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 five foundational courses

• AM 100: Mathematical Methods for Engineers
• AM 112: Introduction to Partial Differential Equations
• AM 114: Introduction to Dynamical Systems
• AM 129: Foundations of Scientific Computing for Scientists or Engineers
• AM 147: Computational 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 five non-seminar core courses (i.e., AM 212, AM 213A, AM 213B, AM 224 and AM 160) must be taken for letter grades. Students must also take at least six credits from a combination of three courses in the AM 280 seminar series (i.e., AM 280A, AM 280B, and AM 280C). The combination must include at least one quarter of AM 280B.

• AM 160: Introduction to Scientific Machine Learning
• AM 212: Advanced Methods of Applied Mathematics
• AM 213A: Numerical Linear Algebra
• AM 213B: Numerical Methods for the Solution of Differential Equations
• AM 224: 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

Elective Courses

In addition to these 31 credits, Ph.D. students must complete five additional 5-credit elective courses for a total requirement of 56 credits. As part of the five additional courses, students are required to take (at least) one elective course during their first year (called first-year elective) 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 regular 5-credit AM graduate courses (between course numbers 201 and 279).

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 graduate director. Elective courses can be taken for the Satisfactory/Unsatisfactory grading option or the letter grade option.

Pre-Qualifying Requirements

Under normal progress, Ph.D. students are expected to find a Ph.D. advisor and complete their core course requirements by the end of their second year. Soon after identifying an advisor, they are expected to prepare for a pre-candidacy exam under the guidance of their advisors) to demonstrate their suitability and commitment to research. Students are expected to pass the pre-candidacy exam after completing all core courses and no later than by the end of the summer quarter of their second year, or they will be put on academic notice. The pre-candidacy exam has two parts (written and oral), in which students are expected to review the literature and present a research proposal relevant to the subject area of their intended Ph.D. The written report must not exceed 10 pages and must be submitted to their committee at least one week prior to the planned oral exam date. The oral examination must cover, in no more than 30 minutes, both literature review and planned work. This will be followed by a question-and-answer session between the student and the committee during which the committee members can ask questions on core courses, literature review and proposed research. The committee will assess the student's suitability for future research in the Ph.D. program based on both written and oral performance on the exam.

The committee, which must be formed at least one month prior to the planned pre-candidacy exam date, is approved by the graduate director, and must include:

(i) The student's primary Ph.D. advisor (a ladder-rank faculty member in or outside of AM)

ii) Another ladder-rank AM faculty member, and

(iii) Another ladder-rank UCSC faculty member who is recognized as an expert in the student's research interest area with credentials equivalent to a ladder-rank UCSC faculty member as judged by the graduate director.

Students who do not pass the exam in their first attempt will be allowed to retake it once, and must do so at the very latest before the end of the fall quarter of their third year. 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

Under normal progress Ph.D. students are expected to advance to candidacy anytime between the quarter they pass the pre-candidacy exam and the spring quarter of their third year. A student who has not advanced to candidacy by the start of the fourth year will be subject to academic notice.

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 (for continuity, the members of this committee are ideally the same as those of the pre-candidacy exam committee, augmented with an external member):

• QE Chair: This person must be a tenured AM faculty and cannot be the student's
  Ph.D. advisors).
• 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 advisors).

The candidacy exam also has two parts (written and oral), in which students are expected to review the literature, present a substantially completed research project that is appropriate for submission to an international journal or conference, and present a plan to complete further research projects in pursuit of their Ph.D. The written report, that should not exceed 60 pages, must be submitted to their committee at least two weeks prior to the planned oral exam date. The oral examination must cover, in no more than 45 minutes, both literature review, and existing and planned research. This will be followed by a question-and-answer session between the student and the committee during which the committee members can ask questions on the literature review, and current and proposed research. The committee will assess the student's suitability for advancement to candidacy based on both written and oral performance on the exam.

Students can only advance to candidacy after they have completed all course requirements (including removal of all incomplete grades), passed both pre-candidacy and candidacy examinations, nominated their dissertation reading committee, and paid the advancement to candidacy fee.

Dissertation 

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 advisors) 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.