Denney Research Center 155
(213) 740-2400
FAX: (213) 740-2424
Email: info@math.usc.edu

Chair: Wayne Raskind, Ph.D.

USC Associates Chair in Natural Sciences and University Professor: Michael S. Waterman, Ph.D.

Herbert and Ruth Busemann Visiting Assistant Professorship in Mathematics: Ralph Kaufman, Ph.D.

Professors: Kenneth Alexander, Ph.D.; Richard Arratia, Ph.D.; Peter Baxendale, Ph.D.; Edward K. Blum, Ph.D.; Francis Bonahon, Ph.D.*; Ronald E. Bruck, Ph.D.; Richard S. Bucy, Ph.D. (Aerospace Engineering); Jaksa Cvitanic, Ph.D.; Larry Goldstein, Ph.D.; Solomon Golomb, Ph.D. (Electrical Engineering); Robert Guralnick, Ph.D.; Sheldon Kamienny, Ph.D.; P. Vijay Kumar, Ph.D. (Electrical Engineering); Charles Lanski, Ph.D.; M. Susan Montgomery, Ph.D.*; Paul K. Newton, Ph.D. (Aerospace and Mechanical Engineering); Robert C. Penner, Ph.D.; Pavel A. Pevzner, Ph.D.; Wlodek Proskurowski, Ph.D.; Wayne Raskind, Ph.D.; John E. Rolph, Ph.D. (Information and Operations Management); I. Gary Rosen, Ph.D.; Boris Rozovskii, Ph.D.; Robert J. Sacker, Ph.D.; Hubert Saleur, Ph.D. (Physics); Alan Schumitzky, Ph.D.; Simon Tavaré, Ph.D. (Biological Sciences); Firdaus E. Udwadia, Ph.D. (Mechanical Engineering); Zdenek Vorel, Ph.D.; Paul C. Yang, Ph.D.

Research Professor: George Hajj, Ph.D.

Associate Professors: Nicolai T.A. Haydn, Ph.D.; Igor Kukavica, Ph.D.; Ching Chieh Jay Kuo, Ph.D. (Electrical Engineering); Feodor Malikov, Ph.D.; Fengzhu Sun, Ph.D.; Chunming Wang, Ph.D.

Research Associate Professors: Xiaoqing Pi, Ph.D.; Larry J. Romans, Ph.D.

Assistant Professors: Thomas Geisser, Ph.D.; Sergey Lotofsky, Ph.D.

Research Assistant Professor: E. Robert Kursinski, Ph.D.

Emeritus Professors: Henry A. Antosiewicz, Ph.D.; Theodore E. Harris, Ph.D.*; B. Andreas Troesch, Ph.D.; Paul A. White, Ph.D.

*Recipient of university-wide or college teaching award.

The Department of Mathematics has designed its major to give students an understanding of the several areas of mathematics. The program of study allows students to use electives to prepare themselves for a specific field, whether in industry, teaching or advanced graduate research. The faculty is engaged in a wide variety of research activities and offers courses in many areas.

The department offers the B.S., B.A., and minor in Mathematics; M.S. in Applied Mathematics; M.S. in Mathematical Finance; M.S. in Statistics; M.A. in Mathematics; M.A. in Applied Mathematics; M.S. in Computational Molecular Biology; Ph.D. in Applied Mathematics; Ph.D. in Mathematics; and Ph.D. in Applied Mathematics.

The department strongly advises all mathematics students to take PHYS 151L, 152L and 153L concurrently with MATH 126, 225 and 226.

The university grants four units of credit in mathematics for scores of 3, 4 or 5.

MATH 125, 126, 225 and 226 are required.

Six upper division courses, including MATH 410, 425a and either 434 or 435, are required.

Eight upper division mathematics courses, including:

Four additional courses in natural sciences or computer science, but excluding courses in mathematics, are required. At least two of these must be upper division courses, and each of the four courses must be acceptable for the Bachelor of Science degree in the department in which it is offered.

For each undergraduate degree an overall GPA of 2.0 in all upper division courses taken for the degree is required. In addition, any upper division course specifically listed as required must be passed with a grade of C (2.0) or better (e.g., MATH 410, 425ab and 471 for the B.S. degree).

MATH 125, 126, 225 (or 245), 226 and four upper division courses in mathematics are required, one of which must be from MATH 410, 425a, 435, 440 or 471. The other three upper division courses must each be acceptable for either the B.A. or the B.S. degree in mathematics. These four upper division courses must total at least 16 units.

The honors program is available for mathematics majors. A student must apply to the department for admission. A minimum grade point average of 3.5 is required in the first two years of university work as well as in the lower division mathematics courses MATH 125, 126, 225 and 226.

The students must complete all requirements for the degree program in which they are enrolled. MATH 410, 425ab and 471 are required. The remaining upper division electives must be acceptable for the B.S. degree.

In addition, students in the honors program must register for at least four units of MATH 490x Directed Research.

The student must have an overall GPA of at least 3.5 in all upper division courses.

Those students intending to go on to graduate school should satisfy the language requirement in French, German or Russian.

This program permits exceptional students to receive both a bachelor’s and master’s degree in mathematics within a period of eight- to 10- semesters. It is intended for students with extraordinary secondary school mathematics preparation who demonstrate a superior level of overall scholarship. To complete the program in four calendar years, substantial unit credit upon admission to USC or attendance of summer sessions will be necessary. It is even possible to complete the requirements of the Accelerated Math Program and to minor in another department in an eight- to 10-semester course of study.

The minimum requirements for admission to the program are placement out of MATH 125 and 126 and acceptance into the program at the discretion of the Mathematics Depart-ment. Either the graduate algebra sequence MATH 510ab or the graduate analysis sequence MATH 525a and 520 must be completed in the junior year (with no grade lower than a B) to be considered for admission to the Graduate School for the senior year; thereafter, and as long as they meet standard Graduate School criteria to remain in good standing, participants will hold dual undergraduate and graduate status. A minimum overall GPA of 3.5 is required to remain in good standing in the program. The senior year and any additional semesters are dedicated in part to further work in mathematics. A master’s thesis is required.

University unit requirements for the B.S. and M.A. degrees must be satisfied, and the specific course requirements for completion of the program are as follows:

Successful completion or placement out of MATH 125, 126, 225 and 226.

Successful completion of MATH 410, 425ab and 471 (with a grade of at least C) and at least two additional courses from MATH 407, 408, 430, 432, 435, 440, 445, 458, 465 and 475.

Successful completion of MATH 510a, 510b, 525a and 520 and at least two additional courses from MATH 502ab, 506, 507, 525b, 535ab, 540, 545, 547, and 555ab.

Successful completion or placement out of the sequence PHYS 151L, 152L and 153L and an additional three courses from the natural sciences, physical sciences or computer science; at least two of these three additional courses must be upper-division.

For sample curricula and further information about this program, consult the brochure The Accelerated Math Program available from the Mathematics Department.

All applicants must take the Graduate Record Examinations General Test.

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.

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.

These degrees are under the jurisdiction of the Graduate School. Refer to the Requirements for Graduation section and the Graduate School for general regulations. All courses applied toward the degrees must be courses accepted by the Graduate School.

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 advisor. The advisor serves on the student’s guidance 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.

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 guidance committee.

For this program students are not required to take the screening examination or to satisfy a foreign language requirement.

See Mathematical Finance.

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 which 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.

Thirty units of course work are required, including:

plus at least 12 units of advisor approved courses.

After consultation with the faculty, students may opt for a master’s thesis (and registration in MATH 594ab), or a written examination covering material from MATH 505a and MATH 541ab. The examination will normally be given at the end of the fall semester.

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.

* Both tracks
** Biology track
*** Mathematical science track

Students are required to demonstrate skill in C++, Java or Perl, and to demonstrate knowledge of molecular biology at the level of BISC 311. A substantial report on the commercial or laboratory internship must be submitted (for which enrollment in MATH 592 is required).

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 which are fundamental 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.

At least 24 units are required, including:

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).

At least 24 units are required, including:

The degree is completed with either departmental comprehensive examinations (two examinations, one covering the required component, MATH 525a and 520, and the second covering one of the selected options) or a thesis demonstrating research ability in applied mathematics (the thesis option requires four additional thesis units selected from MATH 594abz).

The program requires a maximum effort by the student for a minimum of four years of full-time work.

The screening examination should be taken by the end of the second semester in the program, with the qualifying examination to follow two semesters later. If the student fails the examination, the department, at its discretion, may permit the student to take it again during the next semester.

The screening examination consists of four one-hour examinations covering the subject content of: MATH 502a Numerical Analysis; MATH 505a Applied Probability (or MATH 507a Theory of Probability); MATH 525a Real Analysis; and MATH 541a Introduction to Mathematical Statistics.

No later than at the end of the first semester after passing the screening procedure the student must form a guidance committee consisting of an advisor and four other faculty members, including at least one from another department.

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.

The student must complete, with no grade lower than B, a minimum of 60 units of courses carrying graduate credit and approved by the guidance committee. These must include MATH 502b, MATH 504ab, MATH 505b, MATH 506, MATH 507b, MATH 509, MATH 520, MATH 525b, MATH 532, MATH 541b, MATH 542L, MATH 545, MATH 555a, MATH 565a, MATH 574, MATH 576, MATH 580, MATH 585.

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.

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.

Following passage of the screening examination and approval of a dissertation topic by the guidance 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.

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.

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.

Appointment of a guidance 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.

The student must complete with no grade lower than B a minimum of 60 units of courses carrying graduate credit and approved by the guidance committee.

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.

The student must demonstrate a reading comprehension of mathematics in two languages (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.

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 guidance committee.

The oral portion of the qualifying examination covers one topic selected from department research areas in mathematics and approved by the guidance committee. The student must demonstrate research potential in this field.

Following passage of the qualifying examination and approval of a dissertation topic by the guidance 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.

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.