spectacularly. A good friend of mine was delivering a short talk at the British Mathematical Colloquium—the UK’s main mathematics conference, held annually—and he was intending to use an overhead projector to show lots of pictures on a projection screen. Unfortunately, when he tried to pull the screen down from the ceiling—it was on a roller and there was a string attached—it fell down on his head. He ended up projecting the pictures onto the wall.

Never believe your hosts when they tell you that all the equipment will work perfectly. Always try it for yourself before the lecture. I was giving a public lecture in Warsaw using a cassette of about eighty 35-mm slides. I was persuaded to hand the slides to the projectionist, who would set everything up for me, while my hosts took me for a quick coffee. As I entered the room to deliver the lecture, the projectionist put the slides into the projector, which was tilted at an alarming angle because of the high position of the screen. The cassette slid right through the projector and fell out the back onto the floor, where its contents were scattered far and wide. Many of the slides were sandwiched between thin sheets of glass, which broke. It took ten minutes to get everything back into some kind of order, in front of five hundred patient people.

Do not confuse the projection screen with a whiteboard and write on it in permanent ink. Many people do this, and their lecture is preserved for eternity, along with their mistake.

The famous physicist Richard Feynman once learned Spanish because he was going to give a lecture in Brazil. Check the local language.

If you are hosting a visitor, and they are giving a talk, make sure that you have the key to the projection room. On one occasion I had to improvise my lecture because the slide projector, though visible to us all inside a wonderfully equipped room, might as well have been located on the moon because the room was locked and no one knew how to get the key.

Do not forget that your visitor may not know the local geography. I was once abandoned inside a Dutch mathematics department building when my hosts went off to the parking garage to go to a restaurant. I had to make my escape through a window, setting off a burglar alarm. I did manage to catch up with them in the garage, which was a good thing because I had no idea where the restaurant was, or even what its name was.

Avoid wandering around strange buildings, especially in the dark. A biologist friend was visiting an institution with a strong marine biology department, and it had an aquarium. The entrance was down a short flight of stairs.
Alone in the building, late at night, he tried to enter the
pitch-black room, felt for the light switch, and accidentally hit the fire alarm instead.

Six fire engines turned up, lights blazing, sirens blaring.

He had called the fire station to explain his mistake, but according to the rules, they could not return to base without checking for a fire.

Committees are another place where you can easily make dreadful mistakes. Universities tend to run as a network of interlocking committees and subcommittees, some with real teeth, some window dressing. Most exist for a reason, and many tackle low-level but essential activities such as grading tests or sorting out course regulations. You will undoubtedly be involved in committee work, and you should be. A university is a complicated place, and it will not function well if everyone is left to make things up as they go along. Every academic has to be something of an administrator, and to some extent the converse holds, especially at senior levels.

Not being a committee animal myself, I can’t offer much useful advice on how to “work” a committee so that it reaches the decision you favor. But I do know how not to. The following story is typical. An important committee was debating a major decision about some particular action. The mathematician on the committee saw immediately that taking the action concerned would lead to disaster, and spent five minutes explaining the logic behind this view, which was unassailable. His
analysis was clear and concise, and left no real room to doubt his conclusions; no one contradicted him. The debate continued, however, because the other members of the committee had not yet had their say. After an hour or more of further discussion—to which the mathematician did not contribute, having already (he assumed) made his point—the committee voted. It decided to take precisely the action that the mathematician had warned it not to.

What was his mistake? It was not in the analysis, nor in its presentation, but in its timing. In any committee discussion, there comes a pivotal moment when the decision can be swayed either way. That is the time to strike. If you make your point too soon, everyone else will have forgotten it; if you are lucky, you may be able to remedy this with a timely reminder. But if you make your point too late, there is no way it can have any effect.

The other thing not to do in committees is to keep making your point when you have already won it. You may lose support merely because you keep banging on about what has by now become obvious to everyone. If you have further ammunition, save it in case it is needed later.

Which advice I will now take myself.