Applied Mathematics Curriculum

Ph.D. Requirements

Foundational Requirements

Students in the AM Ph.D. program must demonstrate mastery in the foundations of scientific computing and applied mathematics, either by producing evidence through undergraduate transcripts, or by taking some or all of the following foundational courses upon entry to the Ph.D. program, by the end of first year:

  • AM 147: Computational Methods and Applications
  • AM 209: Foundations of Scientific Computing OR
    AM 129: Foundations of Scientific Computing for Scientists or Engineers
  • AM 211: Foundations of Applied Mathematics OR
    AM 100: Mathematical Methods for Engineers

Course Requirements

Core Courses

PhD students must complete the core courses described below:

  • 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 280A: Seminar in Mathematical and Computational Biology
  • AM 280B: Seminar in Applied Mathematical Modeling
  • AM 280C: Seminar in Geophysical and Astrophysical Fluid Dynamics

All non-seminar core courses must be taken for letter grades. The AM 280 seminar series needs to be taken three times, including at least one time of AM 280B.

Electives

In addition to these 26 credits, doctor of philosophy (Ph.D.) students must complete six additional 5-credit courses, including a first-year elective (see below), for a total requirement of 56 credits. 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 is a take-home project involving analysis, simulations and writing a formal report. Ph.D. students who do not pass this examination 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. 

After passing the pre-qualifying examination, a Ph.D. student must form an advisory committee by fall of the second year. The advisory committee consists of two or more ladder-rank faculty members, and must include one ladder-rank faculty from within the Applied Mathematics Department. The student shall meet with the advisory committee annually until the advancement.

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.

Upon successful completion of the qualifying examination, a dissertation reading committee will be formed, consisting of the dissertation supervisor and at least two additional readers appointed by the graduate director upon recommendation of the dissertation supervisor. At least one of these additional readers must be in the Applied Mathematics Department. The committee is subject to the approval of the Graduate Division.

Students will advance to candidacy after they have completed all course requirements (including removal of all incomplete grades), passed the pre-qualifying examination and the qualifying examination, nominated their dissertation reading committee, and paid the advancement to candidacy fee. Under normal progress, a student will advance to candidacy by the end of the spring quarter of her/his third year. A student who has not advanced to candidacy by the start of the fourth year will be subject to academic probation.

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

Core Courses

MS students must complete the core courses described below:

  • 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

Electives

In addition to these 22 credits, master of science (M.S.) students must complete four additional 5-credit courses, including a first-year elective (see below), for a total requirement of 42 credits. 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. They must be taken during the first year, and must be selected from the following list:

AM 129: Foundations of Scientific Computing for Scientists and Engineers
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/EART 275: Magnetohydrodynamics
STAT 203: Introduction to Probability Theory

Students cannot receive credit for both AM 129 and AM 209.

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

M.S. 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 ladder-rank faculty members, 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.

AM PhD PLO
  1. Mastery of the fundamental knowledge in applied mathematics.

  2. Ability to use analytical and computational methods to solve a problem.

  3. Ability to develop and apply mathematical methods to model a real-world problem in an application area, and understand its relevance within the research context.

  4. Ability to communicate concepts and results to both other experts in the field and to people outside the field.

  5. Ability to conduct independent research.

Program Learning Outcomes (M.S.)
AM MS PLO
  1. Proficiency with the fundamental knowledge in applied mathematics.

  2. Ability to use analytical and computational methods to solve a problem.

  3. Ability to apply mathematical methods to a real-world problem in an application area.

  4. Ability to communicate concepts and results to those with or without subject matter knowledge.