Princeton University
Department of Mathematics

Mentoring, collaborating, and writing books:
A discussion on the inspirational role of Elias M. Stein
May 2011
Princeton University

Lillian Pierce '02 GS '09
January 2012

I hate to break it to the Hollywood set, but mathematics can be a wonderfully social pursuit. Of course, sometimes we like holing up in solitude while we wrestle with a problem. And maybe we aren’t the classiest dressers on campus. But mathematicians don’t just float around each other like noble gasses. We talk about problems, work together on solutions, give seminars and lectures, write papers and books, advise students, contribute to an oral tradition of anecdotes, and dream about future discoveries; all these interactions serve to build a community of mathematicians. Once in a while someone comes along who is prodigiously accomplished not just in mathematics but also in the more intangible personal skills that build community, and the leadership of such a person can benefit an entire field of mathematics. Last May, a conference at the Math Department celebrated the 80th birthday of just such a leader: Elias M. Stein. Mathematical lectures at the conference illustrated the effect Prof. Stein’s mathematical work has had on the field of harmonic analysis (and beyond), but one more unusual event featured a panel discussion of three arenas in which he has had unprecedented success: mentoring, collaborating, and writing books.

The panelists who participated in this discussion were Ingrid Daubechies (Duke University), Gerald Folland (University of Washington, GS ’71), Loredana Lanzani (University of Arkansas) and Stephen Wainger (University of Wisconsin at Madison, emeritus), and the panel was co-moderated by Rami Shakarchi (GS ’09) and myself (’02, GS ’09).

We probably all recognize good mentoring when we see it—in fact, we probably have all benefited from a mentor at some point in our lives. But it’s a bit harder to define mentoring in the abstract. A mentor need not be someone in the official role of teacher or advisor; the advice they give need not have anything to do strictly with mathematics, but may come more from life and career experience. One mentor might play a role at just one key moment in your development, but with another you might build up a decades-long relationship. At the Stein conference, the panelists discussed how to be a good mentor, and several key ideas stood out. At its heart, mentoring aims to increase the awareness of all the possible opportunities and goals available. In particular, mentors can be role models, and this aspect of mentoring can be a motivation for someone who aims to encourage young mathematicians who don’t fit the stereotypical image of a mathematician. As a role model, it may be important to analyze the actual image you are projecting, and perhaps adjust how you present yourself professionally or personally to students, depending on what type of inspiration you think the students need. Mentors can convey informal knowledge about the practice of mathematics—not technical points per se, but more of a philosophical or aesthetic sense of how math works, and what makes a question interesting. The altruistic mentality of mentoring need not be dissociated from other professional relationships one has in the context of collaborating, teaching, advising. Being a good mentor, a good colleague, or a good teacher, all involve, in some form, nurturing the members of the community. As one panel member said, we should treat everyone with kindness. One is never too old to benefit from a mentor—nor too young to be a mentor. In fact, mentoring people who are at an earlier stage of their mathematical life is a great way to build a nurturing community from the ground up, and also to gain perspective on how far one has come already. It also isn’t solely the responsibility of the mentor to initiate and sustain the relationship—if you are feeling in need of some guidance, inspiration, or encouragement, look around: someone you know already is probably ready to help.

In many courses in the math department, students are encouraged to collaborate on solving problem sets. Collaboration becomes even more important and rewarding when you start doing original research rather than solving known problems. How to be a good collaborator? Panelists described the singular pleasure of collaborating with Prof. Stein, who conveys such genuine appreciation of each collaborator’s contribution, whether large or small, that it creates a positive and trusting environment that in turn stimulates new ideas. Collaboration with a more senior mathematician can be an important form of mentoring in itself, but how does a younger mathematician get the courage to collaborate with someone much more knowledgeable? Just remember that a more senior mathematician probably also has many more responsibilities, and is probably very happy to have a collaborator who will flesh out the main ideas with detailed computations. Of course, mathematical research is not all smooth sailing, and any collaborative project will encounter challenging headwinds. For advice on how to weather the storm and create a long and fruitful collaboration, one can turn back a century to the four axioms outlined by J. E. Littlewood, who had a famously successful and amicable collaboration with G. H. Hardy, which they carried on in an epistolary fashion (yes, by paper mail). (1) The writer of the letter is under no obligation to check that the letter is correct; (2) The recipient of the letter is under no obligation to reply to (or even read) the letter; (3) It is preferable that both collaborators not think about the same technical point at the same time; (4) No matter who contributes what to the project, both are credited as equal co-authors. Perhaps these rules should be modified to fit the etiquette and velocity of the email age, but the key themes of confidence, freedom, trust, and appreciation, remain. While one is very fortunate when one can learn math from a mentor or with a collaborator, one often must turn back to the basics: reading books. Books function differently in mathematics than in the humanities or experimental sciences. Typically the first move that a recent PhD graduate makes in the humanities is to publish the PhD thesis as a book. In math, most mathematicians go their whole career without writing a book—in fact, the traditional wisdom might point people away from writing books, rather than research papers. On the other hand, in contrast to other sciences, books (and papers) in math are cherished for much longer. What neuroscientist today goes back to publications from the 1920’s, much less the 1820’s, to glean ideas for current research? Good books in math don’t have an expiration date. How does one write a good book? First one must determine if the book you’d like to write is actually needed. Then you must hold in mind your intended audience, and shape the book so that it tells a story. There is always a balance that must be reached, between what one wants to write about, and what other people want to read; in particular, the mathematics must be presented in a way that is clearly understandable to the intended audience (for example, with intuitive, consistent notation). Sometimes, there is a need for a book that presents recent research in an expository fashion, so that new ideas become accessible to a wider audience. Some books codify a broad field of research, and reading them becomes a rite of passage for young researchers. Other books are true textbooks, meant to introduce students to classical mathematics. In the case of the Princeton Lectures in Analysis series by Stein and Shakarchi (Fourier Analysis, Complex Analysis, Real Analysis, and Functional Analysis), the texts present classical material but in a new, unified, fashion. Readers ranging from undergraduate students to senior mathematicians point to the many strengths of this series: the variety of interesting applications of the theory (and unconventional proofs of theorems); a large selection of good problems of many different levels; a convincing presentation of the historical origins of ideas. Writing a book, especially a good book, isn’t easy: it takes a tremendous amount of effort and time. But we should all be grateful to people who expend this effort to give us good books to read! And the authors of good books may take some satisfaction in knowing that the number of hours devoted by people all over the world to reading their books will quickly outnumber the hours put into creating the books!