I. Description of Programs

The department offers programs of study leading to M.S. and Ph.D. degrees in Mathematics, M.S. and Ph.D. degrees in Applied Mathematics and the M.S. degree in Statistics.

The master's programs demand 30 credit hours of graduate courses with at least 18 hours at the 400 level. Up to six hours may be replaced by thesis. Courses are chosen in consultation with a faculty advisor or a member of the department's Graduate Committee.

The doctoral programs require a minimum of 72 credits of graduate courses beyond the bachelor's degree. Students who have earned a master's degree at another institution must register for a minimum of 48 credit hours of graduate courses.

Intially, courses are chosen in consultation with a faculty adivosr or a member of the department's Graduate Committee. After completing all department requirements needed for admission to candidacy, the plan of study is directed by the student's thesis advisor and the doctoral committee. A full-time graduate student normally registers for 9 credit hours per semester. A student who has completed the minimum course credits toward the degree normally registers for only 3 credits per semester and submits a Full-Time Certification form each semester at registration. If the student has applied for and been admitted to Doctoral Candidacy (this has several requirements, including some paperwork), then the student normally registers for 1 credit of "Maintenance of Candidacy." Students should consult the College of Arts and Sciences Graduate Student Handbook and the Lehigh University Catalog for more specific information concerning rules and requirements (including deadlines).

II. Description of Non-course Requirements
a) The Comprehensive Examination

Every candidate for the M.S. degree in Mathematics, M.S. degree in Applied Mathematics, Ph.D. degree in Mathematics or Ph.D. degree in Applied Mathematics, must take and pass a Comprehensive Examination. The purpose of the examination is to test the student's knowledge of basic mathematics in the areas of calculus, linear algebra and ordinary differential equations. Candidates for the M.S. degree in Statistics must also take and pass a Comprehensive Examination. The statistics examination tests the student's knowledge in elementary calculus, linear algebra, probability and statistics.

The examination is normally given in January and March. Full-time students entering in August must take the examination before completing eighteen hours of graduate level course work at Lehigh. Hence it is expected that full-time students entering in August, will take the examination in January, and in the event of failure, again in March. Full time students entering in January take the examination in August, and in the event of failure in November.  Part time students must take the examination as soon as possible after completing 15 hours. However, part time students are encouraged to consider taking the examination earlier.

The comprehensive examinations  consist of  ten problems of a general and elementary nature in the topics described above. Correct solutions to six problems will assure a passing grade. Partial credit is not awarded.

A student who passes the examination and satisfies all other program requirements for the M.S. degree will be eligible for a master's degree in the appropriate program. The examination serves the additional purpose of helping the department decide whether further graduate study is appropriate for the student. Only students who pass the examination may proceed to further graduate study.

b.) The Qualifying Examination

Every candidate for the Ph.D. in Mathematics or Ph.D. in Applied Mathematics must take and pass a Qualifying Examination. The purpose of the examination is to test the breadth and depth of the student's knowledge of graduate level topics and to assess the student's readiness to begin thesis work.  The Qualifying Examination consists of three three-hour written tests in different areas of mathematics. The three tests must be taken and passed during one examination period. For a Ph.D. in Mathematics, a student chooses three tests from the following list. We list courses which are roughly associated to the contents of each examination. The Qualifying Examinations are not tied directly to the content of the courses as they are taught in any particular year. The examination syllabi provide more detailed information.

Algebra (327, 428)
Analysis (316, 401, 402)
Differential Equations (205, 320, 322, 405)
Differential Geometry (423, 424)
Logic and Set Theory (303, 304, 329; see Prof Stanley for a tailoring of your exam)
Numerical Analysis (230, 430)
Probability (309, 310, 463)
Statistics (309, 334, part of 338, statistical inference from 461)
Topology (307, 408)
Discrete Structures, Combinatorial Analysis, and Graph Theory (446, 448)
Financial mathematics (467,468)

For a Ph.D. in Applied Mathematics a student chooses three tests from the following list:

Applied Probability (309, 310, 463, applied emphasis)
Differential Equations (205, 320, 322, 405, applied emphasis)
Discrete Structures, Combinatorial Analysis and Graph Theory (446, 448, applied emphasis)
Linear Algebra and Linear Programming (see Prof Isaak)
Numerical Methods (230, 430, applied emphasis)
Statistical Methods (312, 338, 462)
Financial mathematics (467, 468, applied emphasis)
An area in the mathematical or physical sciences chosen by the applicant and approved by the department

A detailed syllabus is available for the test in each of the above subjects.

The examination is given three times a year, in August, January and May.   In the event of failure, the student should arrange a meeting with the mathematics graduate advisor and coordinator within two weeks of failing.  Students who do not pass the examination will no longer be able to continue study in the department.   A student may take the Qualifying Examination any time after satisfactorily passing the Comprehensive Examination.  A student who fails the Qualifying Examination twice will be allowed a third attempt if and only if, on the second attempt, the student passes two of the three tests.  A full-time student must pass the qualifying examination by the examination date immediately following the sixth semester of study in order to continue in the program.  A part-time student must pass the qualifying examination by the examination date immediately following the semester in which the student has attempted a total of 54 credits in the mathematics program. 

Normally, a full-time student who enters the program in August and does not pass the Qualifying Examination by the January immediately following the fifth semester of study (i.e. the January of the
third year of study) in the program will not be awarded financial support for the following fall and subsequent semesters of study; although the student will be allowed to continue in the program through the spring semester immediately following the fifth semester of study.

Normally, a full-time student who enters the program in January and does not pass the Qualifying Examination by the May immediately following the fifth semester of study in the program will not be given financial support for the following fall and subsequent semesters of study; although the student will be allowed to continue in the program through the semester immediately following the fifth semester of study.

c.) The General Examination

A Ph.D. student must pass the General Examination no later than 7 months before the degree is conferred. More details can be found in the College of Arts and Sciences Graduate Student Handbook.

d.) Doctoral Candidacy.

A Ph.D. student who plans to graduate in May must enter candidacy by the preceding August. This involves some work such as forming a Doctoral Committee and submitting a proposal. More details can be found in the College of Arts and Sciences Graduate Student Handbook.

e.) The Doctoral Dissertation and Defense

f.) The Language Requirement

There is no formal departmental language requirement.  However, every candidate for a Ph.D. in Mathematics is encouraged to develop a reading knowledge of the mathematical literature in French, German, Russian, or Chinese. Some advisors require that their Ph.D. students demonstrate that they have developed such a reading knowledge.

III. Special Courses and Programs

a.) Reading Courses

Students interested in intensive study of a field of mathematics not covered in courses listed in the catalog can petition to enroll in Mathematics 409(Fall), 410(Spring) Mathematics Seminar or Mathematics 450, Special Topics. A written description of the proposed reading course must be submitted to the Graduate Committee indicating the topics to be studied and the professor(s) directing the course. Reading courses must be approved by the Department's Graduate Committee. Students who have not yet passed the Qualifying Examination may take at most one reading course (or up to three credits) per semester. While Mathematics 409/410 and 450 may be repeated for credit, students are not permitted to take a reading course in a subject or on a topic which is covered in a course regularly offered by the department and for which the students has already received credit.

c.) Teaching Fellowships

Students who are near completion of the dissertation may be considered for the Teaching Fellowship program. Teaching Fellows are assigned to teach one lecture section of a first year course (Mathematics 0, 51, 52, 61, 75, 76) in which a regular faculty member is in charge. The professor and the Teaching Fellow meet biweekly to discuss the conduct of the course and the professor observes the Teaching Fellow in the class at least twice during the semester. An evaluation of the Teaching Fellow is written by the professor at the end of the semester.

To qualify as a Teaching Fellow, a student must have served successfully as a teaching assistant for at least three years.

Typically, students are nominated by the faculty to serve as Teaching Fellows. However, any student may request consideration. Students are recommended by the Graduate Committee and appointed by the department chair.

IV. Financial Support

a.) Teaching Assistantships

Many graduate students are employed as teaching assistants in the department. Teaching Assistants typically are assigned to teach recitation sections of freshman and sophomore level courses. The assigned duties include preparation and presentation of material, grading of student work and holding office hours. These constitute a commitment of twenty hours per week. Teaching Assistants receive a stipend (approximately $17,000 for 2009-2010) and tuition remission. The number of Teaching Assistants appointed in the Mathematics Department varies from year to year but has been at least twenty over the last five years.

Students, new and continuing, who wish to be considered for a Teaching Assistantship should apply to the Graduate Committee. The Graduate Committee, after reviewing all applicants, makes a recommendation to the chair of the department. Preference is given to students currently enrolled in the program and supported by the department who are making satisfactory progress towards a degree. A ranked list of new students and students not currently supported is prepared by the committee and presented to the Chair. The Chair uses this list to make new appointments.

b.) Graduate Fellowships

A few graduate fellowships are available. These are awarded on a competitive basis.

Any students, new or continuing, who wish to be considered for a Graduate Fellowship should apply to the Graduate Committee. The Graduate Committee after reviewing all applicants, makes a recommendation to the chair of the department. The primary criteria used is merit.