Graduate Student Handbook | Department of Mathematics

FOREWARD

This handbook is intended for graduate students in the Department of Mathematics at the University of Alabama. The Department wants your graduate study to be as pleasant and free of hassles as possible, and we hope that this handbook will help to make it so. We have included a lot of what we hope is useful information, including specific requirements, departmental policies and procedures, and general philosophy of graduate work. There are many other sources of information about the University and its graduate programs, and most of these are mentioned in the following pages. This handbook was revised in August of 1999. Further information can be found at the University of Alabama Web site http://www.ua.edu/.

Warning: This handbook does not contain everything you need to know.

TABLE OF CONTENTS

Useful People and Offices
The Graduate and Undergraduate Program Directors
Degree Requirements
I. The Master's Degree in Mathematics
II. The Doctor of Philosophy Degree in Mathematics
III. The Doctor of Philosophy Degree in Applied Mathematics
Employment and Financial Aid
International Students
Appendix One: The Qualifying Examination in Mathematics
Appendix Two: The Joint Program Examination in Applied Mathematics
Appendix Three: Current Members of the Graduate Mathematics Faculty

USEFUL PEOPLE AND OFFICES

Dean of the Graduate School: Dean Ronald Rogers, 102 Rose Administration, 348-5921

Department Chairperson: Prof. Z. J. Wu, 345 Gordon Palmer, 348-5090, zwu@as.ua.edu

Graduate Program Director: Prof. Vo Liem, 332C Gordon Palmer, 348-4898, vliem@as.ua.edu

Undergraduate Advisor, Jon Carson, 348-9883, jcorson@as.ua.edu

Director of Introductory Mathematics: Samuel Evers, 339 Gordon Palmer, 348-9883, severs@as.ua.edu

Administrative Specialist: Mrs. Michele Farley, 345 Gordon Palmer, 348-5074, mfarley@as.ua.edu

Graduate School: 102 Rose Administration; 348-5921.

English Language Institute: 121 B. B. Comer Hall; 348-7413.

Office of Housing and Residential Life: 1st floor, Mary Burke East; 348-6676.

Office of International Student Affairs: 207 Osband Hall; 348-5402.

Office of Graduate Student Services: 229 Osband Hall; 348-6796.

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THE GRADUATE AND UNDERGRADUATE PROGRAM DIRECTORS

Your graduate advisor will be the Graduate Program Director. The Graduate Director, a graduate faculty member, and the chairman of the department constitute the Graduate Advisory Committee (GAC), which has the responsibility of making sure that each graduate student receives the best possible education. You must meet with your advisor to decide on the courses you plan to take each term. Your advisor will probably also want to discuss with you your overall objectives and how well you are progressing toward them. The Graduate Director is familiar with university and departmental regulations, and can help you to follow them. However, please be aware that it is not your advisor's job to make sure that you meet all the requirements for your intended degree; that is your responsibility.

You will also probably be assigned to a professor in the department, who will act as your mentor. This professor will help guide you through your graduate career. It is hoped that you and your mentor will discuss your course work, or other mathematics in which you are interested, during regularly scheduled office hours. In time, your mentor may change; for example, if you find one branch of mathematics which is more interesting to you, then it would make sense to have a mentor in that area.

You should feel free to see your advisor about any problems that may arise during your graduate work. You may be having trouble with particular courses; you may decide to re-think your decision about what field to concentrate in; you may have visa difficulties. Your advisor will be able to help take care of any of these troubles with you.

Difficulties that may arise in your teaching duties should be brought to the attention of the Director of Introductory Mathematics. The Director of Introductory Mathematics is the direct supervisor of all Graduate Teaching Assistants, and can help you in the same way that a job foreman can. Are you having trouble assigning grades? Are some of your students disruptive during class? The Director of Introductory Mathematics may be able to help with these sorts of problems.

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DEGREE REQUIREMENTS

The Mathematics Department offers two graduate degrees: Master's and Ph.D. There is also a joint Ph.D. program in Applied Mathematics with the University of Alabama system campuses at Birmingham and Huntsville.

The requirements listed below are all included in the University Graduate Catalog. You should consult a current catalog for complete information about University policies. Incidentally, if University degree requirements happen to change during your stay at the University, you will be subject to those rules that were in force when you enrolled, not to the new rules. Changes made at the departmental level may work retroactively, depending on the circumstances. Students with deficiencies in their preparation may remove them by taking the appropriate undergraduate courses at the University of Alabama. However, these courses may not be awarded graduate credit or count toward the number of hours required to earn a graduate degree. The Graduate Program Director will help you determine if you need to take any undergraduate courses. For example, if necessary, beginning graduate students should take MATH 592, which is an introductory course with topics from Real Analysis and Linear Algebra.

Every graduate student must maintain a grade point average of "B" or better. A student who drops below a "B" average will be placed on academic probation; according to University regulations, a probationary student who cannot return to a "B" average within the next twelve hours of graduate work will be dropped from the program. Furthermore, in order to graduate at least 75% of the hours taken must be completed with grades not less than "B".

You should meet with your advisor before each semester, in order to ensure that you are following a suitable program of study. A complete list of courses offered by all departments is in the Graduate Catalog.

The graduate school requires all students for an advanced degree to submit an "Admission to Candidacy" form. Master's students can only do this after receiving 12 hours of graduate credit and Ph.D. students must have passed the qualifying examination. All students must apply for their degree using a different form called the "Application for Degree" form. The forms, available from the Graduate School office, must be approved by the end of the registration period for the semester in which the student expects to graduate. If you fail to graduate on time, a new application for degree form needs to be filed.

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I. THE MASTER'S DEGREE IN MATHEMATICS

DESCRIPTION OF THE PROGRAM

Candidates for the Master's degree in Mathematics can specialize in pure mathematics or applied mathematics, or do a combination of both. A total of 30 hours of graduate work is required to obtain a Master's degree in Mathematics.

Candidates for the Master's degree may choose either of two plans. One plan (Plan I) requires successful completion of 24 semester hours of course work, plus a thesis. The other plan (Plan II) requires no thesis, but 27 semester hours of courses plus 3 hours of work devoted to a project supervised by a member of the graduate faculty in Mathematics.

At least 21 of the course hours must be taken in Mathematics; courses in related areas, such as physics, finance, or computer science, may be taken with the approval of the Graduate Advisory Committee. At the conclusion of course work, a comprehensive examination (usually oral) will be given. This examination will cover the project and also (possibly) coursework. If a thesis has been written, the examination may cover the thesis as well as course material.

Students in Plan I usually should start their thesis work at the start of their second year. Students in Plan II usually will do their project during the first semester of their second year. Ideally, time should be spent on the project (or thesis) during the preceding summer.

Projects typically involve writing a major paper (not necessarily original research) in some area of mathematics. A master's thesis is much more formal and usually lengthier than a project. Typically, a thesis is expository, but based on a substantial body of mathematics. Students are responsible for finding their own thesis or project advisor, although the graduate advisor is usually very helpful to suggest names.

Candidates for the Master's Degree in Mathematics must complete three of the following four core courses:

MATH 510 Numerical Linear Algebra

MATH 532 Graph Theory and Applications

MATH 580 Real Analysis I

MATH 585 Introduction to Complex Calculus

and one of the following two courses (but not both):

MATH 598 Research Not Related to Thesis

MATH 599 Thesis Research

Concentration Area Requirement

Pure Mathematics Concentration

Prerequisites for the Pure Mathematics track include a good knowledge of standard undergraduate mathematics, including several courses where proofs are required. Students should have had an undergraduate course in Mathematical Analysis, which covers rigorously the proofs of theorems which appear in Calculus, and which examines in detail such subjects as continuity, limits, differentiation, and (Riemann) integration. A beginning course in abstract Algebra, covering the notions of groups, rings and fields, is also necessary.

Candidates for the area of concentration in pure mathematics must complete the following three courses:

MATH 565 Introduction to General Topology

MATH 573 Abstract Algebra I

MATH 583 Complex Analysis I

The algebra course covers the topic of Group Theory, which has applications to the solution of polynomial equations, and related subjects. The topology course examines the notion and properties of topological spaces, including ones with quite surprising properties. Complex Analysis introduces the beautiful and powerful classical theory of functions of a complex variable.

The 30 hour requirement thus leaves three other elective courses, with which students may choose to specialize in some particular field of mathematics, or branch out to obtain a broad background. Suggested courses include MATH 588 (Differential Equations), MATH 566 (Algebraic Topology), MATH 681 (Real Analysis II), and MATH 571 (Principles of Modern Algebra II).

Applied Mathematics Concentration

Prerequisites for the Master's degree in Applied Mathematics include a calculus sequence, a course in differential equations and a course in linear algebra.

Candidates for the area of concentration in Applied Mathematics must complete two sequences (four courses) of the following five sequences of special topics:

MATH 511, 512 Numerical Analysis I and II

MATH 520, 521 Optimizations I and II

MATH 541, 542 Boundary Value Problems (MA 541), Integral Transforms and Asymptotic (MA 542)

MATH 545, 546 Theoretical Foundations of Fluid Dynamics I and II

MATH 554, 555 Mathematical Statistics I and II

The 30 hour requirement thus leaves two other elective courses, which may allow the student to gain more depth or more breadth in their coursework, subject to the approval of the Graduate Advisory Committee.

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II. THE DOCTOR OF PHILOSOPHY DEGREE IN MATHEMATICS

The Doctor of Philosophy degree in Mathematics is intended as a research degree and is awarded on the basis of scholarly proficiency (as demonstrated by course work and the Qualifying Examination) and the ability to do independent, original research (demonstrated by the Ph.D. dissertation). Briefly, a successful student must:

A) Complete 48 hours of graduate-level courses.

B) Pass the Ph.D. Qualifying Examination in two areas of mathematics, to the satisfaction of the Graduate Advisory Committee.

C) Complete at least 24 semester hours of dissertation research.

D) Write a dissertation consisting of the student's own original research.
E) Give an oral presentation (the "dissertation defense") of the dissertation results to a committee appointed by the Graduate Advisory Committee.

These requirements are discussed in more detail below. For University rules regarding transfer credit, residence requirements, and other policies, deadlines, etc., you should refer to a current Graduate Catalog.

Course Work

Students must complete 48 hours of course work in order to qualify for the Ph.D. Most of the courses required for a Master's Degree are part of the approved collection. The student's program for the Ph.D. in Mathematics must be approved by the department. Ph.D. students in mathematics normally take three two-course sequences in mathematics. Typical sequences might include:

MATH 565 and 566 General and Algebraic Topology

MATH 573 and 674 Abstract Algebra I and II

MATH 580 and 681 Real Analysis I and II

MATH 583 and 684 Complex Analysis

MATH 511 and 512 Numerical Analysis

MATH 520 and 521 Optimization

MATH 541 and 542 Classical Applied Mathematics

MATH 545 and 546 Fluid Dynamics

MATH 554 and 555 Mathematical Statistics

Students with deficiencies in their undergraduate background may be advised to take other courses before proceeding with the program above; for example, some students may be asked to complete MATH 592 before taking MATH 580. Only courses with numbers above 500 are accepted for graduate credit; however, some courses have dual numbers so that they can be taken for either undergraduate or graduate credit (e.g., MATH 565 is the same class as MATH 465).

Typically, a Ph.D. requires between four and six years of full-time study. Normally students take two or three courses per semester, and (if they are employed as Graduate Teaching Assistants) teach two classes. However, the total (course load plus teaching) must not exceed 18 hours, nor fall below 12 hours. We normally expect that a student will finish the required courses above in the first two years of graduate study--or the first three years, if background holes must be filled. Of course, the required sequences comprise only 18 of the necessary 48 hours, so there is a lot of room for students both to specialize and broaden their studies. Normally, past the second or third year, students will specialize in the areas in which they intend to write a dissertation.

Some students in mathematics may want to have a strong minor concentration in some area such as computer science, finance, or physics. If you have any intention of working outside an academic setting, such a concentration is well worth thinking about. The Computer Science Department has recognized the need for a course that carries graduate credit, but is intended for students in other departments with little computer background. The courses CS 511, CS 512 (both 3 hours) and CS 513 (6 hours) are designed to fill these needs.

The Ph.D. Qualifying Examination

In order to maintain good standing in the graduate program, every student is expected to continue to make "acceptable progress" toward the intended degree. Early in a student's career, the courses taken and the grade point average measure acceptable progress. In addition, every student is expected to take and pass the Ph.D. Qualifying Examination in a timely fashion. Every student who enters the Ph.D. program in Mathematics must take and pass the Qualifying Examination by the end of their third year of full-time graduate study. Students who do not do so will be dropped from the program. The Qualifying Examination is usually offered once per year, and is given during the summer so students may concentrate on the Exam without having the distractions of courses and teaching duties. Current practice is to give the Exam during the first or second week of July; rarely, the Qualifying Exam may also be given during January, but only in case of high demand.

To pass the Qualifying Examination each student must take and pass examinations based upon two course sequences. These examinations are four hours in length. Students normally will attempt these examinations at the end of their first or second year of full-time graduate study. A first-year student may attempt one or two examinations; a second-year student who has not attempted any area of the qualifying examination previously, normally should attempt two areas. Once an area has been passed it need not be retaken, even if another area has been failed.

Each student can attempt at most four area examinations. If a student fails an examination in one area, then the student may substitute the remaining area at a future examination. Such future examinations normally should take place within one year of failing a previous examination. A subsequent failure will mean that the student will be dropped from the program.

The two-course sequences which you take are designed to prepare students for the Qualifying Examination. However, since courses may vary from year to year and from instructor to instructor, it is conceivable that your courses may not cover all of the topics covered on an examination. It is always worth checking with your instructors whether additional material, not covered in your particular courses, will be examined on the qualifying exam. Included as an appendix to these notes is a list of topics on which certain of the qualifying exams are based. Copies of some old qualifying examinations are available from Rita Reese or can be obtained from the department's web site. One of the hallmarks of a good research mathematician --- indeed, of any good scholar --- is the ability to learn independently; here is an opportunity to exercise those skills.

The Ph.D. Dissertation

Upon successful completion of the Qualifying Examination and required course work, students will be "admitted to candidacy for the Ph.D. degree." This phrase is University jargon, but the word "candidacy" carries an important message. In some sense, all that has gone before in a student's academic career is preparation for the writing of a doctoral dissertation. Every student must, of course, demonstrate a broad knowledge of mathematics. However, the Qualifying Examination and the courses leading up to it really represent very basic mathematics --- mathematics that has been around for rather a long time. The main aim of a Ph.D. student is to become a creative and independent mathematician; this is where the dissertation comes in.

After passing the Qualifying Examination, you should decide on a major area of specialization and try to find an advisor in that area. Your major area of specialization must normally be in an area connected to one of the areas in which you passed a qualifying examination. The Graduate Program Director may be able to help you choose an advisor. An appendix to this document lists the current faculty members and their interests. However, be advised that faculty interests may change, and faculty members may already have other Ph.D. students; it is possible that your first choice may not be available.

You will still need to do course work, and much of this should be completed in your main area of specialization. Other courses can be done in related areas. While you will still be seeing your Graduate Program Director, particularly at the beginning of each semester to choose courses, you will now be guided more by your advisor concerning courses and research requirements.

At first, your advisor will probably ask you to do a good bit of reading --- in books and research papers of interest. After this has been going on for some time, your advisor may suggest a problem to work on, or you may think up or come across such a problem on your own. As you think about difficult problems, other questions will invariably arise, and (we hope...) solutions will be found. There is a gradual evolution that goes on in the writing of a dissertation. The original problem may turn out to be impossible, at least with presently known techniques. However, related problems may be accessible, or partial results may be available. Your advisor may guide you toward the most accessible problems, and may be able to suggest ways of attacking a problem, but it is unlikely that your advisor knows the answer already; if so, you should be looking for a different problem!

You should note that the twenty-four hours of dissertation research required by the Graduate Catalog represents a minimum, not a guarantee; your work will not stop once this minimum is achieved. Good results are required before one has a dissertation. We emphasize that the dissertation represents original work --- work that has not been done before. The work should also be of such a quality that it can be published in a recognized research journal.

The Dissertation Defense

Each student is required to have a Ph.D. committee consisting of five faculty members; one of these is to be chosen from outside this department (possibly from another university). Your advisor and the Graduate Advisory Committee will usually suggest names of people for the committee. The committee's purpose is two-fold: first, to make useful suggestions about your dissertation; second, to administer a final oral examination (the dissertation defense). Because of the first purpose, the committee should be kept closely informed of your progress as you work toward a degree. Once your dissertation is written, you must provide copies to your committee, giving them at least a month to read your work before your dissertation defense. The defense will be oral and may involve both a general presentation from you concerning your work and questions from the committee. The committee's questions are not necessarily restricted to the dissertation, but may involve related topics. Your advisor can help you to prepare for the defense.

Once the dissertation is written, and assuming that all goes well and you are deemed to have passed the oral defense, all requirements for the Ph.D. degree will have been satisfied. After the defense, official copies of your dissertation are to be presented to the University. You should take careful note of the endless regulations involving official copies of Ph.D. dissertations. The type of paper is specified; so is the size of the margins. The work must be bound; and so on... . Please check with the Graduate School for publications regarding the preparation of dissertations. You should be aware that typing and preparing an official copy of a dissertation is not a short-term project; you should probably allow your typist at least two months for the job.

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III. THE DOCTOR OF PHILOSOPHY DEGREE IN APPLIED MATHEMATICS

The Ph.D. program in Applied Mathematics is a joint endeavor, conducted with the Mathematics Departments at the University of Alabama campuses in Birmingham and Huntsville. Even if you are not interested in Pure Mathematics, we suggest that you read at least the part of section II that describes the Ph.D. dissertation; that section discusses some of the philosophy of the Ph.D. degree that is common to both programs.

The following are the minimum requirements for the Ph.D. in Applied Mathematics:

A) Complete 42 hours of graduate-level mathematics courses.

B) Pass the Joint Program Examination.

C) Satisfy the residency requirement of one continuous full-time academic year after passing the Joint Program Examination.

D) Satisfy the language requirement.

E) Complete an acceptable Program of Study which includes at least four graduate- level courses in a minor area of concentration outside the Department.

F) Pass a Comprehensive Qualifying Examination associated with the Program of Study.

G) Complete at least 24 semester hours of dissertation research and defend a research dissertation, the results of which are publishable in a nationally recognized journal.

Joint Program Examination

It is anticipated that the student will take the Joint Program Examination after the first year of graduate studies. This examination will cover topics from graduate courses in Linear Algebra (MATH 592), Numerical Linear Algebra (MATH 510), and Real Analysis I and II (MATH 580 and MATH 681), and will be administered in two parts: (1) Real Analysis, and (2) Linear Algebra and Numerical Linear Algebra. (See Appendix Two for a list of topics on which the Joint Program Examination is based.)

The examination may be taken at most twice. On either the first or second attempt, students must pass both parts of the Joint Program Examination by the end of their second year of full-time graduate studies; those who do not will be dropped from the program.

Program of Study

Each Program of Study will stress breadth, depth, and research competence, as well as an understanding of the relationship of the major area to its applications, and will be individualized to meet the student's needs and requirements of the joint Ph.D. program. It will be permissible for a student to complete a Program of Study at one campus, but students will be encouraged to visit campuses other than their own. The three departments will arrange for lecture courses over the T.V. network which links the campuses. Hence, Programs of Study will share the combined expertise of the three campuses.

Programs of Study require prior approval by the Joint Program Committee. A Program of Study will consist of at least 54 semester hours at the graduate level, including

(a) courses needed to prepare for the common core portion of the Joint Program Examination;

(b) a major area of concentration consisting of at least six courses in addition to those taken in (a), selected so that the student will be prepared to conduct research in an area of applied mathematics;

(c) a body of support courses giving breadth to the major area of study;

(d) an outside minor that is designed to support the major area of concentration and that consists of at least four related graduate courses in an area of science, engineering, operations research, or applied statistics.

Comprehensive Qualifying Examination

It is anticipated that the student will take the Comprehensive Qualifying Examination after three years of graduate studies. The examination will cover the program of study, with a written and an oral component, and will be jointly prepared and graded by the student's Graduate Study Supervisory Committee. This will consist of six faculty members: the student's advisor (serves as Committee Chairman); two others from student's home department; one faculty member from each of the mathematics departments at UAB and UAH; and one from outside the department in the student's minor area. The written component will consist of three parts; two parts will be devoted to the student's major area, and one part will be devoted to his minor area. Three hours will be allowed for each part. The oral portion will cover the entire program of study. Copies of old exams can be available from Rita Reese or can be obtained from the department's web site.

If the judgment of the Supervisory Committee is that the student's performance on the test is not satisfactory, then they may, at their discretion, and without obligation, elect to give the test at most one additional time. The second test, if given, will conform to the above policies for the first test. Students must pass both the written and oral component by the end of their fourth year of full-time graduate studies; those who do not will be dropped from the program.

Language Requirements

The language requirement for each student will be set by the Joint Program Committee with the approval of the appropriate Graduate Dean.

Dissertation Defense

The Graduate Study Supervisory Committee serves as the student's Ph.D. committee. The committee's purpose is two-fold: first, to make useful suggestions about your dissertation; second, to administer a final oral examination (the dissertation defense). Because of the first purpose, the committee should be kept closely informed of your progress as you work toward a degree. Once your dissertation is written, you must provide copies to your committee, giving them at least a month to read your work before your dissertation defense. The defense will be oral and may involve both a general presentation from you concerning your work and questions from the committee. The committee's questions are not necessarily restricted to the dissertation, but may involve related topics. Your advisor can help you to prepare for the defense.

If finances permit, there will be an external examiner who is a faculty member in a mathematics department other than those in the University of Alabama system. This examiner, to be approved by the Joint Program Committee, will have experience in a well-established Ph.D. program, and will have expertise in the area of the dissertation. The examiner will attend the dissertation defense, will advise the Graduate Study Supervisory Committee as to the quality of the dissertation, and will file a report with the Joint Program Committee.

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EMPLOYMENT AND FINANCIAL AID

Financial assistance is available to all graduate students on a competitive basis. Most aid comes in the form of a teaching assistantship. All teaching assistants are given a full tuition waiver. First-year students normally do not teach their own sections but help with grading and lab sections. The College of Arts and Sciences and the Graduate School have additional merit-based fellowships. Teaching is usually available in the summer and summer research fellowships worth up to $2000 may also be available, especially for Ph. D. students. Students who have less than 18 hours of graduate work in mathematics can not be in charge of teaching their own sections. International students with teaching assistantship are required to participate in the international teaching assistantship program (ITAP); only after satisfactorily completing the ITAP Course and passing the ITAP Proficiency Exam will international students be allowed to teach.

The current graduate stipend is $9045 for a 9-month teaching assistantship. All students who are allowed to teach their own sections, after removing the above restrictions, will receive an extra $250 per course taught during the fall and spring semesters (up to a maximum of $500 per semester) in addition to the 9-month stipend. Students who have passed one exam in the Qualifying Examination in Mathematics may receive an additional stipend of $1000; students who have passed both exams in the Qualifying Examination in Mathematics may receive an extra $2000 stipend. Students who have passed the Joint Program Examination may receive an additional stipend of $2000; students in the Joint Program may receive an additional stipend of $1000 when they pass the Comprehensive Qualifying Examination. The maximum additional amount of money that any one student can receive from the Department of Mathematics, on top of the regular teaching assistantship, is $5000 per year. The additional money that teaching assistants may receive comes from the Henry Miller Fellowship Fund, and is due to the generosity of Dr. Henry Miller, a retired faculty member. All additional money, above and beyond the regular graduate stipend of $9045, is subject to availability of funding, continued good progress towards obtaining a degree and a good teaching record.

Graduate Teaching Assistantships (GTA's) require students to teach classes in addition to studying. The University measures the amount of work expected of students in terms of 40-hour work week. A student who is expected to work for 40 hours each week is said to be assigned a 1.0 FTE (standing for Full Time Equivalency); a student working for 20 hours each week has a 0.5 FTE, and so on. In the Mathematics Department, most employed graduate students are assigned a 0.5 FTE, which allows for half of the time to be spent working, and half studying. Of course, work loads vary during the semester, and you may find yourself working 25 hours some weeks and 15 during others. It is important to remember, however, that although the University is acting as your employer, your main duty at the University is that of a student pursuing an advanced degree.

Any student with an assistantship of 0.5 FTE or greater is awarded a full tuition grant, which pays the full amount of that student's tuition. You will never see this money; it just means that you won't have to pay tuition fees at the beginning of each semester. Graduate Assistants are eligible for a variety of other benefits, including health services (not insurance, however), and membership in the Alabama Credit Union. You should consult the booklet, "Graduate Assistant Guide" (put out by the Graduate School) for a listing of all benefits.

Typically, GTA's teach undergraduate courses chosen from among Introductory Algebra (MATH 005), Intermediate Algebra (MATH 100), Precalculus Algebra (MATH 112 and 113), Finite Mathematics (MATH 110), Precalculus Algebra and Trigonometry (MATH 115), or Calculus & Its Applications (MATH 121). Your duties may consist of teaching your own section of a course, or of conducting problem sessions for courses that are taught in large lecture sections. If you are teaching your own sections, the normal load is two courses; assignments for problem sessions may vary.

Courses are assigned to each GTA by the Director of Introductory Mathematics; for teaching duties, she is your immediate supervisor, and will be happy to help you whenever possible. You should take note of the fact that there are minimum and maximum course loads for graduate assistants; if you have an award of 0.5 FTE, you must enroll for a total of between six and twelve hours of academic subjects --- that is, two to four courses of three hours each. You must maintain a minimum of six hours of graduate course work each semester to be eligible for continued support.

Graduate Teaching Assistants normally have responsibility for the mathematical education of somewhere between forty and eighty undergraduates, depending on enrollment. Undergraduates often relate well to GTA's, partially because GTA's tend to be closer to them in age. You can have a very positive influence on your students, and you should take the responsibility seriously. Sometimes, however, teaching duties can start to interfere with your own studies; this is particularly likely toward the end of the semester, when everyone is giving important exams. GTA's must learn to divide their time between their duties as a student and as a teacher, and not allow one responsibility to conflict with the other.

GTA's are paid on a monthly basis on the last working day of the month. All prospective employees must complete an Immigration and Naturalization Service I9 form regardless of citizenship. See the Administrative Specialist, Mrs. Reese, in the Mathematics Department to complete the paperwork.

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INTERNATIONAL STUDENTS

Students from other countries often have slightly different problems from American students. Visa difficulties can sometimes crop up, for example. There are many different ways that this can happen, and you should always speak immediately to the Graduate Program Director or the Department Chairman if there is any question about your visa.

If your native tongue is not English, you have the added burden of taking courses, and (probably) teaching courses in an unaccustomed language. The University has established certain guidelines and procedures to ease the problems of non-native English speakers. All international students are required to take the Test of English as a Foreign Language (TOEFL) before being admitted; the Graduate School has established a minimum of 550 (as of August 1991) on the TOEFL for admission. These guidelines are not intended as roadblocks or filters for graduate students, but are primarily to protect prospective students. Graduate study in mathematics is hard enough without the added problems of trying to cope with an unfamiliar language!

The English Language Institute (ELI) was established on the University of Alabama campus a number of years ago to help international students to master English, and to certify their proficiency in the language. Before being allowed to teach, every non-native speaker must take and pass the appropriate test given by the ELI. Before the fall semester begins, the ELI conducts a training program for prospective graduate teaching assistants. The program focuses on three main areas of study: pronunciation, teaching methods, and U.S. culture. Every non-native speaker must complete the appropriate course and pass the proficiency exam in a timely manner. Failure to do so may result in the loss of the assistantship. [In very exceptional cases, the examination may be waived by the Graduate School, for example, if a student has spent two or more years as an undergraduate studying at another American university.] The ELI gives three kinds of passing grades: "full," "trial full," and "conditional." A student who receives a conditional pass will normally be assigned problem sessions by the Undergraduate Director. Normally, before students arrive at the University, they will have received a letter offering them a Teaching Assistantship. The award is contingent upon the completion of the ELI course and the appropriate grade on the proficiency examination. In order to qualify for the introductory course, students must have a TOEFL score of 550 or above. The ELI course usually begins during the last week of July (you will be informed of the date), so international students should expect to arrive in Tuscaloosa well before that time. International students who fail or receive a conditional pass in the ELI examination are required by this department to take courses at ELI and retake the examination at the next opportunity. Failure to do so will result in the loss of the GTA award.

In addition to the required courses, the ELI also offers a number of (non-required) short courses during the school year, intended to help students improve their spoken English, writing skills, and cultural knowledge. The ELI will be glad to provide you with a schedule of these opportunities.

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APPENDIX ONE
THE QUALIFYING EXAMINATION IN MATHEMATICS

ALGEBRA: Properties of groups: Elementary theory of groups, automorphisms, split extensions, Sylow theorems, examples including dihedral groups, quaternion groups and other groups of small order, p-groups, nilpotent groups, solvable groups and simple properties of such groups, abelian groups, quasicyclic groups, finitely generated groups, free groups and their construction, wreath products, groups with the maximum condition or the minimum condition, simple groups. Properties of rings: Descending chain condition and Artinian rings, Wedderburn's Theorem, idempotents, polynomial rings, matrix rings, modules over semi-simple rings, ascending chain condition and Noetherian rings, finitely generated modules, prime ideals, primitive ideals, primitive rings, direct sums of modules, free and projective modules, invariant basis number.

The Qualifying Examination is based on the material in the courses MATH 573 and MATH 674, which include some of the topics listed above. Most of the above material can be found in the books of D.J.S. Robinson, A Course in the Theory of Groups, and D. S. Passman, A Course in Ring Theory. The algebra exam usually includes definitions, statements and proofs of theorems, examples, and standard exercises.

ANALYSIS: Basic concepts of set theory, measurability and Lebesgue measure; measurable functions, Lebesgue integration; bounded, monotone, and Lebesgue convergence theorems for integrals, Fatou's lemma; absolute continuity and functions of bounded variation; Hilbert space, bounded linear operators and their adjoints: Lp spaces, Holder and Minkowski inequalities; Banach spaces, dual spaces, Hahn-Banach Theorem, closed graph theorem, uniform boundedness theorem, and applications; Ascoli's Theorem and applications.

The courses preparatory to the analysis exam are MATH 580 and MATH 681. Most of the above material can be found in Royden's book, Real Analysis. Students should be familiar with a substantial collection of examples and counterexamples, and with the proofs of standard theorems.

TOPOLOGY: Topological spaces, metric spaces, Baire Category Theorem, separation and countability axioms, compactness and related concepts, connectedness and related concepts, continuous functions, Urysohn's Lemma, Tietze's Extension Theorem, spaces of functions, Tychonoff's Theorem, quotient spaces, homotopies of continuous functions, fundamental group, covering spaces and lifting criteria, singular homology, Hurewitz Theorem, exact sequences, Euler Characteristic, and computations of certain fundamental and homology groups.

Courses preparatory to the topology exam are MATH 565 and MATH 566. Most of the material can be found in Fred H. Croom's, Principles of Topology, chapters 2 through 8. This does not mean that the problems on the exam will be taken from this text exclusively; however, a student who demonstrates a reasonably high ability to apply this material will typically pass the exam.

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APPENDIX TWO
THE JOINT PROGRAM EXAMINATION IN APPLIED MATHEMATICS

NUMERICAL/LINEAR ALGEBRA: Vector spaces over a field. Subspaces. Quotient spaces. Complementary subspaces. Bases as maximal linearly independent subsets. Finite dimensional vector spaces. Linear transformations. Null spaces. Ranges. Invariant subspaces. Vector space isomorphisms. Matrix of a linear transformation. Rank and nullity of linear transformations and matrices. Change of basis. Equivalence and similarity of matrices. Dual spaces and bases. Diagonalization of linear operators and matrices. Cayley-Hamilton theorem and minimal polynomials. Jordon canonical forms. Real and complex normed and inner product spaces. Cauchy-Schwartz and triangle inequalities. Orthogonal complements. Orthonormal sets. Fourier coefficients and the Bessel inequality. Adjoint of a linear operator. Positive definite operators and matrices. Unitary diagonalization of normal operators and matrices. Orthogonal diagonalizational of real symmetric matrices. Bilinear and quadratic forms over a field. Triangular matrices and systems. Gaussian elimination. Triangular decomposition. The solution of linear systems. The effects of rounding error. Norms and limits. Matrix norms. Inverses of pertubed matrices. The accuracy of solutions of linear systems. Orthogonality. The linear least squares problem. Orthogonal triangularization. The iterative refinement of least squares solutions. The space Cn . Eigenvalues and eigenvectors. Reduction of matrices by similarity transformations. The sensitivity of eigenvalues and eigenvectors. Hermitian matrices. The singular value decomposition. Reduction to Hessenberg and triadiagonal forms. The power and inverse power methods. The explicitly shifted QR algorithm. The implicitly shifted QR algorithm. Computing singular values and vectors. The generalized eigenvalue problem.

REAL ANALYSIS: Lebesgue measure on R1 : Outer measure, measurable sets and lebesgue measure, non-measurable sets, measurable functions. Positive functions and general functions. Comparison with the proper and improper Riemann integral. Differentiation and integration: Monotone functions, functions of bounded variation, absolute continuity, the fundamental theorem of calculus. Definition of a positive measure. Measure spaces. Measurable functions. The integral with respect to a positive measure. Convergence theorems for positive measures: Monotone and dominated convergence. Lp spaces for positive measures with p - 1, 2, ¥, definition, completeness. Product measures, Lebesgue measure on Rk , Fubini's theorem.

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APPENDIX THREE
CURRENT MEMBERS OF THE GRADUATE MATHEMATICS FACULTY

Note: If you need more information about contacting our faculty members, see our Faculty/Staff directory.

ALLEN, PAUL J., Ph.D. (Texas Christian), Professor and Undergraduate Mathematics Director, 1967, 1975
Current Research Interest: Algebra
E-mail: Pallen@as.ua.edu........Phone: 348-1966

BELBAS, STAVROS, Ph.D. (Brown), Associate Professor, 1984,1989
Current Research Interest: Optimal Control
E-mail: Sbelbas@gp.as.ua.edu........Phone: 348-1992

CORSON, JON M., Ph.D. (Michigan), Associate Professor, 1990, 1996
Current Research Interest: Algebra and Topology
E-mail: Jcorson@gp.as.ua.edu........Phone: 348-1965

DAVIS, ANTHONY M. J., Ph.D. (Cambridge), D.Sc. (London), Professor, 1985,1987
Current Research Interest: Fluid Dynamics
E-mail: Adavis@euler.math.ua.edu........Phone: 348-1991

DIXON, MARTYN R., Ph.D. (Warwick), Professor, 1981, 1994
Current Research Interest: Algebra
E-mail: Mdixon@gp.as.ua.edu........Phone: 348-5154

EVANS, MARTIN, Ph.D. (Wales), Associate Professor, 1989, 1993
Current Research Interest: Algebra
E-mail: Mevans@gp.as.ua.edu........Phone: 348-5301

FRENKEL, ALEXANDER, Ph.D. (Moscow), Professor, 1989, 1992
Current Research Interest: Fluid Dynamics
E-mail: Afrenkel@gp.as.ua.edu........Phone: 348-5434

HADJI, LAYACHI, Ph.D. (Illinois at Urbana-Champaign), Associate Professor, 1990, 1995
Current Research Interest: Fluid Dynamics
E-mail: Lhadji@gp.as.ua.edu........Phone: 348-5153

HALPERN, DAVID C. M. J., Ph.D. (Arizona), Professor, 1993, 2004
Current Research Interest: Fluid Dynamics
E-mail: Dhalpern@euler.math.ua.edu........Phone: 348-1977

HOPENWASSER, ALAN, Ph.D. (Pennsylvania), Professor, 1976, 1983
Current Research Interest: Analysis
E-mail: ahopenwa@gp.as.ua.edu........Phone: 348-1979

HSIA, WEI-SHEN, Ph.D. (Rice), Professor and Chairperson, 1974, 1983 Current Research Interest: Optimization
E-mail: Whsia@gp.as.ua.edu........Phone: 348-1967

LAURIE, CECELIA A., Ph.D. (California-Berkeley), Professor, 1980, 1995
Current Research Interest: Analysis, Operator Algebras
E-mail: Claurie@gp.as.ua.edu........Phone: 348-1976

LEE, TAN-YU, Ph.D. (California-Santa Barbara), Associate Professor, 1980, 1988
Current Research Interest: Optimization
E-mail: Tlee@gp.as.ua.edu........Phone: 348-5303

LIEM, VO THANH, Ph.D. (Utah), Professor, 1979, 1986
Current Research Interest: Topology
E-mail: Vliem@gp.as.ua.edu........Phone: 348-4898

MAI, TSUN-ZEE, Ph.D. (Texas, Austin), Associate Professor, 1988, 1993
Current Research Interest: Numerical Analysis
E-mail: Tmai@gp.as.ua.edu........Phone: 348-1974

MOORE, ROBERT L., Ph.D. (Indiana), Professor, 1978, 1986
Current Research Interest: Analysis
E-mail: Rmoore@gp.as.ua.edu........Phone: 348-1978

NEGGERS, JOSEPH, Ph.D. (Florida State), Professor, 1967, 1979
Current Research Interest: Algebra, Combinatorics and Graph Theory
E-mail: Jneggers@gp.as.ua.edu........Phone: 348-5304

ROEHL, FRANK, Ph.D. (Hamburg), Associate Professor, 1986,1991
Current Research Interest: Algebra
E-mail: Froehl@gp.as.ua.edu........Phone: 348-1990

ROSEN, HARVEY, Ph.D. (Florida State), Professor, 1969, 1993
Current Research Interest: Topology
E-mail: Hrosen@gp.as.ua.edu........Phone: 348-1964

SUN, MIN, Ph.D. (Wayne State), Professor, 1986, 1994
Current Research Interest: Optimal Control
E-mail: Msun@gp.as.ua.edu........Phone: 348-1986

TRACE, BRUCE S., Ph.D. (California-Los Angeles), Associate Professor, 1984-1988
Current Research Interest: Topology
E-mail: Btrace@gp.as.ua.edu........Phone: 348-1962

TRENT, TAVAN T., Ph.D. (Virginia), Professor, 1979, 1987
Current Research Interest: Analysis
E-mail: Ttrent@gp.as.ua.edu........Phone: 348-1973

WANG, JAMES L., Ph.D. (Brown), Professor, 1976, 1985
Current Research Interest: Analysis
E-mail: Jwang@gp.as.ua.edu........Phone: 348-1972

WANG, PU P., Ph.D.(Lehigh), Professor, 1990, 2002
Current Research Interest: Applied Stochastic Processes
E-mail: Pwang@gp.as.ua.edu........Phone: 348-5302

WU, ZHIJIAN, Ph.D. (Washington), Professor, 1990, 1999
Current Research Interest: Analysis
E-mail: Zwu@gp.as.ua.edu........Phone: 348-1963

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