### Guide for Graduate Students

# I. Introduction

This document contains information about the UT Department of Mathematics and Statistics, about some of the opportunities and facilities available to graduate mathematics students, and descriptions of the various graduate programs offered by the department. Along with additional information, announcements and useful links, this information is also accessible on the department web page.

# II. The Department and its Facilities

There are currently about thirty faculty members in the Department of Mathematics and Statistics and approximately forty full-time graduate students. The range of academic backgrounds and mathematical interests among the faculty is quite broad and, because of the favorable faculty-student ratio and the informal atmosphere in the department, graduate students should find no shortage of opportunities for interaction with faculty.

The main departmental office is on the second floor (Rm. 2040) of the south-east wing of University Hall, and faculty offices are located on the second, third, and fourth floors of the same wing. The main office is generally open on weekdays between 8:30 am and 5:00 pm. Adjacent to the main office is a lounge for faculty and graduate students. Coffee is normally available here at a reasonable cost during the morning hours.

COMPUTING FACILITIES: All students, faculty, and staff at the University are given a UTAD account. This account provides access to a wide variety of computing facilities. See the Information Technology FAQ web pages to learn more.

The department has a computer lab (called the Newton Lab) located in UH 1000. The computers in this lab (which can be accessed using your UTAD account) have a number of department specific pieces of software including maple, matlab, python, and others.

LIBRARY FACILITIES: The UT library collection is housed in the William S. Carlson Library, located immediately south of the Student Union. More than a million volumes and five thousand periodicals are available to students for both research and recreational reading. The library follows an open stack policy for books and periodicals and offers a wide range of modern library services and facilities, including catalog searches and renewal of borrowed items through the library web page. (This page is linked to the department web page.) The library currently receives approximately three hundred mathematics journals. A list of titles is available from the department.

Mathematics books are arranged under the Library of Congress call letters "QA" on the fifth floor of the library. Bound periodicals are found on the second and third floors. Current (unbound) periodicals are displayed in alphabetical order by title in the basement of the library. Periodicals (bound and unbound) may not be removed from the library, but pay photocopy machines are available. New periodicals are circulated to the Department for one week and are available in the Department lounge. Students may also obtain library privileges through a reciprocal arrangement at Bowling Green State University and at the University of Michigan.

SEMINARS, COLLOQUIA AND OTHER EVENTS: An important aspect of any graduate program is the opportunity it affords students to become involved in the various research activities in the department. Graduate students are strongly encouraged to participate in departmental seminars and are required to attend the Friday department colloquia. The latter are talks or presentations (often by distinguished mathematicians from outside the department) which are normally intended for a general mathematical audience with no specialized knowledge.

Colloquium announcements will appear, usually a week or so in advance, on several bulletin boards around the department and on the colloquium web page. Following each colloquium, there is an opportunity to meet informally with the speaker in the department lounge over refreshments.

The academic year at UT runs from late August through the middle of May and is divided into two 15-week semesters. The department also tries to offer a limited number of graduate "topics" courses during the summer months in areas suggested by faculty and selected according to student interests.

Several purely social events are usually organized during the year for faculty, staff, students and their families. These include a party at the end of the Fall Semester and usually a Fall and Spring picnic in one of the local Metroparks.

# III. Graduate Programs in Mathematics

The department offers the following graduate programs:

- Master of Arts (M.A.)
- Master of Science (M.S.) - Applied Mathematics
- Master of Science (M.S.) - Statistics
- Master of Science and Education (M.S.Ed.)
- Doctor of Philosophy (Ph.D.)

## General Requirements

- All students must satisfy the requirements of the College of Arts and Sciences for their particular program.
- All students are required to attend the Friday department colloquium. A grade of satisfactory or unsatisfactory will be assigned based on attendance.

## Advising

Each entering graduate student will, according to his or her program, be assigned a faculty adviser to assist in course selection and registration. Currently, these advisers are:

In subsequent semesters, the students will be expected to seek guidance from the adviser in matters related to his or her program, and to obtain the adviser's signature on registration forms. When a student passes his or her written exams, this role is then assumed by the student's oral exam adviser, and following the oral exams by the thesis adviser.

## Masters Programs

The M.A. and M.S. programs of this department have distinct aims and are oriented towards students with different career goals.

The M.A. program is designed to expose students to a significant body of rigorous introductory graduate-level pure mathematics.

The primary aim of the M.S. programs is to offer students an opportunity to broaden their knowledge of mathematics and its applications, and to develop mathematical and statistical skills that will be useful to them in industry, business, government and other areas. It is possible for students in either the M.A. or M.S. programs to begin course work toward a doctoral degree. However, such students must apply to join the Ph.D. program after completing a Master's degree.

Degree RequirementsCandidates must satisfy the masters requirements of the College of Arts and Sciences.The specific departmental requirements for each Masters degree are as follows:

### Master of Arts

For the Master of Arts degree in mathematics, students must meet the following requirements (but refer also to The Doctoral Program description):

- A minimum of 30 hours of graduate credit must be completed. Math 6930 and 6940 do not count toward the 30 hours. The following courses are prerequisites and will not count toward the degree: Linear Algebra I (5300) and Advanced Calculus (5780).
- The total graduate and undergraduate programs must include the following: Abstract Algebra I and II (5330, 5340), Topology I and II (5450, 5460), Real Analysis I and II (5820, 5830) and Complex Analysis (5880). Students may substitute the doctoral core courses in Algebra, Analysis or Topology for their 500-level counterparts, subject to the approval of the adviser. At most 20 credits from the above count toward the 30 hours requirement for the M.A. degree.
- At least one two-semester sequence in a single discipline at the 6000 level must be taken from the following: Algebra I and II (6300, 6310), Topology I and II (6400, 6410), General Topology I and II (6420, 6430), Differential Geometry (6450,6460), Algebraic Topology I and II (6470, 6480), Real Analysis I and II (6800, 6810), Functional Analysis I and II (6820, 6830), Complex Analysis I and II (6840, 6850) or Ordinary Differential Equations I and II (6500, 6510).
- The remaining course work must be chosen from the following electives: Topology I (5460), Introduction to Differential Geometry I and II (5540, 5550), Ordinary Differential Equations (5800), Partial Differential Equations (5810), Calculus of Variations I and II (5860, 5870) or any of the courses from (3).
- The student must either pass comprehensive examinations or write a Master's thesis. The comprehensive examinations will be in Algebra, Analysis, and one of Topology, Differential Equations, or Probability and Statistics. If a thesis is elected the student must take an oral examination on the general areas of the thesis. As an alternative the student may take two of the three written doctoral examinations. In this case, if the student does not pass both at the Master's level it will not be considered as an attempt provided the regular M.A. exams are taken in the same semester.

### Master of Science (Applied Mathematics)

A minimum of 30 graduate credits must be completed. Math 6930 and 6940 do not count toward the 30 hours.

Preparatory courses are Linear Algebra I (5300), Applied Linear Algebra (5350), Introduction to the Theory of Probability (5680), Mathematical Statistics (5690), and Advanced Calculus (5780). These may be taken for credit by applied master's students, but cannot be used to fulfill the course requirements for the Master of Science degree in the Applied Mathematics option.

The following courses or their equivalents are required: Numerical Analysis I, II (5710, 5720), Introductory Real Analysis I, II (5820, 5830), Complex Analysis (5880), Introduction to Differential Equations I, II (6500, 6510). In the case where there are corresponding 6000 level courses, students may substitute (with consent of the adviser) such courses to fulfill these requirements.

Elective courses may be chosen from: Applied Functional Analysis (6150), Functional Analysis I, II (6820, 6830), Methods of Mathematical Physics (6720, 6730), Introduction to Differential Geometry I, II (5540, 5550), Differential Geometry I, II (6440, 6450), Calculus of Variations I, II (5860, 5870), Linear and Nonlinear Programming (6180), Infinite Dimensional Optimization (6190), Theory of Computation (5390) and Discrete Structures and Analysis of Algorithms (5380), Partial Differential Equations I, II (6540, 6550), Dynamical Systems (6520, 6530), Applied Probability (5660), Linear Statistical Models (5620), Operational Mathematics (5850) or (for students not in the Ph.D. track), a one-semester course in an applied or pure science as approved by the adviser.

To complete the program a student may either elect to write a master's thesis under the supervision of a faculty adviser or pass comprehensive exams in analysis and differential equations. For full-time students it is expected that either of these requirements will be completed by the end of the student's second year of study.

Options:

- The option of writing a master's thesis is recommended to students who enter the program with a good background in mathematics and who do not intend to pursue doctoral studies. If a student chooses to write a master's thesis he or she must be assigned a thesis adviser by the end of the student's first year of study.
- Students choosing to write comprehensive examinations may either take the M.S. comprehensive examinations offered at the end of the Fall and Spring semesters, or pass the corresponding doctoral qualifying exam at the master's level. If a student does not pass either doctoral qualifying exam and takes the M.S. level examinations in the same semester, the first attempt will not be counted.

### Master of Science (Statistics)

Prerequisites to the Statistics program are Calculus, Linear Algebra, Introductory Probability and Statistics and some proficiency in Computer Programming.

- A minimum of 35 graduate credits must be earned, including at least 9 at the 6000 level; 6930, 6940 may not be applied toward this minimum.
- Preliminary courses 5300 or 5350, 5680 and 5780 must be taken immediately upon enrolling in the program if they or their equivalents have not been previously taken. At most one of these courses may be counted toward the 35 hours required.
- Required statistics courses include 5600, 5610, 5620, 5630, 5640, 6600, 6610, 6620, 6630, 6650 and 6690. Also required is one elective course numbered 5600 or 6600.
- A two-part comprehensive examination, one part in probability and statistical theory and one in applied statistics, must be passed.

### Master of Science and Education

- A minimum of 32 hours of graduate credit must be completed. Math 6930 and 6940 do not count in the 32 hours. At least 18 hours must be in Mathematics and 9 in Education, with an additional 6 hours to be assigned in consultation with advisers. As part of the additional 6 hours the student may elect to write a paper in Mathematical Education or one of an expository character in mathematics.
- The total undergraduate and graduate programs must include the following : at least 6 hours of abstract algebra and/or linear algebra, 6 hours in geometry, statistics, probability and/or computer programming, 3-6 hours of analysis (beyond calculus), 3 hours of complex analysis, and one course in logic or foundations.
- The student must pass comprehensive examinations in 3 of the areas of study in mathematics. The exact areas are to be arranged with the advisers.
- For specific education requirements, consult the College of Education and Allied Professions.

### Masters Comprehensive Examinations

The Masters Comprehensive Examination (required of all Masters students except those M.A. and M.S. applied students who do not plan to continue for the Ph.D. at UT and who elect to write a thesis) is scheduled regularly in the Fall and in the Spring. Students are expected to take the examination within two years of their entrance into the program.

The Graduate Student Affairs Committee (G.S.A.C.) will appoint the examining committees and oversee the implementation of examination procedures and policies.

Copies of some previous examinations are on file in the department office and may be obtained for perusal or copying. Some may also be found in the online graduate exam archive.

Detailed descriptions and a list of the regulations governing the Masters Comprehensive Examinations appear in the Graduate Exam Syllabi web page. Students should note that the rules vary somewhat from program to program.

Included with each set of regulations is a syllabus and list of relevant core courses for each examination. It is important to realize that the list of courses is provided only as a guide; it is the syllabus that defines the scope of the examination.

## The Doctoral Program

Mission Statement.The doctoral program in Mathematics seeks to nurture and cultivate students' talents and interest in Mathematics, Statistics and their applications. The faculty with ongoing research programs will encourage and train our students to independently investigate mathematical programs at the forefront of our discipline and so, to the greatest extent possible, develop the necessary skills and background to enable them to pursue successful careers in Academia, Education, Government or Industry, allowing them to make significant contributions to society, whether in the city of Toledo, the state of Ohio or elsewhere.</div>Doctoral degrees are offered with emphasis in pure and applied mathematics and statistics. The feature of the doctoral degree which distinguishes it from all other graduate degrees is the requirement of a dissertation demonstrating the candidate's ability to successfully undertake an independent program of significant research. The Ph.D. degree is not awarded as a result of coursework or examination.

### Qualifying Exam

Students choosing to work in pure or applied mathematics will be examined in two of the four areas of algebra, real analysis, topology, or differential equations. Students choosing to work in statistics will be examined in real analysis and probability and statistics. (See the Graduate Exam Syllabi for more information.)

Doctoral students in pure or applied mathematics who have successfully completed the doctoral core courses in Algebra, Topology, and Real Analysis, and meet the university's credit requirement for the M.A. degree and who have passed the Ph.D. qualifying examinations at the master's level, may obtain a M.A. degree in pure mathematics by applying to the M.A. program. Doctoral students who have completed the course requirements for the M.S. degree in Applied Mathematics and have passed the Real Analysis and Differential Equations qualifying examinations at the master's level may obtain an M.S. degree in Applied Mathematics by applying to the M.S. program. For more details see the graduate adviser.

### Oral Examination

The student must pass an oral examination in the general area of the intended dissertation work. The intention of the oral exam is to demonstrate the student's ability to engage in mathematical research. (See the Graduate Exam Syllabi web page for more information.)

### Degree Requirements

Candidates must satisfy the doctoral requirements of the College of Arts and Sciences.

The departmental requirements for the Ph.D. are as follows:

- A minimum of 90 hours of graduate credit must be completed. Math 6930 and 6940 do not count in the 90 hours. Of the 90 hours, at least 18 but no more than 36 shall be allotted for the dissertation.
- As a general rule, in the first year of study for the Ph.D. students must enroll in Abstract Algebra (6300, 6310), Real Analysis (6800, 6810), and Topology (6400, 6410). (In any event, successful completion of these or equivalent courses is required for the degree.)
- The student must successfully complete the year-long sequence in Complex Analysis (6840, 6850), and two other regularly scheduled year-long 6000/8000 sequences excluding (6720, 6730).
- Students must pass the qualifying and oral examination. Full-time students must pass the qualifying examination within the first two years of study and pass the oral exam within a year of passing the qualifying examination.
- The student must demonstrate the ability to read mathematical literature in one foreign language, ordinarily chosen from among French, German and Russian. Another language, e.g., Polish, Spanish, Italian or Japanese, may be substituted if it is necessary for the student's specific program. The language requirement should be fulfilled within one year after passing the qualifying exams.
- All doctoral students are expected to spend two consecutive semesters in supervised teaching. This requirement should be met before dissertation research is begun.
- The student must write a Ph.D. dissertation under the direction of a faculty member. Before completing the dissertation the student must report on it in an open seminar. A completed dissertation must be approved by an outside examiner, who should be an authority in the field, and the student must defend it before a faculty committee. This committee is to be chosen by the adviser in consultation with the student and must consist of at least three faculty members including the adviser. The choice of the committee must be approved by the Graduate Student Affairs Committee. Normally this committee will be formed when the student's efforts in research have begun to bear fruit.
- The student will present a bound copy of the dissertation to the department.

# IV. Information for Teaching Assistants

All teaching assistants should receive a copy of theGraduate Assistant Handbookwhich contains a description of the Graduate School's policies and procedures and also some useful teaching tips. There is an abridged version of the GA Handbook available online.

## Summer Support

While the Department cannot guarantee summer financial support to teaching assistants, it has been possible in recent years to provide some level of support to almost all T.A.s who requested it. In addition to assuming some teaching or grading responsibilities, T.A.s who receive summer support must enroll in a specified number of hours in mathematics according to the level of support.

## Policy on Outside Employment

It is the policy of the Department that graduate students with assistantships are engaged in a full-time occupation and that significant time and energy spent in additional employment are likely to be detrimental to both academic performance and teaching effectiveness. Therefore, T.A.s who wish to take on other employment during the academic year (whether on or off campus) must obtain formal approval on a semester-by-semester basis from the Graduate Student Affairs Committee. The Committee will consider each such request individually but, in any case, will not grant approval unless the applicant has completed at least one year of full-time graduate study at Toledo and has obtained a grade-point average of at least 3.50 in the semester (excluding summer) immediately preceding the employment. At most seven hours per week of such additional employment will be approved. Requests by T.A.s to be included on the Department's list of tutors will also be subject to these restrictions.