Applied Mathematics Ph.D.


Graduate studies in applied  mathematics (AM) at UC Santa Cruz focus on developing skillsets in mathematical modeling, analysis, and scientific computation applied to a broad range of science and engineering disciplines, including fluid mechanics, high-performance computing, mathematical biology, dynamical systems, stochastic processes, control, and optimization. The overarching goal of the AM graduate programs at UC Santa Cruz is to underscore the application of mathematics in solving real-life problems. In particular, the doctor of philosophy (Ph.D.) program prepares the students with the foundational tools of applied mathematics, enabling them to perform independent research that transcends the disciplinary boundaries in science and engineering.

Advancement to Candidacy

Course Requirements

Foundational Courses

Students in the AM Ph.D. program must demonstrate mastery in the foundations of scientific computing and applied mathematics, either by submitting evidence of undergraduate transcripts to the AM graduate director for approval or by taking all of the following foundational courses upon entry to the Ph.D. program, or a combination of the two, by the end of their first year:

All of the following three courses
AM 100Mathematical Methods for Engineers


AM 129Foundations of Scientific Computing for Scientists and Engineers


AM 147Computational Methods and Applications


These foundational courses can be taken for letter grades or the Satisfactory/Unsatisfactory (S/U) option.

Core Courses

All AM Ph.D. students must complete the following core courses. All non-seminar core courses (i.e., AM 213A, AM 213A, AM 213B, and AM 214) must be taken for letter grades. The AM 280 seminar series (i.e., AM 280A, AM 280B, and AM 280C) needs to be taken at least three times in a combination of the three courses. The combination must include at least one time of AM 280B.

AM 212AApplied Partial Differential Equations


AM 213ANumerical Linear Algebra


AM 213BNumerical Methods for the Solution of Differential Equations


AM 214Applied Dynamical Systems


AM 280ASeminar in Mathematical and Computational Biology


AM 280BSeminar in Applied Mathematical Modeling


AM 280CSeminar in Geophysical and Astrophysical Fluid Dynamics


Elective Courses

In addition to these 26 credits, Ph.D. students must complete six additional 5-credit courses for a total requirement of 56 credits. As part of the six additional courses, students are required to take (at least) one elective course (first-year elective) during their first year to help them engage in research as early as possible. First-year electives are designed to prepare students for their ultimate research emphasis within applied mathematics. These electives can be selected from any non-core 5-credit AM graduate courses (level 200 or above), and must be approved by either the student's first-year advisor,  official advisor, or the graduate director.

Ph.D. students 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 advisor toward their advancement to candidacy.

All elective courses, including the AM courses and non-AM courses, must be approved by the student's first-year advisor, official advisor, or the graduate director. Elective courses can be taken for the Satisfactory/Unsatisfactory grading option or the letter grade option. 

Transfer of Credit

Up to three UC Santa Cruz courses (15 credits) fulfilling the degree requirements of the Ph.D. degree may be taken during students' undergraduate study at UC Santa Cruz. Note that this 15-credit limit does not apply to the requirements of the three foundational courses (AM 100, AM 129, and AM 147), which may all be taken prior to the start of the Ph.D. program without penalty. Ph.D. students who have previously earned an M.S. degree in a related field at another institution may substitute courses from their previous university with the approval of the student’s official advisor and the graduate committee.

Petitions should be submitted along with the transcript from the other institution or from their UC Santa Cruz undergraduate study. For courses taken at other institutions, copies of the syllabi, exams, and other coursework should accompany the petition. Such petitions are not considered until the completion of at least one quarter at UC Santa Cruz. At most, a total of three courses may be transferred.

Students who complete at UC Santa Cruz the M.S. degree in AM or the M.S. degree in Scientific Computing and Applied Mathematics (SciCAM) and continue on to the Ph.D. program in AM at UC Santa Cruz 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 UC Santa Cruz Graduate Division.

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. 

Ph.D. students whose native language is not English must take the Test of English as a Foreign Language (TOEFL) exam or the International English Language Testing System (IELTS) exam and score a minimum of 26 on the spoken portion of the Internet-based TOEFL or an overall 8 on the IELTS before being considered for a position of teaching assistant. The English-language requirement for teaching assistants is may be waived for students who have received an M.S. degree from an English-speaking institution and with the experience as a teaching assistant at that university. International students may also participate in the Graduate Preparation Program (GPP) (the registration deadline is mid-July annually) prior to starting the graduate program. Successful completion of the GPP satisfies the English-language requirement for teaching assistants. See more details on the English language waiver for teaching assistantship.

Pre-Qualifying Requirements

At the end of the first year, all Ph.D. students will take a pre-qualifying examination in June covering all (non-seminar) core and foundational courses, also called the Ph.D. First-Year Exam (FYE). This examination is a take-home project involving analysis, simulations, and writing a formal written report. Ph.D. students who do not pass this examination will be allowed to retake it in mid-late September 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 form a Qualifying Exam (QE) committee in preparation of the exam. This committee must consist of at least four members as follows:

  • QE Chair:  This person must be tenured AM faculty and cannot be the student's Ph.D. advisor(s).
  • Member 1:  Ladder-rank AM faculty member
  • Member 2:  Ladder-rank UCSC faculty member (any department).
  • External Member: This person must be tenured UCSC faculty from outside AM or tenured faculty in the same discipline at another institution or a recognized non-faculty expert in the student's research area with credentials equivalent to a ladder-rank UCSC faculty member as judged by the graduate director and the dean of Graduate Studies. The External Member cannot be the student's faculty advisor(s). 

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 at the latest. The oral proposal defense is a public seminar (open session) followed by an oral qualifying examination (closed session) given by the qualifying committee. The student’s oral presentation must be approximately 45 minutes in length. Ph.D. students will also be required to submit a substantial written document to the qualifying examination committee, describing their research to date as well as their Ph.D. proposal ahead of time.

Students will advance to candidacy after they have completed all course requirements (including removal of all incomplete grades), passed the pre-qualifying examination (i.e., FYE) and the qualifying examination (i.e., oral proposal defense), 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 the student's third year. A student who has not advanced to candidacy by the start of the fourth year will be subject to academic probation.


Upon successful completion of the qualifying examination, a dissertation reading committee must be formed consisting of the dissertation supervisor and at least two additional readers appointed by the graduate director upon recommendation of the dissertation supervisor. Students should consult their advisor(s) about the membership of their committee.

The committee must include the following:

  • Committee Chair: This person should be an AM faculty member (usually the student’s Ph.D. advisor). 
  • Member 1: Ladder-rank AM faculty member who is not the student's primary Ph.D.  advisor.
  • Member 2: Ladder-rank UCSC faculty (may be the student’s Ph.D. advisor) or a recognized expert in the student's research area with credentials equivalent to a ladder-rank UCSC faculty member as judged by the graduate director and the dean of Graduate Studies.

At least two members of the committee must be AM faculty. If a student has two co-advisors, they should both be listed as co-chairs. Additional members may be added to the committee. The committee is subject to the approval of the Graduate Division.

Post-Qualifying Requirements

After the advancement, Ph.D. students shall consult with their dissertation committee to present and discuss progress made and receive comments and suggestions from their committee members.

Relationship of AM Master's Program and AM Doctoral Program

The M.S. and Ph.D. programs are freestanding and independent, so that students can be admitted to either program. Students completing the M.S. program may request to transfer into the Ph.D. program (provided they pass the pre-qualifying examination). Students in the Ph.D. program can receive a non-terminal M.S. degree upon completion of the 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. The defense consists of a public presentation (open session) of the research followed by a private examination (closed session) by the reading committee. Successful completion of the dissertation defense is the final requirement for the Ph.D. degree.

Review of Academic Progress

Each year, the faculty reviews the progress of every student in the graduate programs. Students who do not make adequate progress toward completion of degree requirements are subject to dismissal from the program (see the UC Santa Cruz Graduate Handbook and the AM Graduate Handbook for the policy on satisfactory academic progress). Also, please refer to the Graduate Division's specific guidelines on annual student reviews. In addition, students will work with their academic advisors (e.g., first-year advisor or official Ph.D. advisor) to fill out an AM advising form at least once a year to discuss the overall academic progress, research relationship, and students’ welfare.

Applying for Graduation

All candidates for a degree must submit an application for the Ph.D. degree to the Baskin Engineering Graduate Advising Office by the date stated in the Academic and Administrative Calendar for the quarter you wish to receive the degree.  The deadline for degree applications is typically in the second week of the quarter.

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 staff advisor to determine which option fits their situation. For more information about applying for graduation, visit the Baskin Engineering Graduate Studies website