Undergraduate Degrees

Graduate Degrees

Courses of Instruction

Denney Research Center 155
(213) 740-2400
FAX: (213) 740-2424
E-mail: mathdept@math.usc.edu

Chair: M. Susan Montgomery, Ph.D.

Faculty

USC Associates Chair in Natural Science: Michael Waterman, Ph.D.

Professors: Kenneth Alexander, Ph.D.; Henry A. Antosiewicz, Ph.D.; Richard Arratia, Ph.D.; Peter Baxendale, Ph.D.; Edward K. Blum, Ph.D.; Francis Bonahon, Ph.D.*; Robert Brooks, Ph.D.; Ronald E. Bruck, Ph.D.; Richard S. Bucy, Ph.D. (Aerospace Engineering); Dennis Estes, Ph.D.; Solomon Golomb, Ph.D. (Electrical Engineering); Robert Guralnick, Ph.D.; Eugene Gutkin, Ph.D.; William A. Harris, Jr., Ph.D.; Sheldon Kamienny, Ph.D.; P. Vijay Kumar, Ph.D. (Electrical Engineering); Charles Lanski, Ph.D.; M. Susan Montgomery, Ph.D.*; Robert Penner, Ph.D.; Wlodek Proskurowski, 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.; Alan Schumitzky, Ph.D.; Simon TavarŽ, Ph.D.; Zdenek Vorel, Ph.D.; Michael S. Waterman, Ph.D.*; Paul C. Yang, Ph.D.

Associate Professors: Xianzhe Dai, Ph.D.; Larry Goldstein, Ph.D.; Nicolai T.A. Haydn, Ph.D.; C.C. Jay Kuo, Ph.D. (Electrical Engineering); Alan Nadel, Ph.D.; Wayne Raskind, Ph.D.; Hubert Saleur, Ph.D.; Chunming Wang, Ph.D.

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

*Recipient of university-wide or college teaching award.

Degree Programs

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 Statistics; M.A. in Mathematics; M.A. in Applied Mathematics; Ph.D. in Applied Mathematics; and Ph.D. in Mathematics.

Undergraduate Degrees

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

Advanced Placement Examinations in Mathematics

The university grants four units of credit in mathematics for scores of 3, 4 or 5. The department awards subject credit for MATH 125 for scores of 4 or 5 on the AB version of the exam or for a score of 4 on the BC version. Subject credit for both MATH 125 and 126 is awarded for a score of 5 on the BC version.

Pre-Major Requirements

MATH 125, 126, 225 and 226 are required.

Department Major Requirements for the Bachelor of Arts

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

Department Major Requirements for the Bachelor of Science

Eight upper division mathematics courses, including:

Required courses		Units
MATH 410	Fundamental Concepts of Modern Algebra	4
MATH 425ab	Fundamental Concepts of Analysis	4-4
MATH 471	Topics in Linear Algebra	4

The other four courses must be chosen from:		Units
MATH 407	Probability Theory	4
MATH 408	Mathematical Statistics	4
MATH 430	Theory of Numbers	4
MATH 432	Applied Combinatorics	4
MATH 435	Vector Analysis and Introduction to Differential Geometry	4
MATH 440	Topology	4
MATH 445	Mathematics of Physics and Engineering II	4
MATH 458	Numerical Methods	4
MATH 465	Ordinary Differential Equations	4
MATH 475	Introduction to Theory of Complex Variables	4

Three physics courses are required.

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.

Grade Point Average Requirements

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).

Department Minor Requirements

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.

Honors Program in Mathematics

Admission to the Program

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.

Requirements

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.

Language

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

Accelerated Math Program (AMP)

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 further exam credit 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 Department. 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.

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 and Master of Science in Statistics

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 Requirements for Graduation and the Graduate School 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 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.

Course Requirements

Required courses		Units
PHYS 151L	Fundamentals of Physics I: Mechanics, Waves, and Sound	4
PHYS 152L	Fundamentals of Physics II: Thermodynamics, Electricity, and Magnetism	4
PHYS 153L	Fundamentals of Physics III: Optics and Modern Physics	4

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 guidance committee.

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

Master of Science in Statistics

The object of this program is to provide academic instruction in statistical theory and methods with a fine mathematical base 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. Each student orients 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; in addition, a master's thesis and registration in MATH 594ab is required of all students. MATH 541ab must be taken during the first three semesters; this requirement may be waived for students with sufficient prior education in mathematical statistics. All students are required to take MATH 650. Up to 12 units may be selected from courses outside the department in the chosen field of application or emphasis.

A student in the program must pass a screening examination covering material from MATH 505a and MATH 541a. The examination will be given at the end of the student's first year in regular standing in the M.S. Statistics program.

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 which 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 525a	Real Analysis	3
MATH 520	Complex 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 574abz).

Requirements for the Master of Arts in Applied Mathematics

At least 24 units are required, including:

Required courses		Units
MATH 525a	Real Analysis	3
MATH 520	Complex Analysis	3
and options from A, B and C
(A)
MATH 555a	Partial Differential Equations	3
MATH 565a	Ordinary Differential Equations	3
(B)
MATH 507a	Theory of Probability	3
MATH 541a	Introduction to Mathematical Statistics	3
(C)
MATH 502ab	Numerical Analysis	3-3

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).

Doctor of Philosophy in Applied Mathematics

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

Screening Procedure

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 examination is a written two hour examination covering a choice of the subject content of MATH 510ab Algebra or MATH 525a Real Analysis and MATH 520 Complex Analysis.

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 guidance committee. These must include MATH 794ab Dissertation and eight courses from four of the following groups: MATH 510ab; MATH 525a, 520; MATH 535a, 540; MATH 555a, 565a; MATH 507a, 541b; MATH 502ab.

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 he or she passes the qualifying examination.

Foreign Language Requirement

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.

Qualifying Examination

The written portion of the qualifying examination is comprehensive, consisting of two examinations administered by the department. A student, having selected the analysis option for the preliminary qualifying examination, must select the two examinations from the options: MATH 510ab, Algebra; MATH 535a and 540, Differential Geometry and Topology; MATH 555a and 565a, Partial and Ordinary Differential Equations; MATH 507a and 541b, Probability and Statistics; and MATH 502ab, Numerical Analysis. A student selecting the algebra option for the preliminary qualifying examination must select MATH 525a and 520, Real and Complex Variables, as one of the written examinations and the other examination from the options MATH 535a and 540; MATH 555a and 565a; MATH 507a and 541b; and MATH 502ab.

The oral portion of the qualifying examination covers one topic selected from department research areas in applied mathematics and approved by the guidance 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 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 based on a substantial amount of original research conducted by the student.

Research Areas

Opportunities for research are offered in areas of applied algebra, applied functional analysis, applied topology, biomathematics, control theory, differential equations, discrete mathematics, econometrics, fluid dynamics, numerical analysis, optimization theory, probability, statistics and systems theory.

Doctor of Philosophy in Mathematics

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

Screening Procedure

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 covering a choice of the subject content of MATH 510ab, Algebra, or MATH 525a and 520, Real and Complex Analysis.

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 guidance committee. These must include MATH 794ab Dissertation and the eight courses from four of: MATH 510ab; MATH 525a and 520; MATH 535a and 540; MATH 555a and 565a; MATH 507a and 541b; MATH 502ab.

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 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.

Qualifying Examination

The written portion of the qualifying examination is comprehensive, consisting of two examinations administered by the department. One must cover the subject content of the option, MATH 510ab, Algebra, or 525a, Real Analysis, and 520, Complex Analysis, not selected for the preliminary qualifying examination and the other is selected from the options: MATH 535a, Differential Geometry, and 540, Topology; MATH 555a and 566a, Differential Equations; MATH 507a, Probability, and 541b, Statistics; and MATH 502ab, Numerical Analysis.

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.

Dissertation

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.

Research Areas

Opportunities for research are offered in the area 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.

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