Career paths in research mathematics

What is involved in becoming a research mathematician?

This page describes typical routes to a career in research mathematics, concentrating on pure mathematics and jobs in academia.

Becoming a math professor in one simple paragraph
The first step is lots of school: 4 years as an undergraduate, typically followed by 5 years of graduate school, resulting in a PhD. Then often comes 2 or 3 years of postdoctoral training (the "postdoc"). After the postdoc (or immediately after graduate school if not doing a postdoc), mathematicians who want a career in academia look for tenure track positions at a college or university. Academic jobs involve a mix of teaching, research, and service; the precise mix varies widely among different colleges and universities. After 3-7 years (usually less if you did a postdoc, more if you didn't) you come up for tenure. If you are successful then you have a "permanent" job as a professor.

Many mathematicians work for the government or various sectors of business and industry. The educational training is similar to the description above, except that the choice of classes and specializations may be different, and it is less common to do a postdoc. And the concept of "tenure" does not exist outside of the teaching professions.

Undergraduate studies
Most math majors take one math class each semester for the first year or so, and then two classes for their remaining undergraduate years. The first few classes are usually calculus, differential equations, and linear algebra. After those classes there are some options depending on whether the concentration is going to be in pure or applied mathematics.

In pure mathematics the focus quickly moves to writing proofs. The style of research mathematics is Theorem-Proof-Theorem-Proof, where "Theorem" refers to the precise statement of a mathematical result, and "Proof" refers to the logical argument which establishes the truth of the theorem based on previously established results.

In contrast to the proofs of high-school geometry, the proofs of abstract mathematics have the elegance of a poem -- the good proofs do, anyway.

Basic classes in the pure mathematics stream include abstract algebra, real analysis, topology, and complex analysis.

Basic classes in applied mathematics include partial differential equations, combinatorics, complex analysis (the same topic as in the pure stream, but sometimes taught from a different perspective), and mathematical modeling.

It is becoming more common to do "undergraduate research," often as part of a Research Experience for Undergraduates (REU) program sponsored by the NSF.

Graduate studies
Graduate school in mathematics usually begins with more emphasis on course work, and ends with more emphasis on research as the doctoral candidates work on his or her thesis. The "thesis" is the document which contains the original research of the candidate. In order to receive a PhD in pure mathematics, you have to prove new mathematical results; those results are presented in the thesis. The last step to a PhD is submitting the thesis to an examining committee and giving a public talk describing the results.

A masters degree is usually skipped or viewed as a formality for graduate students pursuing a PhD in mathematics.

In the first year or two, most mathematics graduate programs require comprehensive exams (also called preliminary exams or "prelims") In pure mathematics the (usually written, but sometimes oral) exams cover the core areas of Real Analysis, Complex Analysis, Algebra, and Topology. In applied mathematics programs, there is a greater variety of topics covered in the exams. Many programs have a second round of exams a year later, covering specialized material which is designed to determine if the student is prepared to begin work on a thesis problem. The second round of exams is almost always an oral exam, administered by a committee who keeps asking probing questions until you reach the end of your knowledge.

After (usually before, unofficially) passing all the exams, the doctoral candidate finds a thesis advisor. This requires a complicated balance that, when successful, produces a good match in terms of both mathematical interest and personality. This is a significant decision because the thesis advisor will suggest (or help the student find) a suitable problem whose solution will become the student's thesis. The advisor has a dominant influence on the research direction of the student, will write many letters of recommendation over the coming years, and pretty much is the determining factor on whether the students is successful. After graduation, the student will forever be known as "a student of [thesis advisor]."

After the exams, the student spends the next year or two (or three) working on a thesis. The term "dissertation" is used in many subjects, but in math the more common term is "thesis." It is common to also take a specialized course each semester, and also to participate in a weekly seminar. But working on the thesis is the dominant activity. Note that the verb is working, not writing. The actual writing of the thesis does not begin until after it is clear that the hoped-for results have actually been established. It is not uncommon to find a mistake and have to start over again.

Postdoctoral studies: the "postdoc"
A PhD thesis must contain a new mathematical result that had never been seen before. So what happens next?

It is a significant accomplishment to prove a new mathematical result, but proving one new result is relatively far from a viable research program which continues to make new contributions to the cutting edge of research. A postdoctoral research position, commonly called a "postdoc," can be a useful step in developing a research program.

After graduate school, it is important to start working on something new: something that is not just the logical next step after the thesis work. That might sound counterintuitive, but the point is that a long-term research program will not be viable if it just keeps moving in one narrow direction: it is important to branch out into new areas.

Just like graduate students, postdocs have an advisor, but the relationship is not as close as with the thesis advisor. The postdoc and his or her mentor are colleagues who work together on problems of common interest.

Most major research universities have a few rotating postdoctoral positions that last for 2 or 3 years. It is common for the more prestigious of those positions to have a name like "Lastname Assistant Professor," where Firstname Lastname is a moderately famous dead mathematician who used to be at that university. Although these positions are called "assistant professor," they are temporary positions, not tenure track as described below.

Academic jobs
Faculty positions at colleges and universities are the most common jobs for mathematicians whose research is in pure mathematics.

Most faculty are initially hired as "assistant professors," which means that they are employed provisionally until they receive tenure. The tenure recommendation is usually made by a committee from their department, which is then reviewed by the university administration. If successful, promotion is made to "associate professor." Such a tenured position is effectively permanent (but you can quit or take a different job, of course!).

Promotion to Professor, commonly called "full Professor" typically takes 10 more years, if it ever happens at all. Promotion to full professor is a recognition of significant research accomplishment (and also teaching accomplishment, at some institutions).

The balance between teaching and research varies greatly depending on the institution. At high-powered research universities there is little emphasis on teaching, and many faculty teach just one or two courses each semester. Tenure and promotion at such places is mostly determined by research output.

At smaller universities, a two course teaching load each semester is common, and there may be a greater emphasis placed on teaching quality.

At liberal arts colleges there is a strong emphasis on teaching, and there may also be an emphasis on research. The teaching load is higher than at universities, commonly 3 courses each semester.

At small 4-year colleges and at community colleges there may not be an expectation of research (although many faculty do continue doing research), and the teaching load is much higher -- up to 5 courses each semester.

Nonacademic jobs
There are many nonacademic positions for mathematicians. Most, but not all, tend to be in the more applied areas of mathematics.

The largest employer of mathematicians in the US is the National Security Agency (NSA), which requires sophisticated mathematics to make and break secret codes, and to take care of various other security matters.

The Insurance industry is based on mathematics. The mathematicians who work out things like how much your life insurance should cost (and more generally, determine the financial cost of risk) are called actuaries. There is a long sequence of specialized actuarial exams, covering the many specialized areas in which actuaries work. Typically the first couple of exams are taken while still a student, and passing those exams is necessary to be hired as an actuary. Then the other exams are taken as part of job advancement.

Other areas which rely heavily on mathematics, and which are common jobs for people with a PhD n mathematics, include network analysis, optimization, finance, operations research, physics and astronomy, and many areas of computer science. Recently there has been a huge influx of mathematics into various areas of the biological sciences.

David W. Farmer
May 5, 2009