Department of Mathematics

Princeton has been an internationally known center for mathematics since early in this century, when Dean Henry Fine (after whom Fine Hall is named), with the active encouragement of University President Woodrow Wilson, began to recruit outstanding research mathematicians to the faculty. In addition to having a distinguished senior faculty, the department regularly succeeds in attracting some of the best young mathematicians as junior faculty.

The departmental program is designed to accommodate diverse interests. The only fixed requirement is to take at least one course in each of three basic subjects: real analysis (deeper study of the ideas that come up in calculus), algebra (generalization and abstraction of topics such as linear algebra and the arithmetic of integers, at least in part), and complex analysis (describable as calculus of functions of complex variables, but with a surprisingly different character from ordinary calculus). Beyond this, a great deal of flexibility is possible. Those who want to go on to graduate work in pure mathematics can acquire a strong background with additional courses in the subjects already mentioned, and various other topics such as topology, logic, and differential geometry. By contrast, a program filled out with courses in probability, statistics, optimization methods, and mathematical economics is excellent preparation for business school. A program with emphasis on different kinds of applications could include courses in differential equations, numerical analysis, computer science, or mathematical methods in physics or engineering. The department routinely allows one or two courses with mathematical content taken in other departments to be counted toward the required total of eight departmentals, and the possibility of counting more than two cognates is not completely ruled out.

The ratio of faculty to departmental majors is high and allows for close student-faculty interaction. Most students do their junior independent work in seminars that give the opportunity to work in a small group and gain experience in reading and learning mathematics on one’s own. Seminar topics vary from year to year, but there is practically always a choice between theoretical and applied work.

Because the faculty in the department has a wide range of interests, there is great variety in the kinds of topics that can be pursued in the senior thesis. Also, students interested in working on applications of mathematics often get faculty in other departments to act as their thesis supervisors, and such arrangements have usually worked out well. Theses have been written on subjects ranging from very abstract topics in topology or algebra, to concrete investigations of computational methods, to a mathematical model explaining the shapes of tree leaves, to a strategic analysis of backgammon.

Students sometimes ask how it is possible to write a thesis in mathematics. The question is understandable because standard courses give the impression that everything was worked out centuries ago (which may be largely true of the material that is in the courses). In fact, mathematical discovery (or invention, depending on how you look at it) is going on at a rapid rate. It is true that the six months available for writing a thesis is not long enough for anyone to count on producing a major new theorem, but there is still scope for original and publishable work. A thesis is expected to demonstrate both a thorough understanding of a specialized topic and the ability to explain it in a clear and well-organized way. Moreover, the ready availability of computers makes a new kind of “experimental” thesis possible in mathematics.

Quite a few students enter Princeton with advanced placement in mathematics, but it is not at all necessary for majoring in the subject. Advanced placement provides interested students with an opportunity to take more challenging courses in mathematics (in exceptional cases perhaps including a graduate course), but the standard program provides a solid mathematical training. Satisfactory completion of either Mathematics 215–217–218 (alternatively 215–203–204) or 214–218–203 (alternatively 214-–203–204) is a prerequisite for admission to the department.

When deciding whether to major in mathematics, bear in mind that as one goes on to study more advanced mathematics (theoretical or applied), understanding what is going on becomes more and more important. In mathematics it is possible to get very good grades in high school and fairly good grades in underclass courses on the strength of one’s ability to apply rules in a somewhat mechanical way. Actually, it often makes sense to learn a procedure and practice its application before completely understanding the theory behind it—otherwise it may be hard to see the point of the theory. But to enjoy “higher” mathematics students must have enough interest in why things work to put some effort into understanding them.

People study mathematics for various reasons. Some do so simply because they find it interesting and enjoyable for its own sake. However, training in mathematics is good preparation for a number of careers. As with any subject, there is the possibility of an academic career. Knowledgeable teachers of mathematics and science are badly needed at all levels. Graduate schools of business and departments of computer science and statistics generally look favorably on applications from mathematics graduates, especially those who also have some courses in the relevant specialties. If mathematics appeals to you, there is every reason to take it as part of a liberal education. Also, quite a few graduates of the department have gone into law, medicine, or other careers in which mathematics is not specifically relevant.