Graduate Degree in
Applied + Computational Mathematics
Aims and Scope of the Graduate Program
Graduate Option Rep
Prof. Peter Schroeder
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 fields of application include a wide range of areas such as fluid mechanics, materials science, and mathematical biology, engineering applications, image processing, and mathematical finance. 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, including nonlinear dynamics, computational biology, numerical analysis and scientific computing, computational and theoretical fluid mechanics, theoretical materials science, multiscale computations and homogenization theory, computational methods for electromagnetics and acoustics, statistics, signal and image processing, probability theory and stochastic analysis, and dynamical systems and geometric mechanics. As reflected by the faculty research activities, there is a strong emphasis on computational methods for solving challenging problems arising from engineering and scientific applications.
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 specific areas of physics, biology, engineering, etc.
A regular colloquium provides the opportunity for visitors, faculty, and students to discuss current research.
Each new graduate student admitted to work for the Ph.D. in applied and computational mathematics is given an informal interview on Thursday or Friday of the week preceding the beginning of instruction for the fall term. The purpose of this interview is to ascertain the preparation of the student and assist him or her in mapping out a course of study. The work of the student during the first year will usually include some independent reading and/or research.
All ACM students are required to take a total of 18 nine-unit courses at the graduate level (or the equivalent of 162 units) during their graduate study at Caltech. Among these 18 courses, the following core courses, typically taken during the student’s first year of study, are required: ACM 101 ab, ACM 105, CMS/ACM 107, Ma 108 abc ACM 106 ab, CMS/ACM 117, and an application elective course. The application elective course in the first year is selected, with the recommendation of the student’s adviser, from among a wide range of courses offered by an outside option within the Institute. In addition, CMS 290 is required for all first year CMS/ACM/CDS/CS graduate students during each term (fall/winter/spring). In the second and third years, students are expected to take graduate-level courses appropriate to their chosen research area. The remaining courses towards the 162 unit requirement would normally include graduate-level ACM or CMS courses such as CMS/ACM 113, 201 ab, 210 ab, 216, 217, CDS 140, etc., as deemed appropriate to the student’s research program, and which must be selected in consultation with the student’s research adviser.
Students who have already taken some of the required courses may use them to satisfy the course requirements, even though the units may not be used to satisfy the total unit requirement for the Ph.D. degree.
Students are not admitted to work toward the master’s degree. The master’s degree may be awarded to an ACM student only in exceptional circumstances. Of the 135 units of graduate work required by Institute regulations, at least 81 units of advanced graduate work should be in applied mathematics.
Degree of Doctor of Philosophy
- The Preliminary Examination. The student must pass an oral or written candidacy examination administered by a faculty committee. This examination is offered during the third term of the first year of graduate residence at the Institute. The material covered in this examination is based on the three core sequences described above. For students who have already taken the required courses before coming to Caltech, the examination can also be based on the substituted courses taken in the first year.
- The Candidacy Examination. 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 the student’s research adviser. 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.
- Advising and Thesis Supervision. 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 his or her supervisor as regards the timely completion of the thesis, the student is encouraged to direct his or her 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 his or her concerns through other channels as outlined in Student Problem Resolution Process on page 50.
- Submission of Thesis. On or before the first Monday in April of the year in which the degree is to be conferred, a candidate for the degree of Ph.D. in applied and computational mathematics must deliver a typewritten or printed copy of the completed thesis to his or her research supervisor.
- Final Examination. The final oral examination is held within four weeks after the submission of the thesis. The examination covers the thesis and related areas.
Subject Minor in Applied and Computational Mathematics
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 ACM95/100, 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.