20182019 Graduate Studies Bulletin [Archived Catalog]
Mathematics, Ph.D.


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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.

Degree Requirements (60 Post Baccalaureate Hours)
The PhD 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 PhD 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 PhD 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 threehour 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 oneyear 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 oneyear 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 oneyear 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 wellprepared 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 PhD Comprehensive is an indepth examination consisting of a written part administered in three, threetofour 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 PhD, 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 yearlong 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 yearlong 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 PhD (12 credit hours drawn from subjects not covered by the Comprehensive Examination). A wellrounded 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.
