Graduate Handbook, Math Dept, UCD

Doctor of Philosophy (Ph.D.) in Applied Mathematics (Coordinated with the Department of Applied Mathematics at CU-Boulder). This degree is designed to give candidates a contemporary, in-depth education in applied mathematics and to provide research opportunities in the special fields of computational mathematics, discrete mathematics, optimization, applied statistics, applied probability, computer science and the mathematics of science and engineering.

Master of Science (M.S.) in Applied Mathematics. This degree is designed to prepare a candidate to assume a position as an applied mathematician, a teacher or to continue studies at the Ph.D. level.

Prerequisites for Graduate Studies in Applied Mathematics

Admission to the Graduate School requires at least a 2.75 grade point average (gpa) in all undergraduate course work. The Mathematics Department does not require applicants to submit GRE scores, but strongly recommends that they do so, including the math subject test. Applicants to the doctoral program who have an overall gpa below 3.0 or a gpa in math below 3.2 are unlikely to be accepted without other strong supporting evidence, such as high GRE scores. Students must also have the following preparation: 30 semester hours of mathematics, at least 24 of which are upper division courses with a grade of B- or better. These must include two semesters of advanced calculus; one semester of linear algebra; and one semester of abstract algebra, differential equations or discrete mathematics. Occasionally a student may be admitted with the understanding that certain deficiencies will be made up while the student is formally in the program, but such course work does not constitute part of the graduate degree program. In the case of students who do not have all the prerequisites, or who have a low gpa, the Graduate Committee may recommend provisional admission. Provisional students, however, are ineligible for financial aid, teaching assistantships and/or classification as doctoral students.

Admission of international students requires TOEFL above 500, financial documentation, and certified English translation of all records and references not in English.

Admission Deadlines

A complete application packet (including two official transcripts, at least three letters of recommendation, application part I and II, and application fee) for admission into the graduate program must be submitted to the Graduate Committee of the Department of Mathematics by the following dates in order to enter the program at the beginning of the term indicated:

June 1 for the following fall semester

November 1 for the following spring semester

April 1 for the following summer semester

To be considered for a Teaching Assistantship, the completed packet must be submitted to the Graduate Committee by the following dates:

February 15 for the following fall semester

October 1 for the following spring semester

Advising

Upon acceptance into the graduate program, each student is assigned an adviser. At any time thereafter, by informing the graduate program assistant, the student may request and obtain a change in this assignment to reflect his or her own developing interests.

A required orientation for all graduate students new to the Applied Mathematics program and for all Ph.D. students in residence is held just before the fall semester begins each year. The orientation provides information about the department, the faculty, graduate program requirements, expectations and realities, and an opportunity for students to meet with individual faculty advisers.

During the first semester of graduate study each student is required to set up a tentative plan of study with the adviser's help. The plan of study, which will be part of the graduate student's permanent record, should include a tentative list of courses to be taken in each semester. Both the adviser and the student will sign the plan, which should be reviewed at least once each year with the current adviser. All course substitutions must be noted on the study plan.

Requirements for the M.S. Degree

1. General Requirements:

Students must complete at least 30 semester hours, of which at least 24 hours must be in mathematics courses numbered 5000 or above. At most six hours may be in courses outside math at the 4000 level or above, if approved by the student's adviser and by the Graduate Committee. The overall grade point average must be 3.0 or higher. Courses in which grades below C+ (2.3) are received will not be applied to the M.S. degree (but they do contribute to the overall gpa).

At most 9 hours of transfer credit and 6 hours earned as nondegree student, with grade B- or better, may be applied towards the M.S. degree upon approval of the Graduate Committee. Ten year time limit is in effect on all course credit. Courses already used for another Master's degree cannot be used towards the M.S. degree.

The following courses will not count toward a graduate degree in applied mathematics: Math 5010, Math 5017, Math 5250, or Math 5260.

2. Math Clinic:

All students are encouraged to participate in at least one Math Clinic. With the approval of the student's adviser, a clinic may be used in lieu of a stated area requirement. Details about current and upcoming projects are available from the Clinic Director.

3. Advisory Committee:

By the end of the first year of graduate study, each student will have chosen an adviser to chair his/her Advisory Committee. With this adviser's help, the student will choose two additional members for the Advisory Committee. All three members must be on the graduate faculty, and at least two will be members of the department.

4. Thesis Option:

Each student is encouraged to write a thesis. A student who elects the thesis option must register to from four to six semester thesis hours from the 30 required hours. Although original work is encouraged, the thesis may be expository in nature. The topic should be within a relevant area of applied mathematics and should be chosen with the approval of the adviser. The thesis should be prepared in TeX, LaTeX, or AMSTeX and must comply in format with the specifications of the Graduate School. A typed copy of the thesis will be available to all members of the Advisory Committee at least three weeks before the defense. Eighteen days before the date of graduation, three complete copies of the thesis must be filed with the Graduate School. The student must provide a single postscript file with an attached statement giving the Department the right to distribute the thesis as it wishes.

5. Final Examination:

All students must take and pass a final oral exam. The exam will be given by the student's Advisory Committee. For students choosing the thesis option, the exam will consist of a one hour defense of the thesis. The examining committee may declare that the thesis defense was satisfactory but request that the student make further changes before the thesis is filed with the department and the Graduate School, and specify a deadline and the manner in which the revised thesis will be reviewed. In that case, the student does not need to register for further thesis credit hours, but the requirements for the master's degree are not satisfied until the final version of the thesis is approved by the student's Advisory Committee and the Graduate School. If no member of the committee raises further questions or objections within 30 days after the revised thesis has been received by the chair of the committee, the thesis will be considered approved by the Advisory Committee.

Students choosing the non-thesis option will give a one hour presentation and answer questions on a relatively specific topic which has been selected in consultation with the adviser. A student who does not pass the exam on the first attempt may take it again after three months. Guidelines for oral presentations and thesis defenses are available in the mathematics department.

6. Time Limits for Completion of Degree:

Master's degree students have five years from the date of the start of course work to complete all degree requirements.

7. Course Requirements by Area:

The core courses, Math 5070 (Applied Analysis) or Math 6131 (Real Analysis), and Math 5718 (Applied Linear Algebra) are required of all M.S. students. All optional courses are subject to approval by the student's adviser. In addition, the requirements in one of the following areas must be satisfied. Substitutions may be made with the adviser's written approval. One course cannot be used to fulfill two requirements.

a. Applied Statistics

Math 5387 (Statistical Methods in Research I)

Math 6381 (Mathematical Statistics I)

Math 6382 (Mathematical Statistics II)

Math 6388 (Statistical Methods in Research II)

Strongly recommended: Math 5330 (Workshop in

Statistics Consulting - NOTE: This course can be

taken more than once.)

b. Applied Probability

Math 5310 (Probability)

Math 5792 (Probabilistic Modeling)

Math 5780 (Stochastic Processes)

One of the following two courses:

Math 6131 (Real Analysis)

Math 6381 (Mathematical Statistics I)

c. Computer Science

i) Two foundations courses from the following list:

Math 5446 (Theory of Automata, same as C SC 5464)

Math 5576 (Mathematical Foundations of Artificial Intelligence I)

Math 6420 (Applied Algebra)

C SC 5451 (Algorithms)

ii) One discrete computer-related course from the following:

Math 5405 (Applied Graph Theory)

Math 5410 (Coding Theory and Cryptology)

Math 5793 (Discrete Math Modeling)

Math 6221 (Projective Geometry I)

Math 6409 (Applied Combinatorics)

iii) A continuous computer-related math course from the following:

Math 5595 (Comput Methods/Nonlinear Programming,was Math 5665)

Math 5660 (Numerical Analysis I)

Math 5663 (Intro to Finite Element Methods, formerly Math 7172)

Math 6663 (Numerical Solution of PDE)

Math 6664 (Numerical Linear Algebra)

iv) One additional computer-related course from the following:

One course from i, ii, iii (above), or a sequel to it

Math 5674 (Parallel Computing & Architectures, same as C SC 6551)

Math 5779 (Math Clinic, subject to approval)

Math 5780 (Stochastic Processes)

d. Discrete Mathematics

Four of the following ten courses:

Math 5113 (Modern Algebra I)

Math 5405 (Applied Graph Theory)

Math 5410 (Coding Theory)

Math 5490 (Network Flows)

Math 5793 (Discrete Math Modeling))

Math 6023 (Topics in Discrete Math)

Math 6221 (Projective Geometry I)

Math 6406 (Combinatorial Structures)

Math 6409 (Applied Combinatorics)

Math 6420 (Applied Algebra)

Other suggested courses:

Math 5110 (Number Theory), Math 5593 (Linear

Programming), Math 6114 (Modern Algebra II), Math

7594 (Integer Programming), and Math 7923

(Readings in Discrete Math)

e. Mathematics of Engineering and Science

Three of the following seven courses:

Math 5387 (Statistical Methods in Research I)

Math 5779 (Math Clinic)

Math 5791 (Continuous Modeling)

Math 5792 (Probabilistic Modeling)

Math 5793 (Discrete Math Modeling)

Math 5794 (Optimization Modeling))

Math 6735 (Continuum Mechanics)

Two of the following seven courses:

Math 5660 (Numerical Analysis I)

Math 5661 (Numerical Analysis II)

Math 5663 (Intro to Finite Element Methods, formerly Math 7172)

Math 5733 (Partial Differential Equations, formerly Math 6733)

Math 5743 (Ordinary Differential Equations, formerly Math 6743)

Math 6663 (Numerical Solution of PDEs)

Math 6664 (Numerical Linear Algebra)

f. Numerical Analysis

Math 5660 (Numerical Analysis I)

Math 5661 (Numerical Analysis II)

Three of the following ten courses:

Math 5593 (Linear Programming)

Math 5595 (Computational Methods in Nonlinear Programming)

Math 5663 (Intro to Finite Element Methods, formerly Math 7172)

Math 5667 (Intro to Approximation Theory)

Math 5733 (Partial Differential Equations, formerly Math 6733)

Math 6663 (Numerical Solution of PDEs)

Math 6664 (Numerical Linear Algebra)

Math 6735 (Continuum Mechanics)

Math 7664 (Iterative Methods in Numerical Linear Algebra)

Math 7760 (Math Foundations of Finite Element Methods)

Students in this area are also encouraged to take graduate

level computer science and/or parallel computing courses.

g. Operations Research

Math 5310 (Mathematical Probability) NOTE: This

course is not required for students who completed an

undergraduate, upper division course in probability.

Math 5593 (Linear Programming)

Two of the following thirteen courses:

Math 5387 (Statistical Methods in Research I)

Math 5390 (Game Theory)

Math 5490 (Network Flows)

Math 5595 (Computational Methods in Nonlinear Programming)

Math 5779 (Math Clinic, with approval)

Math 5791 (Continuous Modeling)

Math 5792 (Probabilistic Modeling)

Math 5793 (Discrete Math Modeling)

Math 5794 (Optimization Modeling)

Math 6381 (Mathematical Statistics I)

Math 5780 (Stochastic Processes)

Math 7594 (Integer Programming)

OPMG 6800 (When appropriate, with approval)

Requirements for the Ph.D. Degree

Admission to the Ph.D. Program

The requirements for admission to the Ph.D. program are the same as those given for admission to graduate study, although a master's degree in mathematics may be accepted in lieu of the specific requirements listed there. Ph.D. applicants with course work deficiencies may be accepted into the master's degree program with the understanding that they could possibly be accepted into the Ph.D. program once the deficiencies have been remedied.

Graduation Requirements

There are six phases to the Ph.D. program. A candidate must fulfill course requirements, pass the preliminary examinations, establish a Ph.D. Committee, meet the academic residency and participation requirements, pass the comprehensive examination, and write and defend a thesis.

1. Course requirements:

Students without Masters degree must complete at least 42 credit hours of graduate courses. Students with a Masters degree must complete at least 30 credit hours of graduate courses. Upon approval of the Graduate Committee, at most 18 hours of the credit already applied to the Master's degree and 6 hours earned as nondegree student, with grade B- or better, may be transferred to the Ph.D. degree. All course work must be completed with at least a 3.25 grade point average. Grades below a B- are not acceptable for the Ph.D. Any doctoral student who receives a grade of C+ or lower, or whose overall gpa as a doctoral student falls below a 3.25 gpa, will be reviewed by the Graduate Committee and may be put on probation or suspended. The following courses are required as a part of the above 42, resp. 30 credit hours:

4 PhD level courses, at 3 credit hours each

Math Clinic (Math 5779), 3 credit hours

3 Readings Courses (Math 7921-7926), 1 credit hour each.

Courses at the 7000 level (excluding readings courses and thesis hours) are considered Ph.D. level courses for this requirement, and selected 5000 and/or 6000 level courses may be used in place of 7000 level courses with the consent of the student's adviser. The Readings Courses are one credit hour seminar courses which are announced usually just before the semester offered. All courses should be chosen in consultation with an adviser. Equivalent courses approved by the Graduate Committee, such as a Computer Science Clinic and Readings in Computer Science, are acceptable to satisfy the Math Clinic and Readings requirements.

2. Preliminary examinations:

Each student must pass three preliminary examinations:

1. Applied Analysis

2. Applied Linear Algebra

3. A special area exam in one of

- Discrete Mathematics

- Optimization

- Numerical and Computational Mathematics

- Probability and Statistics

- Science and Engineering Mathematics

- Computer Science

These are three hour written examinations. The first two examinations cover material roughly at the level of first year of graduate study; the special area examination is approximately at the master's level in the respective discipline. The preliminary examinations in applied analysis and applied linear algebra will be offered two times a year, from 10 AM to 1 PM on the first and the second Saturdays of the fall and spring semesters. The special area examination will be scheduled during the first month of each spring semester, but may be offered more frequently upon request. A student wishing to take any preliminary exam must sign up for the exam with the graduate program assistant at least one month prior to the administration of the exam. Students who cannot take the exam as planned must give notice of withdrawal at least one week in advance of the exam. See section 8 for time limits.

Students may take the Applied Analysis and/or Applied Linear Algebra exams without penalty during their first three semesters (first semester only if they are entering the program with a Master's degree in Mathematics) in the program. After that time, a student who has failed one of these exams can petition the Graduate Committee for another attempt. Students not in the Ph.D. program (but in a Master's program at CU-Denver) are limited to at most two attempts for each of these preliminary exams.

There is no penalty free administration of the special area exams and only students in the Ph.D. program may take these exams. The examination committee shall determine appropriate action in cases of failure or low passes of this exam.

3. Ph.D. Advisory Committee:

Each student must choose an adviser and, with the adviser's help, select other members of the Ph.D. committee. The Ph.D. Committee assumes responsibility for advising the student on matters concerning the requirements for the Ph.D. degree, conducting and determining the outcome of the comprehensive exam, and acting as the examining committee for the student's dissertation defense. The composition of this committee must be approved by the Mathematics Graduate Committee at least 30 days before the student takes the comprehensive exam. If the committee membership changes after the comprehensive examination, the adviser must again request approval by the Graduate Committee at least 30 days before the date of the defense.

The Ph.D. Advisory Committee consists of five graduate faculty from the CU system, one of whom is the student's adviser. At least one committee member must be from outside the mathematics department, and at least one must be a regular faculty member of the mathematics department. Committee members may be from outside the CU system, but must be granted special membership on the CU graduate faculty for this purpose. The Chair of the Ph.D. Committee must be a regular member of CU-Denver mathematics faculty. The main role of the Chair is to assure compliance with the rules of the program. Normally the Chair and the thesis adviser are the same person, but it is possible for the thesis adviser to be from elsewhere, e.g., another department at CU, a government lab, another university, or industry. At least three members of the Ph.D. Committee will be chosen to represent the three areas of the student's oral comprehensive examination.

4. Residency and Participation:

By Graduate School requirements, all Ph.D. candidates must accumulate six semesters of residence credit at the University of Colorado beyond the bachelor's degree (although two of these six semesters may be replaced by a master's degree in mathematics earned at another institution). For this purpose a full course load is defined to be five semester hours of course work. The manner in which residence credits may be accumulated is described in the Graduate School Rules.

In order to foster participation in the life of the department and immersion in research, each student is expected to spend at least two consecutive semesters as a full time student without employment outside the university. When this is not feasible, it shall be the responsibility of the student's Ph.D. Advisory Committee to consider the special situation of the student and to respond to a proposal from the student concerning the manner in which the participation requirement shall be satisfied. By whatever means the student's Ph.D. Committee decides to fulfill the participation requirement, it must uphold the spirit of this requirement. In other words, the student is expected to attend department functions, such as Colloquia and Orientation Day.

5. Comprehensive Examination:

Application for candidacy to the Ph.D. program must be made at least two weeks before the comprehensive examination is taken. Candidacy will be granted after at least three semesters of residence have been earned, all preliminary and comprehensive examinations have been passed, and essentially all course requirements have been satisfied.

The comprehensive examination is designed to determine a student's mastery of graduate level mathematics and ability to embark upon thesis research. The examination consists of presentation of a thesis proposal and an oral examination from three areas, chosen from the following list:

1. Applied Statistics

2. Applied Probability

3. Computational Mathematics

4. Discrete Mathematics

5. Optimization

6. Differential Equations and Applications

7. Computer Science

8. Abstract Algebra and Applications

9. Real, Complex, and Functional Analysis

10. An independent area approved by the Graduate Committee

upon the request of the student's Ph.D. Committee at least

30 days before the comprehensive examination.

One of the chosen areas must be the background for the proposed thesis. The extent of the coverage of the chosen areas will be determined by the student's Ph.D. Committee. Preparation for the oral examination should be done in consultation with the members of that committee. The committee will conduct the examination and determine its outcome. The exam is open to all members of the graduate faculty and members of the mathematics faculty will be notified of comprehensive exams at least one week in advance. Students failing the comprehensive exam may petition the Graduate Committee for the second attempt. If the second attempt is granted, the student's Ph.D. Committee will determine the form and extent of the re-examination. See page ten for time limits.

After completion of the comprehensive examination, students become candidates for the doctoral degree and must register during each regular semester of the academic year until graduation for no less than five thesis hours and no more than ten thesis hours. Students may register for thesis hours before the completion of the comprehensive examination with the following limitations. A student may not enroll for thesis hours before the semester in which the last preliminary exam is passed. Prior to the completion of the comprehensive examination, a student may not enroll for more than six thesis hours per semester.

6. Thesis:

Candidates for the Ph.D. degree are required to write a dissertation containing original contributions and evidence of significant scholarship. The thesis is written under the guidance of an adviser who is a member of the graduate faculty of the University of Colorado. The thesis must comply in format with the specifications of the Graduate School and must be prepared in TeX, LaTeX, or AMSTeX. Six weeks before the date of graduation, the Graduate School must be notified by the candidate of the dissertation title. Thirty days before the final thesis defense, the thesis must be available in typewritten form. Eighteen days before the date of graduation, three complete copies of the thesis must be filed with the Graduate School. The student must provide a single postscript file with an attached statement giving the department the right to distribute the thesis as it wishes.

At least thirty days before the date of graduation, the candidate must present and defend the dissertation before an examining committee. The examining committee, denoted above as the student's Ph.D. Advisory Committee, must be approved by the Graduate Committee and the Dean of the Graduate School. The examining committee may declare that the thesis defense was satisfactory but request that the student make further changes before the thesis is filed with the department and the Graduate School, and specify a deadline and the manner in which the revised thesis will be reviewed. In that case, the student does not need to register for further thesis credit hours, but the requirements for the Ph.D. degree are not satisfied until the final version of the thesis is approved by the student's Ph.D committee and the Graduate School. If no member of the committee raises further questions or objections within 30 days after the revised thesis has been received by the chair of the committee, the thesis will be considered approved by the Ph.D. Committee.

Each student must also complete at least 30 hours of thesis credit. Not more than 10 of these hours may be taken in any one semester. Not more than 10 thesis hours taken prior to the semester of the comprehensive examination may be applied to this requirement.

7. Language Requirement:

The University of Colorado permits each department to decide whether or not to implement a foreign language requirement. The Mathematics Department adopted the following:

Recognizing that the need for knowledge of a foreign language to do research in an area varies from no languages in many areas to one or two languages in a few areas, the department has no formal language requirement. Instead, the necessity for some level of proficiency in a foreign language is left to the discretion of each student's major adviser, as is the case for other matters related to a student's preparation for doing research.

8. Time Limits

All examinations and defenses must be scheduled and announced at least two weeks in advance.

Preliminary examinations: Students must pass the Applied Analysis and Applied Linear Algebra exams before the end of their fourth semester in the program or the completion of 24 hours of graduate course work, whichever comes first (third semester, or 18 hours for students entering the program with a Master's degree in mathematics). All preliminary exams must be passed before the completion of 30 hours of graduate coursework (fourth semester, or 24 hours, whichever comes first, for students entering the program with a Master's degree in any area).

Ph.D. Committee: Students must select an adviser and a Ph.D. committee by the end of the semester in which all preliminary exams have been passed.

Comprehensive examinations: Students must pass the comprehensive examination by the end of the seventh semester in the program (sixth semester for students entering the program with a Master's degree in any area), or by the time set by the student's Ph.D. committee.

All requirements for the Ph.D. degree must be completed within four years of passing the comprehensive examination and within seven years of entering the Ph.D. program.

It is recognized that flexibility is necessary, especially for transfer and part-time students; hence petitions for exceptions will be considered by the Graduate Committee. The seven year deadline for completing the Ph.D. is imposed by the Graduate School and exceptions require its approval.

Transitional Rules

All graduate students currently accepted to the applied mathematics program (both master's and doctoral) will have the option of graduating under the rules in effect when they matriculated.

Leave of Absence

Students may request a leave of absence from the Ph.D. program. The student must be in good standing, indicate the return date, give justification for the leave of absence, agree to contact his/her adviser and the Graduate Committee at least once per semester, and agree not to use campus facilities (except e-mail through Mathematics Department computers). Each petition must be approved by the Mathematics Graduate Committee and submitted to the CLAS Dean and the Graduate School for further approval.

A leave of absence does not extend deadlines automatically; this requires a separate petition after returning from the leave of absence. A leave of absence does allow the student to interrupt registration for thesis hours following the comprehensive exam.

Types of Financial Support

Doctoral Fellowships

The Graduate Committee awards a limited number of fellowships to doctoral students based on scholarship and promise of completing the Ph.D. degree. The duration of a fellowship is one year. Each fellowship provides tuition and a stipend, and the recipient is expected to make significant progress toward the Ph.D. by taking courses, studying for and taking the preliminary and comprehensive exams, and conducting research. Additionally the fellow may be required to do grading or teach a class.

Teaching Assistantships

The usual term of a teaching assistantship is an academic year, with the understanding that the appointment for the second semester depends upon the satisfactory completion of duties during the first semester as well as satisfactory progress toward a degree. The duties of a teaching assistant consist of teaching courses and/or assisting in the Mathematics Education Resource Center (MERC computer lab) for various undergraduate courses. Occasionally other duties may be assigned, but the standard work load is 20 hours per week. Details related to teaching assistantships are explained in the document Policies Regarding Teaching Assistants.

A limited number of first year teaching assistants without Colorado residency may be awarded out-of-state tuition differentials. All such students who are U.S.A. residents are expected to gain in-state residency by the beginning of the second year of study.

Research Assistantships

Another type of appointment, dependent upon the contracts and grants of individual faculty members, is that of research assistant. Appointments are made and duties assigned by faculty members with external funding. These appointments are normally not given to new graduate students. The workload for these appointments is also understood to be 20 hours per week.

Lynn Bateman Memorial Teaching Award

The Lynn Bateman Memorial Teaching Award, a cash award of $1,000, has been established to honor the memory of Lynn Bateman by annually recognizing and rewarding a Teaching Assistant in the Department of Mathematics who, in the opinion of the faculty of the Mathematics Department, has excelled in teaching. (See the Policies Regarding Teaching Assistants for further details.)

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UNIVERSITY OF COLORADO at DENVER

GRADUATE PROGRAMS in APPLIED MATHEMATICS

Student Handbook: Graduate Study at CU-Denver

Adopted 6 December 1989

Revised 3 March 1999

Revised 30 April 2000

CONTENTS

Degrees