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.

Learning Outcomes

Students will demonstrate mastery of the core areas of real and complex analysis together with either abstract algebra or numerical anlysis. Mastery includes not only content, but expository skills that will approach the level necessary for the writing of a dissertation.

Students will select a research problem in consultation with the dissertation advisor (major professor), and will write a dissertation consisting of publishable contributions that build on the existing literature.

All students are expected to write cogent and convincing mathematics, using contemporary standards of presentation.

The department expects graduates to have strong oral communication skills.

Students will demonstrate proficient teaching in a variety of settings such as those listed in the Curriculum above.

Existing Program / Major Requirements:

Degree Requirements (60 Post Baccalaureate Hours)

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 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 - MATH 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 - MATH 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 - MATH 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 - MATH 709 (or respectively MATH 701 - MATH 702); but later wishes to specialize in an area for which the other option is more appropriate, then the content of MATH 701 - MATH 702 (or respectively MATH 708 - MATH 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. 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 - MATH 704 and MATH 708 - MATH 709. If admission to candidacy is achieved by passing the exam based upon MATH 703 - MATH 704 and MATH 708 - MATH 709, then it is expected that the student either take MATH 708 - MATH 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. 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. These should be selected in consultation with major professor, doctoral committee, and Graduate Director.

Change Program / Major Requirements:

Degree Requirements (60 Post Baccalaureate Hours)

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.

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 - MATH 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 - MATH 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 - MATH 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 - MATH 709 (or respectively MATH 701 - MATH 702); but later wishes to specialize in an area for which the other option is more appropriate, then the content of MATH 701 - MATH 702 (or respectively MATH 708 - MATH 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. 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 - MATH 704 and MATH 708 - MATH 709. If admission to candidacy is achieved by passing the exam based upon MATH 703 - MATH 704 and MATH 708 - MATH 709, then it is expected that the student either take MATH 708 - MATH 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. 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. These should be selected in consultation with major professor, doctoral committee, and Graduate Director.

There are two primary justifications for removing Math 890 (graduate seminar, 3 credits) from the list of requirements for graduate students who earn a Ph.D in Mathematics: (1) There are better curricular options for these 3 credits, and (2) there are alternative ways to enforce seminar participation.

(1) When the Department no longer requires students to use 3 credits on Math 890, students can use those credits to instead take a 3-credit course focused on specific mathematical content they need for their research.

Mathematics is a subject which builds on itself continuously through history. Mathematicians do not discard results proved 100, 500, or 1000 years ago as outdated relics; rather, old results remain useful and relevant. Old results often serve as the foundation for many important modern research questions. Therefore, Math Ph.D students need to accumulate vast amounts of necessary background to understand modern research questions and to develop strategies for how approach them. While it is unrealistic and impractical for students to obtain all the background knowledge they need through coursework, it is ideal for them to get as much as possible through coursework, taught by experts.

The proposal to remove the 3-credit Math 890 requirement will enable students to take one more course of their choice, which will enhance the knowledge base they need to do research in their chosen area.

(2) There are alternative ways to enforce graduate student participation in seminars than through a course for credit, especially in view of (1).

When the Department no longer requires students to take 3 credits of Math 890, the responsibility for enforcing graduate student participation in seminars will shift to the Ph.D advisor, which is naturally where this responsibility should lie. Most advisors require their students to give at least one seminar presentation separate from the Ph.D thesis defense. As such the Department no longer think that it is necessary to use course credits to enforce seminar participation.

Furthermore, a brief look at graduate mathematics curriculums at peer institutions reveals that most of our peers and peer aspirants do not have this requirement. These include Georgia, UConn, Rutgers, Kentucky, Maryland, North Carolina, Virginia, and Missouri.

In summary, the tenured and tenure-track faculty in the Mathematics Department conclude that removing the 3-credit Math 890 graduate seminar requirement is the correct course of action at this time. The Department conducted a vote on whether to keep or remove the requirement. Out of 37 tenured and tenure-track faculty, 32 voted. The result was 29 - 3 in favor of removing the requirement.

Lastly, I want to clarify we seek to codify this change to the Graduate Bulletin by removing the sentences

``A total of three credit hours of the variable credit doctoral seminar MATH 890 are required; these need not 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."

from the end of the first paragraph of under ``Degree Requirements"

Proposed Effective Term and Year for change to database/bulletin:

The Mathematics Department requests a program change to be reflected in the Graduate Bulletin by removing the sentences,
``A total of three credit hours of the variable credit doctoral seminar MATH 890 are required; these need not 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."
from the end of the first paragraph of the section ``Degree Requirements".

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