When should mathematics students begin publishing in research journals?

The publication resumes of many graduate students in recent years are impressive. It is not unusual for a student to have several refereed journal articles before getting a PhD, and these days even undergraduates face pressures to publish in refereed journals. At the same time, and possibly to support this productivity, eff0rts at research collaboration are starting at an increasingly early stage. There are many positives to the trend. The idea of getting to research frontiers quickly attracts more students to mathematics, there is something very satisfying about an end product like a journal article, and publications are important for rising in the mathematics profession. Added to this, doing mathematics research together in a supportive group is a fun activity in its own right.

Are there downsides to an emphasis on early exposure and quick tangible results? In today’s culture many students feel that fast tracks toward research and publications are the only way to succeed in math, and that this goes hand in hand with the need to build large networks of potential collaborations starting at an early stage. The popular Research Experience for Undergraduates (REU) programs are well-funded and an opportunity for undergraduate math majors to expose themselves to mathematics beyond the usual curricula. On the other hand, getting into a good REU and publishing as a result has become the first defining hurdle for many students (in their self-perceptions as well as in the practical reality of graduate school applications). Aside from the obvious potential dangers of vetting students according to a fixed list of established criteria at such an early stage, there is a need to consider how these judgements are shaping student conceptions of mathematics and their potential place in it.

What is the best way to nurture mathematicians? Opportunities for young mathematicians to learn and do projects together can be healthy, if done in a thoughtful way, but the emphasis on publishing is not always the best way to nurture mathematical potential, especially when one looks, for comparison, at the careers of successful mathematicians. The number of papers published (which can vary immensely) does not determine the influence and importance of a mathematician, and breadth of knowledge developed at a young age aids greater mathematical productivity in the long-term. Is the time-honored tradition of students pouring over books and doing every exercise fading away in the rush to find and solve a problem at the frontiers of knowledge?

Of course, beyond social pressures there are practical issues that affect success as a mathematician. Will I get into a better graduate school if I have a published paper? Will I get a better chance at a postdoc? How many papers do I need to get a tenure-track position? What do I need to do for tenure? Though these questions loom large for many, and do need to be addressed, they are by no means everything. As individuals, mathematicians have choices, mathematics is a personal journey, and ultimately by its very nature mathematics is an endeavor where being an independent thinker is an asset.

Your comments and suggestions are welcome as always!

**Featured Book of the Day**

** The Moduli Problem for Plane Branches** by Oscar Zariski

Moduli problems in algebraic geometry date back to Riemann’s famous count of the 3g-3 parameters needed to determine a curve of genus g.

In this book, Zariski studies the moduli space of curves of the same equisingularity class. After setting up and reviewing the basic material, Zariski devotes one chapter to the topology of the moduli space, including an explicit determination of the rare cases when the space is compact. Chapter V looks at specific examples where the dimension of the generic component can be determined through rather concrete methods. Zariski’s last chapter concerns the application of deformation theory to the moduli problem, including the determination of the dimension of the generic component for a particular family of curves.

An appendix by Bernard Teissier reconsiders the moduli problem from the point of view of deformation theory. He gives new proofs of some of Zariski’s results, as well as a natural construction of a compactification of the moduli space.

“Mathematics is a personal journey, and ultimately by its very nature mathematics is an endeavor where being an independent thinker is an asset”……really the essence! The increased pressure of publishing has affected serious thinking of mathematics since it is much different from the other disciplines!

The issue of early research and publishing in mathematics (especially at undergraduate level) is an issue I have given a lot of thought to recently. I am definitely a strong supported of early involvement in serious research (by which, let’s say, I mean something publishable in mainstream research journals, not in special venues dedicated to “undergraduate research”). Incidentally, I don’t believe there is anything like “undergraduate research”, and unfortunately a lot of REUs give out homework problems dressed up as research, which is not what I am talking about. It is certainly more difficult in mathematics than in other disciplines to involve students early in active research, but it is certainly not impossible: I have done it with over 30 students in the past few years and it nearly always worked out very well. I also believe that, in terms of criteria for graduate school admission, the only thing that correlates well with doing research (which is what we expect graduate students to be able to do) is doing research. Serious research involvement as undergraduates is a much more reliable predictive criterion than any GRE scores or such other things. In that respect, more involvement of undergraduates in actual creative mathematical work will also mean better and more reliable selection for graduate schools. There is of course an issue of opportunities: there may be good students who simply do not have access to good research problems and research mentors. This is a serious problem, but one that can be addressed by creating more opportunities, and serious programs and REUs that really do focus on creative research. In addition to all these considerations, there is the basic fact that if one wants to be a mathematician one wants to be a mathematician which means creating mathematics: it is only natural that students feel the need to be active player and not just passive exercise-solving-machines from as early as possible in their mathematical trajectory.

Thank you for sharing