A research mathematician, like a creative artist, has to be passionately interested in the subject and fully dedicated to it.
MATH IS H(ART)D
The first year or two of research is the most difficult. There is so much to learn. One struggles unsuccessfully with small problems and one has serious doubts about one’s ability to prove anything interesting. I went through such a period in my second year of research, and Jean-Pierre Serre, perhaps the outstanding mathematician of my generation, told me that he too had contemplated giving up at one stage. Only the mediocre are supremely confident of their ability.
You should not regard mathematics as a closed world, and the interaction between mathematics and other disciplines is healthy both for the individual and for society. Collaboration, initially with fellow students or one’s supervisor, has many benefits, and long-term collaboration with coworkers can be extremely fruitful both in mathematical terms and at the personal level. There is always the need for hard quiet thought on one’s own, but this can be enhanced and balanced by discussion and exchange of ideas with friends.
PROBLEM SOLVERS VS THEORISTS
Mathematicians are sometimes categorized as either “problem solvers” or “theorists.” What bearing does this have on a beginning student? Although one has to read books and papers and absorb general concepts and techniques (theory), realistically, a student has to focus on one or more specific problems. This provides something to chew on and to test one’s mettle. A definite problem, which one struggles with and understands in detail, is also an invaluable benchmark against which to measure the utility and strength of available theories.
RESEARCH VIGOUR
The driving force in research is curiosity. When is a particular result true? Is that the best proof, or is there a more natural or elegant one? What is the most general context in which the result holds?
If you keep asking yourself such questions when reading a paper or listening to a lecture, then sooner or later a glimmer of an answer will emerge—some possible route to investigate. Ironically, good ideas can emerge unexpectedly from a bad lecture or seminar. I often find myself listening to a lecture where the result is beautiful and the proof ugly and complicated. Instead of trying to follow a messy proof on the blackboard, I spend the rest of the hour thinking about producing a more elegant proof. Usually, but not always, without success, but even then my time is better spent, since I have thought hard about the problem in my own way. This is much better than passively following another person’s reasoning.
EXEMPLIFICATION
It is essential to be able to test general results by applying them to simple examples. These are examples where one can do concrete calculations, sometimes with elaborate formulas, that help to make the general theory understandable. They keep your feet on the ground.But most of all a good example is a thing of beauty. It shines and convinces. It gives insight and understanding. It provides the bedrock of belief.
AUTONOMY
The art in good mathematics, and mathematics is an art, is to identify and tackle problems that are both interesting and solvable.If you need to learn a new subject, consult the literature but, even better, find a friendly expert and get instruction “from the horse’s mouth”— it gives more insight more quickly. At the start of your research your relationship with your supervisor can be crucial, so choose carefully, bearing in mind subject matter, personality, and track record. Few supervisors score highly on all three. Moreover, if things do not work out well during the first year or so, or if your interests diverge significantly, then do not hesitate to change supervisors or even universities. Your supervisor will not be offended and may even be relieved!
The article is based on the The Princeton Companion to Mathematics, PUB.
18/08/2014
