Dornsife College of Letters, Arts and Sciences
Mathematics
Graduate Degrees
Admission Requirements
All applicants must take the Graduate Record Examinations General Test.
Master of Arts and Doctor of Philosophy in Mathematics and in Applied Mathematics
A substantial undergraduate background in mathematics which includes one year of real analysis (MATH 425ab), one semester of abstract algebra (MATH 410) and one semester of upper-division linear algebra (MATH 471) is required. Students enrolled in one of the department’s master of science or arts programs must complete the Ph.D. screening procedure prior to admission to a Ph.D. program.
Master of Science in Applied Mathematics, in Statistics and in Computational Molecular Biology
A substantial undergraduate background in mathematics which includes one semester of real analysis or advanced calculus and one semester of linear algebra is required.
Regular admission pending completion during the first year of graduate studies of prerequisite undergraduate mathematics may be considered for applicants who otherwise qualify for the program.
Degree Requirements
These degrees are under the jurisdiction of the Graduate School. Refer to the Requirements for Graduation section and the Graduate School section of this catalogue for general regulations. All courses applied toward the degrees must be courses accepted by the Graduate School.
Master of Science in Applied Mathematics
This program is intended for individuals who are seeking or currently hold positions which involve mathematical applications, or for mid-career people wishing to improve their skills in applied areas. Specific options in the program include: biomedicine, discrete mathematics, economics, finance and business economics, fluid dynamics, numerical analysis and computation, and systems and control. In addition, students may design their own option to suit specific needs.
On admission to the program, each student is assigned an option adviser. The adviser serves on the student’s master’s committee and assists the student in determining the courses of study in the selected option. Courses of instruction are drawn from the Department of Mathematics and other participating departments which include: aerospace engineering, biomedical engineering, civil engineering, computer science, economics, electrical engineering, business administration, mechanical engineering, physiology and biophysics, and preventive medicine.
| Required courses | Units | |
|---|---|---|
| MATH 501 | Numerical Analysis and Computation | 3 |
| MATH 505ab | Applied Probability | 3-3 |
| MATH 570a | Methods of Applied Mathematics | 3 |
| MATH 601 | Optimization Theory and Techniques | 3 |
| plus at least 15 units of elected option courses |
In addition, registration in MATH 594ab and a master’s thesis is required for all students. This thesis is the end product of a practicum in the selected option. The practicum is supervised by the student’s master’s committee.
For this program students are not required to take the screening examination or to satisfy a foreign language requirement.
Master of Science in Mathematical Finance
See Mathematical Finance.
Master of Science in Statistics
The object of this program is to provide academic instruction in statistical theory with a solid mathematical foundation while emphasizing applications to real world problems. Some probability theory is included to provide a rigorous foundation. The program is intended for individuals who are seeking or currently hold positions that involve statistical methodology and practice. A student may orient his or her course of study toward a particular field of application through appropriate selections from the program listings plus elective courses from other disciplines.
Course Requirements
Thirty units of course work are required, including:
| Required courses | Units | |
|---|---|---|
| MATH 541ab | Introduction to Mathematical Statistics | 3-3 |
| MATH 650 | Seminar in Statistical Consulting | 3 |
| and one from each of options A, B, C: | ||
| (A) | ||
| MATH 505a | Applied Probability | 3 |
| MATH 507a | Theory of Probability | 3 |
| (B) | ||
| MATH 542L | Analysis of Variance and Design | 3 |
| MATH 545L | Introduction to Time Series | 3 |
| (C) | ||
| MATH 501 | Numerical Analysis and Computation | 3 |
| MATH 502a | Numerical Analysis | 3 |
| PM 511a | Data Analysis | 4 |
| plus at least 12 units of adviser-approved courses |
Students may opt for a master’s thesis (and registration in MATH 594ab) or a written examination covering material from MATH 505a or MATH 507a, and MATH 541ab. The examination will normally be given at the end of the fall semester. Students must pass MATH 505a or MATH 507a, and MATH 541ab with a grade of B or higher. If a student receives a grade of B- or lower in any of these courses, the requirement can be waived upon passing the screening exam for the course at the master’s level or higher.
Master of Science in Computational Molecular Biology
The computational molecular biology program is designed to attract recent graduates in either mathematics, statistics, biology or computer science, or scientists and engineers interested in retraining. A commercial or laboratory internship is required. Students will be prepared for employment in the rapidly expanding areas of computational molecular biology and bioinformatics. The program has two tracks, appropriate for different undergraduate backgrounds: biology and mathematical science. The required courses for each track are indicated below.
| Required Courses | Units | |
|---|---|---|
| Biological Sciences | ||
| BISC 403*** | Advanced Molecular Biology | 4 |
| BISC 478** | Computational Genome Analysis | 4 |
| BISC 505* | Genomics and Molecular Genetics | 4 |
| BISC 542* | Seminar in Molecular Biology | 3 |
| BISC 577ab* | Computational Molecular Biology Laboratory | 2-2 |
| Computer Science | ||
| CSCI 485** | File and Database Management | 3 |
| CSCI 570*** | Analysis of Algorithms | 3 |
| Mathematics | ||
| MATH 407** | Probability Theory | 4 |
| MATH 408** | Mathematical Statistics | 4 |
| MATH 505a*** | Applied Probability | 3 |
| MATH 541a*** | Introduction to Mathematical Statistics | 3 |
| MATH 578a*** | DNA and Protein Sequence Analysis | 3 |
| MATH 592* | Computational Molecular Biology Internship | 3 |
| MATH 650* | Seminar in Statistical Consulting | 3 |
| Total units | 32-33 | |
Students are required to demonstrate skill in C++, Java or Perl, and to demonstrate knowledge of molecular biology at the level of BISC 320L. A substantial report on the commercial or laboratory internship must be submitted (for which enrollment in MATH 592 is required).
Master of Arts in Mathematics and Master of Arts in Applied Mathematics
The objective of the Master of Arts program is to prepare students for research, teaching and other professional careers in mathematics and applied mathematics, respectively. In addition to the algebra requirement and differential geometry/topology option for the Master of Arts in Mathematics, the two programs differ in emphasis: the Master of Arts in Mathematics emphasizes the core courses in pure mathematics, and the Master of Arts in Applied Mathematics emphasizes courses in mathematics and affiliated fields that are fundamental in applied mathematics.
Relationship to Ph.D. Programs in Mathematics and in Applied Mathematics
The two year M.A. program is an expansion of the first year of graduate studies in the Ph.D. program in mathematics (respectively, the Ph.D. program in applied mathematics). The program provides a rigorous foundation in mathematics (applied mathematics) while affording students additional time for preparatory training. The comprehensive examinations for the M.A. program can serve as the preliminary qualifying examination for either Ph.D. program, and the written Ph.D. qualifying examinations serve as comprehensive examinations for the corresponding Master of Arts degree.
Requirements for the Master of Arts in Mathematics
At least 24 units are required, including:
| Required courses | Units | |
|---|---|---|
| MATH 510ab | Algebra | 3-3 |
| MATH 520 | Complex Analysis | 3 |
| MATH 525a | Real Analysis | 3 |
| and one option from A, B, C or D: | ||
| (A) | ||
| MATH 535a | Differential Geometry | 3 |
| MATH 540 | Topology | 3 |
| (B) | ||
| MATH 555a | Partial Differential Equations | 3 |
| MATH 565a | Ordinary Differential Equations | 3 |
| (C) | ||
| MATH 507a | Theory of Probability | 3 |
| MATH 541b | Introduction to Mathematical Statistics | 3 |
| (D) | ||
| MATH 502ab | Numerical Analysis | 3-3 |
The degree is completed with either departmental examinations (two written examinations selected from the two required components and the optional component) or a thesis demonstrating research ability in pure mathematics (the thesis option requires four additional thesis units selected from MATH 594abz).
Requirements for the Master of Arts in Applied Mathematics
At least 24 units are required, including MATH 525a Real Analysis (3), and at least three from these courses:
| MATH 502a | Numerical Analysis | 3 |
| MATH 502b | Numerical Analysis | 3 |
| MATH 505a | Applied Probability, or | |
| MATH 507a | Theory of Probability | 3 |
| MATH 505b | Applied Probability, or | |
| MATH 506 | Stochastic Processes, or | |
| MATH 507b | Theory of Probability | 3 |
| MATH 541a | Introduction to Mathematical Statistics | 3 |
| MATH 541b | Introduction to Mathematical Statistics | 3 |
| MATH 555a | Partial Differential Equations | 3 |
| MATH 565a | Ordinary Differential Equations | 3 |
Other elective courses, including those from other departments, have to be approved by the program adviser.
The degree is completed with either departmental comprehensive examinations (two examinations, one covering the required component MATH 525a, and the second covering one of the elective MATH courses) or a thesis demonstrating research ability in applied mathematics (the thesis option requires four additional thesis units selected from MATH 594abz).
Doctor of Philosophy in Applied Mathematics
Application deadline: January 1
The program requires a maximum effort by the student for a minimum of four years of full-time work.
Screening Procedure
The screening examination consists of four one-hour examinations covering the subject content of: MATH 502a Numerical Analysis; MATH 505a Applied Probability (or, at the student’s discretion, MATH 507a Theory of Probability); MATH 525a Real Analysis; and MATH 541a Introduction to Mathematical Statistics.
The department offers the examinations twice a year, at the end of summer and at the beginning of the spring semester. All four parts of the screening examination must be attempted by the end of the third semester (not counting summer sessions) in the program. The students may take each of the exams as many times as they wish, but three of the exams must be successfully completed by the end of the fourth semester (not counting summer sessions) and all four must be successfully completed by the end of the sixth semester in the program. The qualifying examination should follow two or three semesters after the successful completion of the screening procedure.
Qualifying Exam Committee
No later than at the end of the first semester after passing the screening procedure the student must form a qualifying exam committee consisting of an adviser and four other faculty members, including at least one from another department.
Qualifying Examination
The written portion of the qualifying examination consists of a Ph.D. dissertation proposal. This document should include: introduction, statement of the problem, literature survey, methodology, summary of preliminary results, proposed research, references, appendix (including one or two fundamental references).
The oral portion of the qualifying examination consists of a presentation of the Ph.D. dissertation proposal. The student must demonstrate research potential.
Course Requirements
The student must complete, with no grade lower than B, a minimum of 60 units of courses carrying graduate credit and approved by the qualifying exam committee. These must include MATH 794ab and six courses from the following: MATH 502b, MATH 504ab, MATH 505b, MATH 507b, MATH 509, MATH 520, MATH 525b, MATH 530b, MATH 532, MATH 541b, MATH 542L, MATH 545, MATH 555a, MATH 565a, MATH 574, MATH 576, MATH 580, MATH 585.
Transfer of Credit
No transfer of credit will be considered until the screening examination is passed. A maximum of 30 units of graduate work at another institution may be applied toward the course requirements for the Ph.D. A grade of B- (A = 4.0) or lower will not be accepted and, at most, two grades of B will be accepted. A Ph.D. candidate may petition the department for transfer of additional credit, after he or she passes the qualifying examination.
Foreign Language Requirement
The student must demonstrate a reading comprehension of mathematics in one language (other than English) in which there is a significant body of research mathematics (such as Chinese, French, German, Japanese and Russian) by passing a written examination, administered by the Mathematics Department, in translation of mathematical content.
Dissertation
Following passage of the screening examination and approval of a dissertation topic by the qualifying exam committee, the student begins research toward the dissertation under the supervision of the dissertation committee. The primary requirement of the Ph.D. is an acceptable dissertation based on a substantial amount of original research conducted by the student.
Research Areas
Opportunities for research are available from the faculty in several areas of applied mathematics with an emphasis on: computational biology, control theory, financial mathematics, mathematical neurosciences, numerical analysis, optimization, scientific computing, statistical genetics, statistics and stochastic differential equations.
Doctor of Philosophy in Mathematics
Application deadline: January 1
The program requires the maximum endeavor by the student for normally a minimum of four years of full-time work.
The student must choose between two concentrations: Pure Mathematics or Pure and Applied Mathematics.
Screening Procedure
Appointment of a qualifying exam committee and retention in the doctoral program are contingent on passing the preliminary qualifying examination by the end of the second semester. If a student fails the examination, the department, at its discretion, may permit the student to take it again during the third semester of graduate studies.
The preliminary qualifying exam is a written two-hour examination administered by the department. The student must choose between two options: analysis or algebra. Each option approximately covers the content of two one-semester graduate courses, with the precise list of possible topics made available to the student by the department.
Course Requirements
The student must complete with no grade lower than B a minimum of 60 units of courses carrying graduate credit and approved by the qualifying exam committee.
Pure Mathematics Concentration
| Required Courses | units | |
|---|---|---|
| MATH 510a | Algebra | 3 |
| MATH 525a | Real Analysis | 3 |
| MATH 535a | Differential Geometry | 3 |
| MATH 794ab | Doctoral Dissertation | 2-2 |
| Five courses selected from the following: | ||
| MATH 507a | Theory of Probability | 3 |
| MATH 510b | Algebra | 3 |
| MATH 520 | Complex Analysis | 3 |
| MATH 525b | Real Analysis | 3 |
| MATH 532 | Combinatorial Analysis | 3 |
| MATH 540 | Topology | 3 |
| MATH 555a | Partial Differential Equations | 3 |
| MATH 565a | Ordinary Differential Equations | 3 |
Pure and Applied Mathematics Concentration
| Required Courses | units | |
|---|---|---|
| MATH 502a | Numerical Analysis | 3 |
| MATH 510a | Algebra | 3 |
| MATH 525a | Real Analysis | 3 |
| MATH 794ab | Doctoral Dissertation | 2-2 |
| Five courses selected from the following: | ||
| MATH 502b | Numerical Analysis | 3 |
| MATH 507a | Theory of Probability | 3 |
| MATH 520 | Complex Analysis | 3 |
| MATH 525b | Real Analysis | 3 |
| MATH 532 | Combinatorial Analysis | 3 |
| MATH 541a | Introduction to Mathematical Statistics | 3 |
| MATH 555a | Partial Differential Equations | 3 |
| MATH 565a | Ordinary Differential Equations | 3 |
Transfer of Credit
No transfer of credit will be considered until the screening examination is passed. Normally a maximum of 30 units of graduate work at another institution may be applied toward the course requirements for the Ph.D. A grade of B- or lower will not be accepted, and, at most, two grades of B will be accepted. A Ph.D. candidate may petition the department for transfer of additional credit after passing the qualifying examination.
Foreign Language Requirement
The student must demonstrate a reading comprehension of mathematics in one language (other than English) in which there is a significant body of research mathematics (such as Chinese, French, German, Japanese and Russian) by passing a written examination, administered by the department, in translation of mathematical content.
Qualifying Examination
The written portion of the qualifying examination is comprehensive, consisting of two, two-hour examinations administered by the department. These examinations cover two out of the following five options, excluding the option already selected for the preliminary examination: algebra, analysis, geometry/topology, probability/statistics, differential equations. Each option approximately covers the content of two, one-semester graduate courses, with the precise list of possible topics made available to the students by the department. The selection of options must be approved by the qualifying exam committee.
The oral portion of the qualifying examination covers one topic selected from department research areas in mathematics and approved by the qualifying exam committee. The student must demonstrate research potential in this field.
Dissertation
Following passage of the qualifying examination and approval of a dissertation topic by the qualifying exam committee, the student begins research toward the dissertation under the supervision of the dissertation committee. The primary requirement for the Ph.D. is an acceptable dissertation which is based on a substantial amount of original research conducted by the student.
Research Areas
Opportunities for research are offered in the areas of algebraic geometry, arithmetic geometry, combinatorics, complex geometry, control theory, differential equations, differential geometry, dynamical systems, functional analysis, geometric analysis, group theory, K-theory, nonlinear analysis, number theory, numerical analysis, optimization, probability, representation theory, ring theory and topology.