Letters to a Young Mathematician (9 page)

Do they really want to go into the history books as claiming that?

It doesn’t put them off, mind you. No rational argument ever diverts a true trisector from their innate certainty that they are right.

The “degree” invariant also explains why a regular seventeen-sided polygon can be constructed but a seven-sided one cannot. The corresponding degrees turn out to be one fewer than the number of sides: sixteen and six. Because 16 is the fourth power of 2, the seventeen-gon can be constructed by solving four successive quadratics. But 6 is not a power of 2, so no construction exists in that case. In my experience, angle trisectors seldom object to this deduction, although ironically, it implies that a valid trisection of the angle would lead directly to a construction of the regular seven-gon.

There are many other impossible problems in mathematics. Angle trisection is one of three famous “problems of antiquity” credited to ancient Greek geometers, unfortunately without much historical justification, because the limitation to unmarked straightedge and compass was a later addition. The Greeks knew how to solve all three problems using more complex instruments. But it’s true that this was the only way they knew how to solve them. Later mathematicians wondered whether anyone could do better, and eventually realized that they couldn’t.

The other two problems of antiquity are squaring the circle and duplicating the cube. That is, using the traditional methods, construct a square whose area is equal to that of a given circle, or a cube with twice the volume of a given one. In modern terms, these problems ask for constructions for π and the cube root of two, respectively. They can be proved impossible by similar methods. In fact, the cube root of two evidently satisfies a cubic equation: its cube is two. And π satisfies no algebraic equation whatsoever; but that’s another story.

Dear Meg,

Don’t mention it. I am always happy to treat you to a meal whenever we find ourselves in the same city, which, if you’re genuinely intent on pursuing a career in research, could be increasingly often.

But let me play devil’s advocate for a moment. It’s important to ask yourself whether you want to stay at university because that’s where you feel most comfortable. You should not be looking for “comfortable” at your age.

Being a research mathematician is akin to being a writer or an artist: any glamour that’s apparent to outsiders fades quickly in the face of the job’s frustration, uncertainty, and hard, often solitary, work. You can’t expect your occasional moments in the spotlight to make it all worthwhile. Unless you’re more superficial than I believe you are, they can’t possibly. Your satisfaction must come from the high you get when you suddenly, for the
first time, understand the problem you’re working on and see your way to a solution. I use the word “high” advisedly. You need to be something like an addict for this feeling to provide sufficient recompense for all the work.

Here’s the paradox: though much of a mathematician’s work is solitary, even lonely, the most important aspect of your research is not the field you choose or the problems you embark on but how you deal with the people around you.

When you set out to earn a PhD, you do not do so alone. Your fellow students will constitute an important support group; your department will function as your clan within the larger tribe of mathematicians around the world; above all, you will have a thesis adviser (or supervisor, as we say in the UK). Normally he or she will be an established expert with a solid track record in the area you plan to study. Sometimes it will be someone who completed their own PhD only a few years ago, in which case there will probably be a second, more senior adviser to add experience to the mix.

A young adviser is often an excellent choice. They are usually bubbling over with ideas, and having just come through the academic mill themselves, they will probably be more sympathetic to your struggles.

In the April 1991 issue of
The Psychologist
, my sociologist friend Helen Haste analyzed the patterns of gift-giving among the remote and backward peoples known as “academics.” The article was an anthropological spoof,
but it made some telling points. The gifts were copies of research papers, and the article classified academics into a six-rung career ladder, plus one unorthodox role that sticks out sideways.

You are about to join the first rung of the ladder by becoming a DXGS: Dr. X’s graduate student. From there you will, I’m sure, progress rapidly to PYR, promising young researcher, and thence to ER, established researcher. If you elect to remain in academia, the succeeding grades are SS (senior scientist), GOP (grand old person), and EG (emeritus guru).

As a DXGS you will not yet have produced any ritual gifts of your own, and so cannot present them to anyone. You can request them, but normally only from your peers. When performing before the tribe—that is, giving seminars—you will repeatedly invoke two ancestors, a major theorist and your thesis adviser. The PYR is more relaxed and understands the rituals better. He or she will still invoke those two ancestors, but briefly and often as mere footnotes. Astute PYRs invoke SSs instead. They travel to tribal meetings (conferences) so laden with gifts that the journey is more like a pilgrimage, and dispense them liberally. They also feel able to request gifts from their seniors, though not too often and always politely. The ER seldom refers to a major theorist, preferring to mention only ancestors who are currently active, but—tellingly—an ER may also mention progeny, to prove that he or she has them. The ER
does not bring gifts to the tribal meeting, having cleverly dispensed them in advance to the tribe’s inner circle.

The SS invokes a major theorist frequently, with the goal of supplanting him or her by being seen to have made important advances over the major theorist’s ideas. The SS never gives or receives in public but expects to receive many gifts by more covert means. The GOP sits at the pinnacle of the gift-giving hierarchy, offering no gifts but requesting them from everyone, especially juniors. The EG is invoked as an ancestor by almost everyone, but takes part in absolutely no exchanges of gifts.

The role that does not fit into this sequence—indeed, does not fit anywhere, which is its raison d’être, is the maverick guru (MG). Helen has this to say about MGs: “The Maverick Guru has an important symbolic role, having curious magical powers that cause fear and fascination in the community. The MG is outside the mainstream orthodoxy, but engaged in criticizing it. The MG cannot be invoked as an ancestor by junior members of the community who intend to stay within the mainstream . . . An erstwhile MG rapidly becomes a GOP.”

I mention all this because you need to appreciate your place within the tribe, and because your progression from DXGS to PYR to ER depends heavily on your choice of X, who should be either an ER, an SS, or perhaps a GOP.
Do not choose an MG
, no matter how attractive that option may seem, unless you intend your entire
career to operate outside the conventional ladder. And on the whole, I advise you to stay clear of GOPs. Trust me: I wanted to become an MG but I think I’ve ended up as a GOP instead. A GOP will have an impressive record, but much of it will date to the dim and distant past: five years ago, or even longer. The older an academic becomes, the more intellectual baggage he carries. GOPs’ minds tend to run along familiar grooves, and although they do this with impressive ease and confidence, their students may miss out on the really new ideas that are the lifeblood of research. Some GOPs, though, make excellent advisers despite that, usually those who are close to being MGs but aren’t quite.

EGs never have students.

My adviser was an SS in the field of group theory— the formal mathematics of symmetry—named Brian Hartley. He was young, but not too young. I didn’t choose him, and he didn’t exactly choose me either; I chose the field, and the system allocated me to him because he was in that field. There were four or five alternative choices. Any of them would have worked—I later got to know them all well, as colleagues—but my research would have been very different. I was very lucky to get Brian, who put me onto a problem—more an entire program—that really suited my interests and abilities. He was brilliant. He saw me regularly, was always available if I got stuck, and he was hardly ever stumped for an idea.

Brian was, I think, slightly taken aback when I marched into his office on day one of my PhD course and demanded a research problem. Usually it takes students longer to get going. But within half an hour he had given me one—arising from one of his own papers, my first receipt of a gift—and it turned out to be a beauty. The program of research was to study a special type of group that a Russian mathematician, Anatoly Ivanovich Malcev, had associated with a different mathematical structure called a Lie algebra. This structure was first developed over a century ago by the Norwegian Sophus Lie, but (despite its name) it was mainly studied in the context of analysis, not abstract algebra. So Malcev’s purely algebraic version was a new point of view. Like many Russians at the time, he had sketched the ideas but not developed them in detail. My problem was to take Malcev’s thoughts and conjectures, and fill in the necessary proofs and other connections, in effect, to turn a set of sketches and renderings into a finished blueprint of a building. It took me three months, and I got hooked by Lie theory. I ended up writing my entire thesis on Lie algebras.

Brian’s influence did not stop at research problems. He and his wife, Mary, entertained me and other PhD students at their home. Occasionally he invited me to join him at some jazz performance in a local pub. He was an academic father figure, a mentor, and a friend. In 1994 he died, unexpectedly, at the age of 55 while walking in
the hills near Manchester. I wrote his obituary for
The
Guardian
. It ended like this: “I last saw Brian a few weeks ago, at a meeting to mark the sixtieth birthday of a mutual friend. He had just taken up a much-prized fellowship that would relieve him of all teaching duties for five years to concentrate on research. He went out with his boots on, both literally, while walking in his beloved hills, and metaphorically. And that is how we shall all remember him.”

I still find it hard to accept that he’s gone.

As I say, I was lucky. The system assigned me the ideal adviser. But you can do better than that. Don’t leave it up to chance:
choose
your thesis adviser. Read the literature, talk to people in the field, find out who has a good reputation and—crucially—who is good with students. Draw up a short list. Visit them; in effect,
interview
them. Then trust your instincts. And remember, you don’t want a GOP who ignores you; you want a close personal relationship.

Dare I add, not
too
close? The cliché of faculty sleeping with their students exists because it does happen.
Someone once observed that the more subjective the discipline, the better the faculty dressed, and a similar principle seems to apply to illicit sex. Mathematicians, by and large, indulge in rather less of it, perhaps because we dress so badly. In any case, everyone knows it’s unprofessional, and nowadays there are sexual harassment laws. Enough said. For recreation and affection,
restrict yourself to fellow students, or people from off campus, please.

A standard joke states that mathematical ability is typically passed from father to son-in-law (or, these days, from mother to daughter-in-law). The point was that a male PhD student often married his adviser’s daughter. It’s one way to meet people off campus. So your real ancestry can be affected by your mathematical ancestry.

Mathematicians are proud to trace their academic lineage through thesis advisers. Brian was my mathematical father, and Philip Hall my grandfather. Hall was of a generation for whom a PhD was unnecessary as a qualification for a university profession, but the most significant influence on his early work was William Burnside. Burnside can similarly be considered a mathematical son of Arthur Cayley, one of the most famous English mathematicians of Victorian times.

I remember these things and value them. I know where and how I fit into the family tree of mathematical thought. Arthur Cayley is as important an ancestor to me as any of my biological great-great-grandfathers.

Talent must be passed to succeeding generations. I’ve been thesis adviser to thirty students so far, twenty men and ten women. Since 1985, the proportions are fifty percent men, fifty percent women. I
know
women are just as good at math as men because I’ve watched both at close quarters. I am particularly proud of my mathematical daughters, most of whom hail from Portugal, where
mathematics has long been viewed as a suitable activity for women. All of my Portuguese daughters have remained in mathematics. In fact, most of my graduate students have remained in mathematics, and every single one of them earned a PhD. However, one is now an accountant, several work in computing, and one owns an electronics company, or at least he did the last time I heard from him.

The rest of the world is now following Portugal’s lead. In July 2005 the American Mathematical Society released the results of its 2004
Annual Survey of the
Mathematical Sciences
. Since the early 1990s, women have been receiving around 45 percent of all first degrees in math. Women received almost one-third of all U.S. doctorates in the mathematical sciences in the academic year 2003–2004, and one-quarter of those awarded in the top forty-eight math departments. In all, 333 women received math PhDs that year, the largest number ever recorded.

The idea that math is not a suitable subject for women is stone-cold dead. The career ladder is open to both sexes, though it is still unbalanced at the top end.