Applied and Computational Mathematics (ACM)

Caltech’s Computing & Mathematical Sciences department offers an interdisciplinary program of graduate study in applied and computational mathematics leading to the Ph.D. degree. This program is designed to give students a thorough training in fundamental computational and applied mathematics and to develop their research ability in a specific application field. The training essential for future careers in applied mathematics in academia, national laboratories, or in industry is provided by completion of the requirements for a Ph.D. degree in applied and computational mathematics.

The research areas and interests of the applied and computational mathematics faculty cover a broad spectrum representing the use of ideas from across the mathematical sciences to formulate, analyze, and develop algorithms for mathematical models. As reflected by the faculty research activities, there is a strong emphasis on computational methods and theoretical analysis for solving challenging problems arising from applications in engineering, the sciences, and the social sciences.

Reflecting the interdisciplinary nature of the program, several different groups, in addition to the applied and computational mathematics faculty, contribute to the teaching and supervision of research. Students in applied and computational mathematics are expected to combine their basic mathematical studies with deep involvement in some field of application. Basic general courses are listed specifically under applied and computational mathematics, and these are to be supplemented, according to the student’s interest, from the whole range of Institute courses in engineering, the sciences, and the social sciences.

A regular colloquium provides the opportunity for visitors, faculty, and students to discuss current research. In addition there are numerous other specialized seminars linked to ACM.

Each new graduate student admitted to work towards the Ph.D. in applied and computational mathematics selects course work in consultation with their advisor based on the student’s background and interests. The work of the student during the first year will usually include some independent reading and/or research.

During their graduate study, all ACM students are required to take a total of 12 graduate courses of 9+ units each, selected in consultation with their advisor. The courses ACM/IDS 104, ACM/EE/IDS 116, IDS/ACM/CS 157, IDS/ACM/CS 158, ACM 190, and CMS 290 will not count toward the Ph.D. requirements. The 12 classes must include no fewer than 3 from the core set (ACM/IDS 101ab, ACM/EE 106ab, CMS/ACM/IDS 107ab, CMS/ACM 117, ACM 118, CMS/ACM/EE 122); no fewer than 6 from the the core set together with any other ACM 100+ classes; and no fewer than 9 from the previous sets together with any 100+ level classes in EE/CMS/CS/Math. All classes must be taken for grades. The remaining classes will be selected among all level 100+ classes from any option. Three of the core classes are selected by the student, in consultation with their advisor, as the subject matter considered for the qualifying examination. This written exam is administered shortly before the start of the beginning of the second year of study, typically during the month of September (see below). In addition, CMS 290 is required for all first year ACM graduate students during each term (fall/winter/spring).

Only students who expect to pursue the Ph.D. degree will be admitted to the option. Ph.D. students in good standing may petition to add the M.S. degree after completing all course requirements. Research units may not be used to satisfy the course requirements for such an M.S. degree. The awarding of such an M.S. degree requires fulfillment of the Institute requirements for a master’s degree, satisfaction of the core course requirements, passing the ACM preliminary exam, and completion of at least 135 units of graduate work required by Institute regulations, including at least 81 units of advanced graduate work in applied mathematics. An ACM student who does not pass the preliminary exam may still be awarded an M.S. degree in exceptional cases at the discretion of the ACM Option Representative.

Before the beginning of the second year, all first-year students must take a preliminary examination administered by the faculty. Its purpose is to ensure a solid and broad knowledge in applied and computational mathematics, and in the event of a deficiency, to direct the students to necessary courses and reading.

To be recommended for candidacy for the Ph.D. degree in applied and computational mathematics, all students must, in addition to meeting the general Institute requirements and passing the preliminary examination, pass a candidacy examination administered by a committee that consists of at least four faculty, is approved by the option representative, and is chaired by an ACM faculty member who is not the student's primary research advisor. The examination will ascertain the student’s breadth and depth of preparation for research in the chosen area. The examination must be taken within the first three years of residence.

Upon passing the preliminary examination, the student is required to choose a thesis supervisor who assumes the major responsibility in supervising the Ph.D. thesis. At the same time, an advising committee consisting of three faculty members is formed to help oversee the advising process. This committee should be formed no later than the third year of graduate study. The student’s supervisor is part of this committee, but does not chair the committee. The student is encouraged to meet with the committee members informally for advice or suggestions. Joint supervision between two faculty members is also possible, as is seeking a thesis adviser outside the core applied and computational mathematics option, although in this case it is mandatory that an applied mathematics faculty member be nominated as a co-adviser.

Should a disagreement of any kind occur between the student and their supervisor as regards the timely completion of the thesis, the student is encouraged to direct their concerns to the committee chair. If this is not workable, the student should feel free to consult with the option representative, the executive officer, or an applied and computational mathematics faculty member of the student’s choice. If the student’s concerns cannot be resolved through consultation with these individuals, the student is encouraged to pursue resolution of their concerns through other channels as outlined in the Student Problem Resolution Process.

Academic progress will be reviewed regularly, including at least once annually by the option representative.

An ability to explain concepts and to verbally present one’s work is vital to a successful career in research and/or teaching. To this end, students may be required by their advisor and the option representative to occasionally serve as a teaching assistant.

The final oral examination is held within four weeks after the submission of the thesis. The examination covers the thesis and related areas.

The group of courses must differ markedly from the major subject of study, and must include 54 units of advanced courses in applied mathematics and must not be simultaneously used for fulfilling a requirement of the second option. The qualifying courses exclude ACM 95/100, although some flexibility is allowed depending upon the option of origin. The student must pass an oral examination whose subject is directly related to the material covered in the qualifying courses. This oral examination will be waived if the student has received a grade of A in every course.