All students entering the MA and PhD programs in the Fall semester are required to take the Preliminary Examination during Orientation Week. This exam is designed to test the students' mastery of the foundations of the undergraduate mathematics curriculum, principally linear algebra, and advanced calculus, and is used mainly for placement purposes. Based on the performance on the exam, the student may be placed into MATH 7900 (Foundations for Graduate Mathematics) during the Fall semester.

## The PhD Program

**Prerequisites:** To enter the PhD program a student should hold at least a Bachelor's degree in mathematics. The academic record of a student applying to the PhD program should contain substantial evidence that the student will succeed in the doctoral program. In reviewing an applicant's folder, the Graduate Committee gives substantial weight to the applicant's transcripts, letters of recommendation, and GRE scores.

**Requirements:** The PhD degree has the following requirements beyond the residency requirement (however, breadth and depth of knowledge are strongly encouraged). It requires (1) passing written and oral qualifying examinations, (2) writing a dissertation embodying the results of original research, which is acceptable to the student's dissertation committee, and (3) a final oral defense of the dissertation.

The Program of Study must include a minimum of 30 semester hours of course work, including at least 16 hours of 8000- and 9000-level courses not including research, dissertation writing, and directed study. None of the courses GRSC7770, LLED7768, LLED7769, MATH7005, or MATH9005 can be counted on the Program of Study, nor any course with a grade below a C. At least 3 hours of 9300 (Dissertation Writing) must appear on the Program of Study. For additional requirements concerning transfer credit, submission of program of study, admission to candidacy, and regulations concerning Doctoral Final Defense and Doctoral Dissertation, see the current Graduate Bulletin, or consult the Graduate School. The maximum hours of both MATH9000 and MATH9300 in any one semester is 9.

A student's progress towards the PhD degree is initially supervised by a three-person Preliminary Advisory Committee. The student's faculty advisor chooses this committee and is its chair. After the student has passed the Written Qualifying Exams, and before taking the Oral Qualifying Exam, the Advisory Committee is increased from 3 to a minimum of 4 members. The voting members of this committee will be the same as the student's PhD committee.

**The PhD Qualifying Examination System **consists of two parts. The first part consists of four Written Qualifying Exams and the second consists of an Oral Qualifying Exam.

**Written Qualifying Exams **are offered every year in August before the start of Fall semester classes and in January before the start of Spring semester classes. Study guides and copies of previous qualifying exams are available on the Graduate Program website for students to use in preparing for their Written Qualifying Exams.

Written qualifying exams are offered in algebra, complex analysis, numerical analysis, probability, real analysis and topology.

There are three possible grades on each exam: pass, master's pass or fail. Each PhD candidate is required to either: (i) attain pass grades on three written qualifying exams or (ii) attain pass grades on two written qualifying exams and master's pass grades on two written qualifying exams.

The choice of which three or four exams to apply to meet these requirements from the six available exams must be approved by the student's Preliminary Advisory Committee.

Each of the six introductory 8000-level courses (MATH 8000, 8100, 8150, 8200, 8500, and 8600, along with the associated 8xx5 problem session) is designed to help prepare students for the written qualifying exam in the corresponding subject area. However, the final authority for possible topics on the exam lies with the Study Guides; not all topics will necessarily be covered in the introductory courses.

The Written Qualifying Exams may be taken in any order, and more than one exam may be taken at a time. An exam may be repeated until passed; however, timely completion of the Written Qualifying Exams is expected according to the Progress Guidelines. For each written qualifying exam taken by a student, an examining committee decides on a Pass/Masters Pass/Fail recommendation communicated to the studentâ€™s preliminary advisory committee (PAC). The student's PAC may request that the examining committee review its decision. In case of disagreement between the examining committee and the PAC, the PAC may appeal the examining committee's decision to the Graduate Committee. It is expected that requests from the PAC to the examining committee to review the pass/fail decision will be based on substantive grounds such as a factual error in the questions or grading of the exam.

**The Oral Qualifying Exam** is based on the student's anticipated area of specialization. In it, the student is expected to present material from one or a few research papers and to answer general questions about the area of specialization. Timely completion of the Oral Qualifying Exam is expected according to the Progress Guidelines. To begin preparing for the Oral Qualifying Exam, the student decides upon a thesis advisor. At this time the student's committee will increase from 3 to a minimum of 4 members. The oral qualifying exam is designed to assess the student's readiness to begin work on a thesis. The student prepares by reading research papers in the area, and the student, advisor, and committee agree upon a body of material for which the student will be responsible. The exam will consist of a presentation on the prepared research papers, followed by a question period covering the presentation and the agreed upon body of material. Oral exams are only open to faculty members.

## The Master of Arts Program

**The purpose of the MA program **in Mathematics is to offer students who hold a Bachelor's degree in mathematics or a closely related field an opportunity to broaden their knowledge in several areas of mathematics and its applications. This program will prepare a student for teaching at junior colleges or for careers in business, government, or industry. An inadequately prepared PhD applicant may be admitted to the MA program with the possibility of transferring later to the PhD program if he or she makes sufficient progress.

**Prerequisites: **To enter the MA program a student should have a strong Bachelor's degree in mathematics or a closely related field. The student should have had training at the junior/senior level in courses requiring reading and writing proofs, preferably including at least two from modern algebra, topology, and real analysis. Additional courses in pure and applied mathematics, probability, statistics, physics, and computer science are desirable.

**MA Program Requirements:** The MA program in mathematics is offered under two plans: (1) MA *with thesis,* and (2) MA *without thesis. *The general Graduate School requirements include a minimum of 30 semester hours of course work of which at least 12 hours must be in courses open only to graduate students (exclusive of 7000 and 7300 but including 6000 level courses and 3 hours of 8850). A maximum of 6 hours of 7000 and 3 hours of 7300 may be applied toward the 30 hours. GRSC7770, LLED7768, LLED7769, MATH7005, MATH9000 and MATH9005 **cannot** be counted on the Program of Study. For additional requirements concerning transfer credit, submission of program of study, admission to candidacy, and regulations concerning preparation of theses, see the current Graduate Bulletin, or consult the Graduate School.

Departmental requirements are as follows.

- Candidates for the MA degree
*with thesis*are required to take 30 credit hours of mathematics-related coursework, and to write a thesis. The course work must include 9 hours in 8000-level MATH courses, (not counting 8xx5, 8800, 8900-8980, or more than one semester of 8850) and 3 hours of MATH 7300 (Masterâ€™s Thesis). It is desirable that the thesis should present original research. However, the thesis may be expository in nature, in which case it should be a synthesis of several research articles and books. The student must give a final oral defense of the thesis, and it must be approved by a committee of three members including the thesis advisor. - Candidates for the MA degree
*without thesis*are required to take at least 33 credit hours of mathematics-related coursework. The course work must include 12 hours in 8000-level MATH courses, (not counting 8xx5, 8800, 8900-8980, or more than one semester of 8850). Candidates are also required to take Comprehensive Examinations in three areas as specified below.

A student's progress towards an MA degree is supervised by a 3-person Master's committee, formed at the beginning of their graduate career. The student's faculty advisor chooses this committee and is its chair.

**The three MA Comprehensive Exams** taken by students in the MA *non-thesis *program must be chosen from three different areas among (1) Analysis, (2) Algebra, (3) Topology, and (4) Applied. See the Appendix for a list of courses, grouped by area. The course groups corresponding to the four areas are (1) A and E, (2) B and F, (3) C and G, and (4) D and H. At least one exam must cover an 8000-level course. Master's Comps are two hours in length and must initially be taken in a one-week period, ordinarily at the end of the candidate's second year of study. The examiner marks the exam and makes a pass/fair recommendation, but success is ultimately determined by the student's committee; if the student's work is not satisfactory, the committee may recommend "fail" or administer another exam.

Students in the MA *non-thesis* program are given credit for Master's Comps if they have passed at least two or more PhD Written Qualifying Exams, with grades of Masters Pass or Pass, in two different areas from (1) - (4) above.

## The Master of Applied Mathematical Science (MAMS) Program

**The purpose of the MAMS **program is to provide mathematical training for students who wish to work in business, government, or industry. It is designed to produce applied mathematical scientists who can solve quantitative and qualitative problems arising in practical applications (for example, in areas such as computer aided industrial design, operations research, engineering or systems analysis). The MAMS program is intended for people who wish to sharpen their mathematical skills for use in applied situations.

The MAMS degree offered in the Mathematics Department is inherently interdisciplinary in nature. A principal feature of the MAMS program is that the student works on an individual problem. This problem can come from any applied area of study (for example, physics, agricultural engineering, ecology, marine sciences, or finance). Some upper-level course work in that area may be included in the student's program of study. The project results are written up by the student in a substantial technical report. The student also gives an oral presentation of the report to the faculty. The technical report should clearly describe the problem, detail the mathematical analysis and results, and interpret the results in terms of the original problem.

**Prerequisites:** In order to be admitted to the MAMS program, a student must have taken courses in multivariate calculus, linear algebra, and ordinary differential equations. Students should also have had some experience with computers.

**Course Work:** The course work in students' programs of study should broaden their knowledge and skills in applied mathematics. To obtain a MAMS degree the student must pass 33 credit hours of approved course work, including either

Real Analysis (MATH 6100) or Complex Variables (MATH 6150)

and either

Probability (MATH 6600) or Introduction to Partial Differential Equations (MATH 6720).

At least 9 credit hours of 8000-level mathematics courses must be included in the student's program of study (not counting 8xx5, 8800, 8900-8980, or more than one semester of 8850) with at least one course taken from each of any two of the following areas:

**NUMERICAL ANALYSIS**

Advanced Numerical Analysis (MATH 8500, 8510, 8520)

Special Topics in Numerical Analysis (MATH 8550)

**PROBABILITY**

Probability (MATH 8600)

Stochastic Processes (MATH 8620)

Stochastic Analysis (MATH 8630)

**DIFFERENTIAL EQUATIONS**

Industrial Mathematics (MATH 8700)

Variational Methods/Perturbation Theory (MATH 8710)

Ordinary Differential Equations (MATH 8740)

Introduction to Dynamical Systems

Partial Differential Equations (MATH 8770)

In addition, students may take up to nine hours of course work in other departments in an area related to the technical report project.

**Technical Report:** A distinguishing feature of the MAMS program is the writing and presentation of a technical report following an investigation into a real-world applied mathematics problem. This report, written under the guidance of a faculty advisor, consists of three parts:

- The introduction, in which the problem is explained clearly in non-technical terms.
- The description of the mathematically formulated problem and the mathematical analysis performed.
- The summary, which relates the results of the problem and explains any conclusions.

The report and presentation may be viewed as training for a real job situation where one communicates the results of a project and any relevant conclusions to a manager or a client.