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.
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. Theses sometimes do contain publishable original research, but it is difficult to get that kind of result in the time available. Quite a few students enter Princeton with advanced placement in mathematics, but it is not at all necessary for majoring in the subject. 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.