Applied Mathematics Ph.D.


Advancement to Candidacy

Course Requirements

Required Core Courses

All applied math Ph.D. students must complete the following core courses:

AM211Foundations of Applied Mathematics


AM212AApplied Mathematical Methods I


AM213ANumerical Linear Algebra


AM213BNumerical Methods for the Solution of Differential Equations


AM214Applied Dynamical Systems


AM280BSeminar in Applied Mathematical Modeling



In addition to these 31 credits, doctor of philosophy (Ph.D.) students must complete five 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.

Transfer Credit

Up to three School of Engineering courses fulfilling the degree requirements of the Ph.D. degree may be taken before beginning the graduate program through the concurrent enrollment program. Ph.D. students who have previously earned a master's degree in a related field at another institution may substitute courses from their previous university with approval of the adviser and the graduate committee. Courses from other institutions may not be applied to the M.S. degree course requirements. Petitions should be submitted along with the transcript from the other institution or UCSC Extension. For courses taken at other institutions, copies of the syllabi, exams, and other course work should accompany the petition. Such petitions are not considered until the completion of at least one quarter at UCSC. At most, a total of three courses may be transferred from concurrent enrollment and other institutions.

Students who complete the M.S. degree in Applied Mathematics or the M.S. degree in Scientific Computing and Applied Mathematics at UCSC and continue on to the Ph.D. program in Applied Mathematics at UCSC can transfer all applicable courses taken during the M.S. to the Ph.D. program, provided that such students meet the minimum residency requirement for Ph.D. programs, as specified by the UCSC Graduate Division.

Foreign Language Requirements

Applicants whose native language is not English must take the TOEFL or IELTS exam and submit an official score report with the admission application. For admission purposes, the minimum passing score for the TOEFL exam must be 570 on the paper-based, 230 on the computer-based, or 89 on the internet-based test. The minimum passing score for the IELTS exam is 7. If you have received a degree from an English-speaking institution or have lived in the United States for more than four years, you are eligible for a TOEFL exemption.

To be considered for teaching assistant support you must have scored a minimum of 26 on the spoken portion of the Internet-based TOEFL or an overall 8 on the IELTS. International students may also participate in the Graduate Preparation Program (GPP) prior to starting the graduate program. Successful completion of the GPP satisfies the English-language requirement for teaching assistants.

Teaching Requirement

Ph.D. students will be required to serve as teaching assistants for at least two quarters during their graduate study. Certain exceptions may be permitted for those with extensive prior teaching experience, for those who are not allowed to be employed due to visa regulations, or for other reasons approved by the graduate director.

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 will have two parts: an in-class written examination, followed by a take-home project. 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.

Qualifying Examination

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.

Relationship of Master's and Doctoral Programs

The M.S. and Ph.D. programs are freestanding and independent, so that students can be admitted to either. Students completing the M.S. program may proceed into the Ph.D. program (provided they pass the pre-qualifying examination), and students in the Ph.D. program can receive a non-terminal M.S. degree upon completion of M.S. requirements, including the capstone research project. Each Ph.D. student will be required to have knowledge of applied mathematics equivalent to that required for the M.S. degree. In addition, Ph.D. candidates will be required to complete coursework beyond the M.S. level.



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). 

Dissertation Defense

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.

Academic Progress

Each year, the faculty reviews the progress of every student in the graduate programs. Students not making adequate progress toward completion of degree requirements are subject to dismissal from the program (see the Graduate Handbook for the policy on satisfactory academic progress). Also, please refer to specific guidelines on the annual student reviews.

Applying for Graduation

All candidates for a degree must submit an Application for Doctor of Philosophy Degree to the Graduate Advising Office by the date stated in the Academic and Administrative Calendar for the quarter you wish to receive the degree.  Failure to declare candidacy by the deadline means that you cannot be considered a candidate until the next term.

A student is required to be registered or on Filing Fee Status, whichever is applicable, during the quarter in which the degree is conferred. Students should consult the department adviser to determine which option fits their situation. For more information about applying for graduation, visit the Baskin School of Engineering Graduate Studies website