## Guide to the Program

**Overview**

The Department of Mathematics has evolved into one of the premier centers in the Southeast for mathematics research and education. Its masters and doctoral programs have been cited for excellence by the S.C. Commission on Higher Education. With its internationally renowned faculty and supportive atmosphere, the department provides a stimulating environment for graduate studies. As the face of mathematics changes, the department responds with appropriate curriculum additions and revisions.

The department's degree programs provide first the core fundamentals, and then the specialized expertise and interdisciplinary skills required of the modern mathematician. Training for those who wish to pursue a career in teaching, those who plan mathematics-related careers in business, government, or industry, and those who wish to obtain the intensive training that will lead them into the contemporary research community is available.

The Department of Mathematics offers programs leading to the Master of Arts, Master of Science, and Doctor of Philosophy, including a Ph.D. option of a concentration in Applied and Computational Mathematics. This concentration emphasizes core mathematics leading to the frontiers of research both within applied and computational mathematics and cuts across disciplinary boundaries.

The department also offers programs leading to the Master of Mathematics and, in conjunction with the College of Education, a program leading to the degree of Master of Arts in Teaching. A description of the basic M.A.T. requirements appears in the College of Education section of the Graduate Studies Bulletin.

Inquires concerning individual cases should be directed to: Director of Graduate Studies, Department of Mathematics, University of South Carolina, Columbia, SC 29208; email: graddir@math.sc.edu.

**Admission (all degree programs)**

For admission into the M.S., M.A., or Ph.D. degree programs, applicants must have a bachelor's degree from an approved institution and should have an undergraduate foundation in mathematics equivalent to that of a major in mathematics at the University of South Carolina. At a minimum, this should include a course in abstract algebra (equivalent to MATH 546) and one in advanced calculus (equivalent to MATH 554). A one year sequence in each is desirable. A minimum B (3.0) average in all college-level math courses is required for full admission. Applicants who do not have this preparation may be conditionally admitted and placed in such undergraduate courses as necessary to strengthen their backgrounds.

Applicants should submit an official transcript from each school or college previously attended, at least two letters of recommendation from persons familiar with their abilities in mathematics, and an official report of scores achieved on the GRE. A GRE score of at least 700 on the quantitative portion is expected. Applicants whose native language is not English are also required to submit a satisfactory score on the iBT TOEFL exam. The minimum score for admission to the program is 88. A minimum iBT TOEFL score of 100 is required for consideration for a teaching assistantship; there are also minimum levels for each sub-category (listening, speaking, reading, writing), which can be viewed here. The GRE Mathematics Subject Exam is not required, but a strong score enhances the probability of admission with assistantship and the possibility of a supplemental fellowship.

For admission to the M.M. or M.A.T. degree programs, applicants must have a bachelor's degree from an approved institution and have completed multivariable calculus (Calculus III, equivalent to MATH 241). Further, it is desirable that they have completed six credit hours in mathematics beyond multivariable calculus. At least a B (3.0) average for all college level mathematics courses is expected. Applicants with background deficiencies may be admitted on a conditional basis and placed in certain dual undergraduate/graduate courses to strengthen their foundation. Course work below the 500-level can not be used toward these degrees. Applicants should submit an official transcript from each school or college previously attended, at least two letters of recommendation from persons familiar with their abilities in mathematics, and a report of scores achieved on the GRE. A combined GRE score of 1000 is expected, with at least 550 on the quantitative portion.

Application materials should be submitted as much as possible online to the Graduate School, or be mailed to: The Graduate School, University of South Carolina, Columbia, SC 29208. More details can also be found at the link to Admissions Requirements at the left hand side of this page.

**Degree Requirements (general)**

There are certain requirements imposed by the Graduate School on all programs. We reiterate only the most pertinent ones here; others appear elsewhere in Graduate Bulletin, and are routinely fulfilled over the course of the program of study.

The M.S. and M.A. degrees require 30 approved credit hours of course work, at least half of which (excluding the thesis) must be taken at the 700 level or above. In addition, a Comprehensive Examination taken upon conclusion of the program is required. Both the M.S. and the M.A. degrees require a thesis (3 credits of MATH 799).

Each candidate for the Ph.D. degree is required to complete a minimum of 60 hours of course work beyond the baccalaureate degree, including 12 credit hours of graduate course work separate from the course work covered by the Admission to Candidacy and Comprehensive Examinations (see below) and 12 credit hours of dissertation work (MATH 899). The Ph.D. program has three examinations: Admission to Candidacy, Comprehensive, and Doctoral Defense.

Note that "credit hours" are not earned if a course is taken on an "Audit" basis. Courses labeled 7xx-I may not be used to satisfy M.S., M.A., or Ph.D. requirements except in rare circumstances, and only by special permission. These courses are designed for the M.M. and M.A.T. programs.

**Master of Science in Mathematics (in detail)**

The M.S. is designed primarily for students who seek broad and intensive preparation for teaching in a junior college or working in industry.

The M.S. degree requires a thesis and 30 approved semester hours of graduate course work, including satisfactory completion of the three-credit thesis course MATH 799, MATH 703, and at least one of MATH 701, 708, and 709. The courses in the student's program should be numbered 700 and higher. However, in special circumstances some 500-level courses, or 7xx-I courses, may be approved for a student's program if the courses supplement 700-level course work. In general, a student's M.S. program should be fairly broad in scope and should include courses of both a pure and applied nature.

The thesis for this degree is generally a short monograph (to be bound and delivered to the department), the content of which is drawn from several research papers in an area of interest to the student. The thesis is subject to the approval of the thesis committee, consisting of the major professor and a second reader.

Upon conclusion of the program, each M.S. degree candidate either undergoes an oral examination administered by the thesis committee (the "defense", which includes an oral presentation of the thesis and also serves as the Masters Comprehensive Exam), or obtains a pass on the Masters Comprehensive Examination (a "master's pass" on the Admission to Candidacy Examination). Students who follow the second path are invited to present the thesis in a seminar format.

**Master of Arts in Mathematics (in detail)**

The M.A. is designed primarily for students who wish to enter a Ph.D. program in mathematics. A student's program of study for this degree is usually narrower than the M.S. in scope but more intense in content. Course work for the degree is regarded as preparatory for the Ph.D.

The M.A. degree requires a thesis and 30 approved semester hours of graduate mathematics course work, including the three-credit thesis course, MATH 799. All courses in the student's program must be numbered 700 and above (excluding 7xx-I courses) and must include a one-year sequence in real and complex analysis (MATH 703-704) and one of the one-year sequences in abstract algebra (MATH 701-702) or in the foundations of computational mathematics (MATH 708-709). These courses form the core of the student's program and provide the topics upon which the Masters Comprehensive Examination (Admission to Candidacy) is based; a "master's pass" or "pass" is required.

The thesis for this degree is generally a short monograph (to be bound and delivered to the department), the content of which is drawn from several current research papers, possibly including the student's original contributions, which could lead to topics of suitable depth for a Ph.D. dissertation. The thesis is subject to the approval of the thesis committee, consisting of the major professor and a second reader. The student is invited to present the thesis to the department in a seminar format.

**Doctor of Philosophy Degree (in detail)**

The Ph.D. is designed to produce a skilled, professional mathematician who is trained to conduct research in mathematics, function effectively as a classroom teacher at the college level, or become a professional practitioner in an industrial or national laboratory setting.

Each candidate for the Ph.D. degree is required to complete a minimum of 60 hours of course work beyond the baccalaureate degree, including 12 credit hours of dissertation research and writing (MATH 899). Students are advised by a doctoral committee. This committee is generally chaired by the major professor (dissertation supervisor) and consists of at least five members, one from outside the department. The core members are writers of the student's Comprehensive Exams. A total of three credit hours of the variable credit doctoral seminar MATH 890 are required; these need not all be taken at once, rather credit is determined by the extent and intensity of participation. Students may earn these doctoral seminar credits by presentation of contemporary research in a student/faculty seminar in their research area.

Students pursuing the Ph.D. degree in mathematics are required to take three examinations: the Admission to Candidacy, Comprehensive, and Doctoral Defense Examinations.

The Admission to Candidacy Examination in mathematics consists of two three-hour written examinations and is administered with two options. The first examination for both options is based primarily, but not exclusively, on the content of the one-year sequence in real and complex analysis (MATH 703-704). The second examination for the first option is based primarily, but not exclusively, on the subject matter of the one-year sequence in abstract algebra (MATH 701-702). The second examination for the second option is based primarily, but not exclusively, on the subject matter of the one-year sequence in the foundations of computational mathematics (MATH 708-709). Two attempts of the Admission to Candidacy Examination are allowed. The first attempt should occur after the first year of graduate study and within the first two years of graduate study. The second attempt must be made at the next scheduled examination. Exceptions to the time constraint for unusual cases may be petitioned to the Graduate Director. Note that the exams are based upon the content of the various courses; it is not required that the well-prepared student take all, or even any, of these courses, although it is generally advisable to do so. Students need only be admitted candidacy once: if a student passes the exam based upon one of the options, say MATH 708-709 (or respectively MATH 701-702); but later wishes to specialize in an area for which the other option is more appropriate, then the content of MATH 701-702 (or respectively MATH 708-709) should be learned either by taking these courses or by independent study.

The Ph.D. Comprehensive is an in-depth examination consisting of a written part administered in three, three-to-four hour sessions, and an oral component. The written portion of the examination must either include the subject matter of one-year sequences numbered 710 or higher selected from two (or, exceptionally, three) of the eight areas listed in the Graduate Handbook, or, for the Concentration in ACM, from Groups 1 and 2 as described in the Graduate Handbook. See also the webpages for the Doctoral Program or the ACM Curriculum, respectively. In both cases, the subject matter of the student's research area should be tested in depth. The oral portion of the comprehensive will be based on the student's program of study and may include topics not covered by either the Admission to Candidacy Examination or the written portion of the Comprehensive Examination.

The Comprehensive Examination may be repeated only once. All portions of the examination must be completed within three weeks. As a general rule, the exam is offered twice each year, once in August and again in January, and should be taken after candidates have completed all or most of the courses required in their program, and before commencement of dissertation research. The examination must be completed at least 60 days prior to the receipt of the degree.

To complete the program, the student must write a dissertation (to be bound and delivered to the department), under the direction of a member of the graduate faculty, and defend the content of the dissertation in a final examination before the doctoral committee. It is expected that the content of the student's dissertation will be a significant contribution to the body of current research and will be published in a reputable journal.

To ensure breadth of mathematical training, each student is required to satisfactorily complete (B or better) 12 credit hours of course work in subject areas not covered by the Comprehensive Examination. Directed reading courses (MATH 798) may not be used to satisfy this requirement. Particular courses may be stipulated by the student's doctoral committee. The selection of the courses is subject to approval by the Graduate Director.

**Doctor of Philosophy Degree: Concentration in Applied and Computational Mathematics (ACM)**

Within the course, exam, and dissertation framework of the Ph.D., a student may, by selecting courses with some care, complete a program of study with an ACM Concentration; this will be denoted as an "Area of Emphasis" on the final transcript. It is still possible, of course, to write a dissertation in an ACM area without participating in the formal concentration.

The concentration is distinguished from the ordinary Ph.D. by three year-long sequences (18 credit hours). It is strongly recommended that the Admission to Candidacy Examination be based upon MATH 703-704 and MATH 708-709. If admission to candidacy is achieved by passing the exam based upon MATH 703-704 and MATH 701-702, then it is expected that the student either take MATH 708-709 (6 credit hours) or learn this material independently. The ACM Concentration is also distinguished by the courses upon which the Comprehensive Exam is based. Two year-long sequences (12 credit hours) must be chosen from the ACM areas Groups 1 and 2 as described in the Graduate Handbook and the webpage ACM Curriculum. The third sequence is not restricted.

The breadth requirement for the ACM Concentration is the same as for the ordinary Ph.D. (12 credit hours drawn from subjects not covered by the Comprehensive Examination). A well-rounded program of study will normally encompass four different subjects, as listed in the Graduate Handbook and the webpage ACM Curriculum. These should be selected in consultation with major professor, doctoral committee, and Graduate Director.

**Mathermatics Education (general)**

The department offers two degree programs for students who wish to emphasize secondary and junior college mathematics education--the M.A.T. and the M.M. degrees. Courses at the 700-level specifically designed for these programs are designated by the letter I adjoined to the course number. These courses are generally offered in the late afternoon during the academic year and during the summer to provide area teachers the opportunity to work toward a degree on a part-time basis

**Master of Mathematics Degree (in detail)**

The Master of Mathematics degree is designed primarily for students who seek a broad, thorough training in mathematics which includes course work specifically designed to meet the needs of secondary-school teachers for whom SC certification is not an issue, and for those intending to teach at the junior/community college level.

The M.M. degree requires 30 approved semester hours of graduate course work, up to 6 hours of which may be outside the departments of mathematics, computer science, and statistics. A core of four courses is required of all students: MATH 701-I, 702-I, 703-I, and 704-I.

In addition, students must include in their program (if similar courses have not been taken previously) a course in geometry (chosen from MATH 531 or 736-I) and a course in linear algebra (MATH 526 or 544). To ensure breadth in the program of study, the remaining course work should include courses in discrete mathematics, number theory, and probability and statistics.

Each candidate for the M.M. degree is required to pass a written Comprehensive Examination, which is based primarily on the four core courses. The examination will consist of two, two-to-three hour written examinations. Students should take the Comprehensive Examination immediately upon completion of the core courses.

**Master of Arts in Teaching (in detail)**

The M.A.T. in mathematics is offered by the Department of Mathematics jointly with the College of Education. This degree program is designed specifically for students who wish to obtain teaching certification in mathematics at the secondary level.

The M.A.T. degree requires 30 approved semester hours of graduate-level course work in mathematics and education (exclusive of directed teaching), no less than 6 and no more than 15 of which may be in education, and at least 15 of which must be in mathematics or statistics. The individual student's program is planned according to that student's background and goals. At least half of the student's course work must be numbered 700 or higher.

Each student's program of study must include at least one course in geometry (chosen from MATH 531 or 736-I), algebraic structures (MATH 701-I), real analysis (MATH 703-I), statistics (STAT 509 or STAT 515-516), and number theory (MATH 780-I). If equivalent courses have already been taken, then appropriate substitutions will be made.

Unless previously taken, the student must also take upper division courses in linear algebra (MATH 526 or 544) and discrete mathematics (MATH 574). Normally theses two courses are taken prior to full admission to the program.

Course work in education must include human growth and development (EDPY 705), foundations of education (EDFN 749), a curriculum course (EDSE 770), a reading course (EDRD 518 or 730), and methods of teaching (EDSE 764).

The student must also complete an 18-semester-hour program of methods and internship in mathematics (EDSE 550, 584, 778A and 778B). Students must apply for admission to the professional program and internship through the College of Education's Office of Student Affairs early in the fall or spring semester prior to the semester of Internship B.

Upon admission to the M.A.T. program, the student is assigned a faculty advisor in mathematics to assist in the development of the mathematics portion of the program. Approval of the candidate's program will be granted by a committee of three faculty members, consisting of the faculty advisor in mathematics, the faculty advisor in education, and a faculty member from either mathematics or education.

Each student must maintain a B average on all graduate-level course work in mathematics and a B average on all graduate-level course work in education.

Candidates for the M.A.T. degree are required to pass a written Comprehensive Examination covering their program of study and emphasizing the theoretical underpinnings of calculus, the basic forms of mathematical reasoning, argumentation, and proof, a repertoire of fundamental examples and counter-examples, problem solving, and insight into how these can inform the teaching of secondary mathematics. Geometric and statistical reasoning will frequently be called upon; students will generally be free to draw on their knowledge of any of analysis, algebra, discrete mathematics, or number theory as they see fit to demonstrate forms of mathematical argumentation and proof.