Applied Mathematics Curriculum

Ph.D. Requirements

Course Requirements

PhD students must complete the core courses described below:

  • AM 211: Foundations of Applied Mathematics
  • AM 212A: Applied Mathematical Methods I
  • AM 213A: Numerical Linear Algebra
  • AM 213B: Numerical Methods for the Solution of Differential Equations
  • AM 214: Applied Dynamical Systems
  • AM 280B: Seminar in Applied Mathematical Modeling (3x)

In addition to these, PhD students must complete five additional 5-unit courses from the approved list, for a total requirement of 56 units. All elective courses must be approved by the student’s official adviser.

First-year electives are designed to prepare students for their ultimate research emphasis within applied mathematics. These electives can be selected from any 5-credit graduate AM courses (level 200 and above). 

Ph.D. students who already have an M.S. degree (or equivalent) will be allowed to substitute up to two elective courses with corresponding numbers of credits of independent study (i.e., 5 or 10), during which they conduct research with their adviser toward their advancement to candidacy.

Pre-Qualifying Requirements

At the end of the first year, all Ph.D. students will take a pre-qualifying examination covering the (non-seminar) core courses. This examination consists of a take-home project involving analysis, simulations, and writing a formal report. Students who do not pass this exam will be allowed to retake it before the start of the following fall quarter; if they fail the second examination they will not be allowed to continue in the Ph.D. program, but will have the option to continue into the M.S. program and exit with the M.S. as the terminal degree.

Qualifying Exam

Ph.D. students must complete the oral proposal defense, through which they advance to candidacy, by the end of the spring quarter of their third year. The proposal defense is a public seminar followed by an oral qualifying examination given by the qualifying committee. The student’s oral presentation must be approximately 45 minutes in length. Applied mathematics students will also be required to submit a substantial written document describing their research to date as well as their Ph.D. proposal ahead of time to the qualifying examination committee.

Additional information on the qualifying exam process, including links to required forms, can be found on the Advancement to Candidacy web page.

Dissertation Requirements

A dissertation is required for the Ph.D. degree. The dissertation will consist of a minimum of three chapters composed of material suitable for submission and publication in major professional journals in applied mathematics (or related subject areas of application). The completed dissertation will be submitted to the reading committee at least one month before the dissertation defense, which consists of a public presentation of the research followed by a private examination by the reading committee. Successful completion of the dissertation defense is the final requirement for the Ph.D. degree.

NOTE: This is an abbreviated version of the program requirements. Please review the Program Statement for a full explanation of all program requirements.

M.S. Requirements

Course Requirements

MS students must complete the core courses described below:

  • AM 211: Foundations of Applied Mathematics
  • AM 212A: Applied Mathematical Methods I
  • AM 213A: Numerical Linear Algebra
  • AM 213B: Numerical Methods for the Solution of Differential Equations
  • AM 214: Applied Dynamical Systems
  • AM 280B: Seminar in Applied Mathematical Modeling

In addition to these 27 units, MS students must complete three additional 5-unit courses, including a first-year elective from the approved list, for a total requirement of 42 units.

First-year electives are designed to prepare students for their ultimate research emphasis within applied mathematics. They must be taken during the first year, and must be selected from the following list:

AM 129 or AM 209, Foundations of Scientific Computing
AM 216, Stochastic Differential Equations
AM 217, Introduction to Fluid Dynamics
AM 227, Waves and Instabilities in Fluids
AM 229, Convex Optimization
AM 230, Numerical Optimization
AM 231, Nonlinear Control Theory
AM 232, Applied Optimal Control
AM 238, Fundamentals of Uncertainty Quantification in Computational Science and Engineering
AM 250, An Introduction to High Performance Computing
AM 260, Computational Fluid Dynamics
AM 275, Magnetohydrodynamics
STAT 203, Introduction to Probability Theory

M.S. students will be allowed to substitute one elective course for an independent study course with their required research project (see capstone requirement).

MS Capstone Requirements

A capstone project is required for the M.S. degree. For the M.S. degree, students will conduct a capstone research project. Students must submit a proposal to the potential faculty sponsor. If the proposal is accepted, the faculty member will become the sponsor and will supervise the research and writing of the project. The project will involve the solution of a problem or problems from the selected area of application. When the project is completed and written, it will be submitted to and must be accepted by a committee of two individuals, consisting of the faculty adviser and one additional reader. Additional readers will be chosen appropriately from within the Applied Mathematics Department or outside of it. Either the adviser or the additional reader must be from within the Applied Mathematics Department.

NOTE: This is an abbreviated version of the program requirements. Please review the Program Statement for a full explanation of all program requirements.