Graduate Program

Requirements

The Program in Applied and Computational Mathematics offers a select group of highly qualified students the opportunity to obtain a thorough knowledge of branches of mathematics indispensable to science and engineering applications, including numerical analysis and other computational methods.

It will be the student’s responsibility to choose three areas in which to be examined out of a list of six possibilities to be specified below. This choice of topics should be achieved by the end of October. The director of graduate studies, in consultation with the student, will then appoint a set of advisers from among the faculty and associated faculty. The adviser in each topic will meet regularly with the student, monitor progress and assign additional reading material. They can be any member of the University faculty, but normally would be either program or associated faculty. The first-year student should choose three topics from among the following six applied mathematics categories:

  • Asymptotics, analysis, numerical analysis and signal processing;
  • Discrete mathematics, combinatorics, algorithms, computational geometry and graphics;
  • Mechanics and field theories (including computational physics/chemistry/biology);
  • Optimization (including linear and nonlinear programming and control theory);
  • Partial differential equations and ordinary differential equations (including dynamical systems); and
  • Stochastic modeling, probability, statistics and information theory.

Other topics as special exceptions might be possible, provided they are approved in advance by the director of graduate studies. Typically, students take regular or reading courses with their advisers in each of the three areas, completing the regular exams and course work for these courses. 

Preliminary Exam

At the end of the first year, first-year students will also take a preliminary exam, consisting of a joint interview by their three first-year advisers. Each student should discuss with their first-year advisers which of these courses are relevant for their areas.

In order to assess whether they have sufficient preparation, or whether it would be good to take a particular course, it is a good idea to obtain some typical homework or a final exam from a previous year.  If the student fails the preliminary examination or a part thereof the first time, they make take it a second time.  

General Examination

Before being admitted to a third year of study, students must pass the general examination. The general examination, or generals, is designed as a sequence of interviews with assigned professors that takes place during the first year and covers three areas of applied mathematics. The generals culminate in a seminar on a research topic, usually delivered toward the end of the fourth term.

A student who completes all departmental requirements (coursework, preliminary exams, with no incompletes) but fails the general examination may take it a second time. If the student fails the general examination a second time, then Ph.D. candidacy is automatically terminated. 

Master of Arts

The Master of Arts degree is normally an incidental degree on the way to full Ph.D. candidacy, but may also be awarded to students who for various reasons leave the Ph.D. program. Students who have satisfactorily passed required coursework including the resolution of any incompletes and have passed the preliminary exam, may be awarded an M.A. degree. Students must complete the required “Advanced Degree Application form” upon learning the Department’s determination of their candidacy in order to receive the M.A.

Doctoral Dissertation

The doctoral dissertation may consist of a mathematical contribution to some field of science or engineering , or the development or analysis of mathematical or computational methods useful for, inspired by, or relevant to science or engineering.

Satisfactory completion of the requirements leads to the degree of Doctor of Philosophy in applied and computational mathematics.