Advice to a Young Mathematician #5 (Peter Sarnak, 2010)

I have guided quite a number of Ph.D. students over the years, which perhaps qualifies me to write as an experienced mentor. When advising a brilliant student (and I have been fortunate enough to have had my fair share of these) the interaction is a bit like telling someone to dig for gold in some general area and offering just a few vague suggestions. Once they move into action with their skill and talent they find diamonds instead (and of course, after the fact one cannot resist saying “I told you so”). In these cases, and in most others as well, the role of a senior mentor is more like that of a coach: one provides encouragement and makes sure that the person being mentored is working on interesting problems and is aware of the basic tools that are available. Over the years I have found myself repeating certain comments and suggestions that may have been found useful. Here is a list of some of them.

  1. When learning an area, one should combine reading modern treatments with a study of the original papers, especially papers by the masters of our subject. One of the troubles with recent accounts of certain topics is that they can become too slick. As each new author finds cleverer proofs or treatments of a theory, the treatment evolves toward the one that contains the “shortest proofs.” Unfortunately, these are often in a form that causes the new student to ponder, “How did anyone think of this?” By going back to the original sources one can usually see the subject evolving naturally and understand how it has reached its modern form. (There will remain those unexpected and brilliant steps at which one can only marvel at the genius of the inventor, but there are far fewer of these than you might think.) As an example, I usually recommend reading Weyl’s original papers on the representation theory of compact Lie groups and the derivation of his character formula, alongside one of the many modern treatments. Similarly, I recommend his book The Concept of a Riemann Surface to someone who knows complex analysis and wants to learn about the modern theory of Riemann surfaces, which is of central importance to many areas of mathematics. It is also instructive to study the collected works of superb mathematicians such as Weyl. Besides learning their theorems one uncovers how their minds work. There is almost always a natural line of thought that leads from one paper to the next and certain developments are then appreciated as inevitable. This can be very inspiring.
  2. On the other hand, you should question dogma and “standard conjectures,” even if these have been made by brilliant people. Many standard conjectures are made on the basis of special cases that one understands. Beyond that, they are sometimes little more than wishful thinking: one just hopes that the general picture is not significantly different from the picture that the special cases suggest. There are a number of instances that I know of where someone set out to prove a result that was generally believed to be true and made no progress until they seriously questioned it. Having said that, I also find it a bit irritating when, for no particularly good reason, skepticism is thrown on certain special conjectures, such as the Riemann hypothesis, or on their provability. While as a scientist one should certainly adopt a critical attitude (especially toward some of the artificial objects that we mathematicians have invented), it is important psychologically that we have beliefs about our mathematical universe and about what is true and what is provable.
  3. Do not confuse “elementary” with “easy”: a proof can certainly be elementary without being easy. In fact, there are many examples of theorems for which a little sophistication makes the proof easy to understand and brings out the underlying ideas, whereas an elementary treatment that avoids sophisticated notions hides what is going on. At the same time, beware of equating sophistication with quality or with the “beef of an argument” (an expression that I apparently like to use a lot in this context: many of my former students have teased me about it). There is a tendency among some young mathematicians to think that using fancy and sophisticated language means that what they are doing is deep. Nevertheless, modern tools are powerful when they are understood properly and when they are combined with new ideas. Those working in certain fields (number theory, for example) who do not put in the time and substantial effort needed to learn these tools are putting themselves at a great disadvantage. Not to learn the tools is like trying to demolish a building with just a chisel. Even if you are very adept at using the chisel, somebody with a bulldozer will have a huge advantage and will not need to be nearly as skilful as you.
  4. Doing research in mathematics is frustrating and if being frustrated is something you cannot get used to, then mathematics may not be an ideal occupation for you. Most of the time one is stuck, and if this is not the case for you, then either you are exceptionally talented or you are tackling problems that you knew how to solve before you started. There is room for some work of the latter kind, and it can be of a high quality, but most of the big breakthroughs are earned the hard way, with many false steps and long periods of little progress, or even negative progress. There are ways to make this aspect of research less unpleasant. Many people these days work jointly, which, besides the obvious advantage of bringing different expertise to bear on a problem, allows one to share the frustration. For most people this is a big positive (and in mathematics the corresponding sharing of the joy and credit on making a breakthrough has not, so far at least, led to many big fights in the way that it has in some other areas of science). I often advise students to try to have a range of problems at hand at any given moment. The least challenging should still be difficult enough that solving it will give you satisfaction (for without that, what is the point?) and with luck it will be of interest to others. Then you should have a range of more challenging problems, with the most difficult ones being central unsolved problems. One should attack these on and off over time, looking at them from different points of view. It is important to keep exposing oneself to the possibility of solving very difficult problems and perhaps benefiting from a bit of luck.
  5. Go to your departmental colloquium every week, and hope that its organizers have made some good choices for speakers. It is important to have a broad awareness of mathematics. Besides learning about interesting problems and progress that people are making in other fields, you can often have an idea stimulated in your mind when the speaker is talking about something quite different. Also, you may learn of a technique or theory that could be applied to one of the problems that you are working on. In recent times, a good number of the most striking resolutions of longstanding problems have come about from an unexpected combination of ideas from different areas of mathematics.

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