# Mathematics, Ph.D.

COLLEGE OF SCIENCE AND TECHNOLOGY

Learn more about the Doctor of Philosophy in Mathematics.

## About the Program

The Ph.D. program in Mathematics prepares students for careers that depend on advanced mathematics. These include broad directions such as advanced research and development, education, government, industry, and national laboratories. For new students, the program offers a repertoire of coursework and research opportunities that ease the transition from undergraduate to advanced graduate studies. The courses provide a sound mathematical background, while helping beginning students to mature mathematically. Naturally, individuals with enough maturity and knowledge need not take these more basic courses. In the Mathematics Department, our philosophy is to participate actively in our students' development as future professionals whose work involves advanced mathematics. We take pride in caring for our students. Our faculty are accessible and willing to talk mathematics with any inquiring student. It is this attitude that most distinguishes our program from other mathematics graduate programs. While requiring excellence, we work hard at providing the environment for achieving it.

**Time Limit for Degree Completion:** 7 years

**Campus Location:** Main

**Full-Time/Part-Time Status:** Students complete the degree program through classes offered before 4:30 p.m. The degree program can be completed on a full- or part-time basis

**Interdisciplinary Study:** The program encourages interdisciplinary coursework, research, and interactions among faculty and students with interest in computer and information sciences, physical and life sciences, statistics, and engineering.

**Affiliation(s):** The Mathematics program at Temple University is affiliated with the American Mathematical Society and the Mathematical Association of America.

**Study Abroad:** Department faculty are active internationally and sometimes travel overseas for conferences and extended research visits. In some cases, students may participate in these activities.

**Ranking:** The Ph.D. program is designed to provide opportunities for education and research that are commensurate with national standards. Faculty are active in professional meetings and initiatives organized by the American Mathematical Society and the Mathematical Association of America.

**Accreditation:** This program adheres to accepted professional standards of mathematics education and research.

**Areas of Specialization:** The department offers a great variety of choices for areas of specialization, with a strong research presence in the following areas:

- Algebra
- Algebraic and analytic number theory
- Combinatorics
- Computational mathematics
- Differential and computational geometry and topology
- Global geometry
- Harmonic analysis
- Mathematical physics
- Mathematics of materials
- Numerical analysis
- Partial differential equations
- Related probability and mathematical statistics
- Several complex variables

Both prospective and matriculated students are encouraged to browse faculty web pages and contact faculty directly for more detailed information regarding areas of specialization and opportunities for further research.

**Job Prospects:** Graduates either continue advanced educational programs or pursue employment in industry, education, or government laboratories and agencies.

**Non-Matriculated Student Policy:** Non-matriculated students must coordinate coursework with the Graduate Chair.

**Financing Opportunities:** Teaching Assistants teach basic undergraduate mathematics courses, ranging from remedial courses through calculus. The standard teaching load is 20 hours per term. In determining the load, credit is given for more difficult and challenging teaching assignments. Research Assistantships are sometimes available, typically through special projects and grants. Support generally includes a stipend and tuition of up to 9 credits per term.

## Admission Requirements and Deadlines

**Application Deadline:**

*Fall:* January 8; December 8 international

For full consideration, applications should be received by the stated deadlines. Applications received after these dates will be considered on a case-by-case rolling basis.

*APPLY ONLINE to this graduate program**.*

**Letters of Reference:**

*Number Required:* 3

*From Whom:* Letters of recommendation should be obtained from individuals who are well acquainted with the applicant's abilities and achievements in mathematics and related areas, particularly former instructors in mathematics courses and projects. Letters from instructors in related areas such as computation or the physical and life sciences are also appropriate. In certain cases, letters from employment supervisors or project leaders may be appropriate as well.

**Coursework Required for Admission Consideration:** Applicants must have completed fundamental undergraduate mathematics courses. Prospective students are encouraged to contact the department to discuss their background.

**Master's Degree in Discipline/Related Discipline:** A master's degree is not required.

**Bachelor's Degree in Discipline/Related Discipline:** All applicants must hold a baccalaureate degree from an accredited college or university.

**Statement of Goals:** Describe your strengths and motivation, the purpose for applying to a graduate program in mathematics, and why you are interested in the intended degree. This forum should be used to make your strongest case for admission and, thus, should be well written.

**Standardized Test Scores:**

GRE General Test and GRE Subject Test in Mathematics: Rigid minimum score criteria is not used. The department considers an applicant's overall record. Students who wish to discuss their scores are encouraged to contact the department directly.

TOEFL score: 79 iBT or 550 PBT minimum

**Resume:** Current resume required.

**Transfer Credit:** Students who have taken graduate courses at other institutions, or at Temple University prior to matriculation, may apply for transfer credit. Applications for transfer credit are not considered until the student has completed at least one term of full-time graduate study or the equivalent, if the student is part-time. All applications for transfer credit are reviewed by the Mathematics Graduate Committee and may be denied if the committee decides that the courses involved are substantially inferior to similar courses offered by the department. No course completed more than five years before the date of application will be awarded credit. Credit for courses substantially similar to courses taken since matriculation will not be awarded. If a course was taken before the bachelor's degree was earned, it cannot be awarded transfer credit. Transfer credit is only available for graduate-level courses in mathematical content. The maximum number of credits a student may transfer is 9.

**Advanced Standing:** A student who has completed a master's degree at another institution may apply for advanced standing. Students are awarded varying numbers of credit of advanced standing. This differs from transfer credit in that the number of credit hours awarded is recorded on the transcript without specific information about the courses transferred. The effect of having advanced standing is to reduce the total number of credit hours the student is required to take at Temple University. Credit for advanced standing can only be awarded to students who have completed a master's degree at another institution. Although there is no requirement that the degree must have been completed recently, the amount of time since the degree has been completed is taken into consideration. The maximum number of advanced standing credits awarded is 30.

**Test Waivers:** An applicant who wishes to have certain admission requirements waived must contact the department directly. Requests are considered by the department on a case-by-case basis. In some cases, an additional appeal to the Graduate School may be required. In such a case, the department makes a preliminary determination for the applicant and, if positive, issues a supporting letter to the Graduate School on the applicant's behalf.

## Program Requirements

**General Program Requirements:**

*Number of Didactic Credits Required:* 48

*Required Courses:*

Students are required to take at least 16 graduate courses, which are chosen with the advice and consent of the student's advisor. These courses should be taken during the first two years of graduate study and include foundational 8000-level courses for the topics in which the student plans to take the Ph.D. Comprehensive Examination. Students who have had graduate courses in these subjects prior to admission may omit some or all of the courses with the consent of their advisor and the Graduate Committee.

Students also take six additional credit hours of:

Code | Title | Credit Hours |
---|---|---|

MATH 9994 | Preliminary Examination Preparation | 1-6 |

MATH 9998 | Pre-Dissertation Research / Elevation to Candidacy | 1-6 |

MATH 9999 | Dissertation Research | 2 minimum |

**Language Examination:** Beginning in Fall 2016, a language requirement is no longer in effect.

**Culminating Events:**

*Written Comprehensive Examination:*

The Ph.D. Comprehensive Examination is a written exam comprised of three separate sections selected from the following areas:

- Algebra
- Applied Mathematics
- Complex Analysis
- Differential Geometry and Topology
- Partial Differential Equations
- Real Analysis

Students can choose any three of these sections for their examination. After exams in three different topics have been attempted, an exam in a fourth topic can only be taken under exceptional circumstances, and only after specific approval by the Graduate Committee. Each section is a three-hour test based primarily on the corresponding two-term 8000-level graduate course sequence. The separate section tests are given on different days, during one two-week period, twice a year: once in August just before the beginning of the Fall term, and once in January during the two weeks prior to the beginning of the Spring term. Students do not have to take all three of the section tests during the same two-week period. Each section test may be repeated once to obtain a higher grade.

Each of the three-hour section tests is further divided into two parts. Part I contains four questions, of which the student is asked to answer three. These questions are designed to test mastery of the facts of the subject. Part II contains three questions, of which two are to be answered. These questions test the ability to solve in-depth problems in the subject.

Students should begin taking the components of the Comprehensive Exam as soon as possible after finishing the corresponding coursework. Students are expected to complete and pass the Comprehensive Examination by August of their second year of study. While some delay in this schedule may be permitted, under exceptional circumstances, students not making good progress toward completing and passing their Comprehensive Examination in a timely fashion will be asked to leave the Ph.D. program.

Incoming students may, with approval of the Graduate Chair, take up to three of the written Ph.D. Comprehensive Examination sections once prior to their first term of enrollment. Upon request by the student, any of these pre-enrollment attempts can be removed from the student's record.

Each section is graded independently by two faculty members, using a scale of 0 to 25. The grades are compared and reconciled in the event of a discrepancy. A total score of at least 60, with a score on each individual section test of at least 13, is required to pass. If a student falls slightly short of this standard, the Graduate Faculty may, at their discretion, recommend a grade of pass based on the whole of the student's academic record.

A student who achieves a total score of at least 40 from the three sections of the Ph.D. Comprehensive Examination, with no individual section below 8, has obtained a master's pass on the examination and has fulfilled the examination requirement for the M.S. degree. If one of the individual exam scores falls below 8 points, that exam may be repeated once, or the exam in a different topic may be attempted once, or the student may take the Master's Comprehensive Examination. As above, such arrangements are subject to approval by the Graduate Committee, and if a student falls slightly short of the required standards, then the Graduate Faculty may, at their discretion, recommend a grade of pass, basing their decision on the student's entire academic record.

*Preliminary Examination:*

The preliminary examination is a two-hour oral exam. It should be taken by the end of the sixth term and must be passed by the end of the seventh term. The Ph.D. preliminary examination can be repeated, in whole or in part, only once. No student is permitted to take the preliminary examination before passing the written Ph.D. Comprehensive Examination and satisfying the foreign language requirement.

The student chooses a chief examiner with the advice and consent of the Mathematics Graduate Committee -- and with the consent of the proposed chief examiner. The chief examiner, in accepting his/her assignment, implicitly offers to be the student's dissertation supervisor if the examination is passed. Approximately one-half of the preliminary examination is conducted by the chief examiner, who asks questions in the area that the student has chosen as a specialty. The other half of the examination is devoted to questions asked by other faculty members on two or more elementary topics related to the area of specialization. The exact description of the elementary topics to be included in the examination is determined by the chief examiner, who will also be responsible for assigning examiners to cover the topics. The examination committee consists of the chief examiner, the examiners for the elementary topics, and any other faculty who choose to attend. All faculty in attendance may vote on the outcome of the examination. The examination is considered passed if the chief examiner and at least one-half of the other faculty present vote in favor of passing.

Students who are preparing to write their preliminary examination should confirm a time and date with the chair of their Doctoral Advisory Committee and register with the Graduate Secretary. The student and chair receive confirmation of the time, date, room, and proctor for the examination.

*Dissertation:*

The candidate's dissertation must be a distinctive and original contribution to research in mathematics. It must be an individual work, with only one author. Previously published work by the candidate may be included, if it represents research done while the student was enrolled in the Ph.D. program in Mathematics at Temple University and was not used to obtain any other degree. Joint work that cannot be attributed to the candidate alone must not be included in the body of the dissertation, but may be attached as an appendix. All previously published work must be logically connected and integrated into the dissertation, with a common introduction, conclusion, and bibliography. Existing copyrights must not be violated.

Preparation of the dissertation is supervised by the student's Doctoral Advisory Committee (DAC). This committee must include at least three Temple graduate faculty, two of whom must be in the Mathematics Department. The chair of the committee must be a member of the Temple University Mathematics Department. The DAC may include members of other Temple University departments; it is also possible for faculty from other universities or expert advisors employed in non-university settings to be included in the DAC.

The first step in preparing the dissertation is to write a dissertation proposal, which must be approved by the candidate's DAC. The proposal is kept on file, and if it becomes necessary to alter the proposal, the changes should be approved by the DAC and filed with the proposal.

The dissertation defense may be attended by faculty and graduate students from Temple University or other institutions, as well as mathematicians or scientists employed in a non-academic setting. The candidate's Dissertation Examining Committee (DEC) must attend the defense. This committee includes the candidate's DAC and at least one additional member, who must be a faculty of some Temple University department other than Mathematics or a faculty member of another university. The DEC meets at the conclusion of the dissertation defense and decides, by majority vote, if the candidate was successful.

If a student needs to change a member of a committee, the new member must be approved by the Mathematics Graduate Committee and registered with the Graduate Secretary and the Graduate School.

When the dissertation is deemed complete by the candidate and the DAC, a defense is scheduled. Students who are preparing to defend their dissertation should confirm a date and time with their DAC and register with the Graduate Secretary at least 20 days before the defense is to be scheduled. The Graduate Chair arranges the date, time, and room. The Graduate Secretary forwards to the student the appropriate forms.

This dissertation defense must be announced in writing at least 10 days in advance of its occurrence. Copies of the announcement must be directed to each member of the candidate's DEC, each faculty member of the Mathematics Department, the Dean of the College of Science and Technology, and the Graduate School. Copies of the announcement are to be posted at the Department Office, the College Office, and on the Graduate School website.

### Courses

**MATH 5000. Special Topics in Math. 3 Credit Hours.**

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may be repeated for additional credit.

**MATH 5001. Linear Algebra. 3 Credit Hours.**

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may not be repeated for additional credits.

**MATH 5041. Concepts of Analysis I. 3 Credit Hours.**

**Department Restrictions:** Must be enrolled in one of the following Departments: CST:Mathematics.

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may not be repeated for additional credits.

**MATH 5042. Concepts of Analysis II. 3 Credit Hours.**

**Department Restrictions:** Must be enrolled in one of the following Departments: CST:Mathematics.

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may not be repeated for additional credits.

**Pre-requisites:**

MATH 5041|Minimum Grade of B-|May not be taken concurrently.

**MATH 5043. Introduction to Numerical Analysis. 3 Credit Hours.**

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may not be repeated for additional credits.

**MATH 5044. Introduction to Numerical Analysis II. 3 Credit Hours.**

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may not be repeated for additional credits.

**MATH 5045. Ordinary Differential Equations. 3 Credit Hours.**

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may not be repeated for additional credits.

**MATH 5061. Fundamentals of Computer Programming for Scientists and Engineers. 4 Credit Hours.**

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may not be repeated for additional credits.

**MATH 5063. Introduction to High-Performance Computing Technology for Scientists. 4 Credit Hours.**

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may not be repeated for additional credits.

**Pre-requisites:**

MATH 5061|Minimum Grade of B-|May be taken concurrently.

**MATH 8001. Candidates Seminar. 1 to 3 Credit Hour.**

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may be repeated for additional credit.

**MATH 8002. Candidates Seminar. 1 to 3 Credit Hour.**

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may be repeated for additional credit.

**MATH 8003. Number Theory. 3 Credit Hours.**

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may not be repeated for additional credits.

**MATH 8004. Number Theory. 3 Credit Hours.**

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may not be repeated for additional credits.

**MATH 8007. Introduction to Methods in Applied Mathematics I. 3 Credit Hours.**

**Department Restrictions:** Must be enrolled in one of the following Departments: CST:Mathematics.

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may not be repeated for additional credits.

**MATH 8008. Introduction to Methods in Applied Mathematics II. 3 Credit Hours.**

**Department Restrictions:** Must be enrolled in one of the following Departments: CST:Mathematics.

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may not be repeated for additional credits.

**MATH 8011. Abstract Algebra I. 3 Credit Hours.**

**Department Restrictions:** Must be enrolled in one of the following Departments: CST:Mathematics.

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may not be repeated for additional credits.

**MATH 8012. Abstract Algebra II. 3 Credit Hours.**

**Department Restrictions:** Must be enrolled in one of the following Departments: CST:Mathematics.

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may not be repeated for additional credits.

**Pre-requisites:**

MATH 8011|Minimum Grade of B-|May not be taken concurrently.

**MATH 8013. Numerical Linear Algebra I. 3 Credit Hours.**

**Department Restrictions:** Must be enrolled in one of the following Departments: CST:Mathematics.

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may not be repeated for additional credits.

**MATH 8014. Numerical Linear Algebra II. 3 Credit Hours.**

**Department Restrictions:** Must be enrolled in one of the following Departments: CST:Mathematics.

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may not be repeated for additional credits.

**Pre-requisites:**

MATH 8013|Minimum Grade of B-|May not be taken concurrently.

**MATH 8023. Numerical Differential Equations I. 3 Credit Hours.**

**Department Restrictions:** Must be enrolled in one of the following Departments: CST:Mathematics.

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may not be repeated for additional credits.

**MATH 8024. Numerical Differential Equations II. 3 Credit Hours.**

**Department Restrictions:** Must be enrolled in one of the following Departments: CST:Mathematics.

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may not be repeated for additional credits.

**Pre-requisites:**

MATH 8023|Minimum Grade of B-|May not be taken concurrently.

**MATH 8031. Probability Theory. 3 Credit Hours.**

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may not be repeated for additional credits.

**MATH 8032. Stochastic Processes. 3 Credit Hours.**

**Department Restrictions:** Must be enrolled in one of the following Departments: CST:Mathematics.

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may not be repeated for additional credits.

**Pre-requisites:**

MATH 8031|Minimum Grade of B-|May not be taken concurrently.

**MATH 8041. Real Analysis I. 3 Credit Hours.**

**Department Restrictions:** Must be enrolled in one of the following Departments: CST:Mathematics.

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may not be repeated for additional credits.

**MATH 8042. Real Analysis II. 3 Credit Hours.**

**Department Restrictions:** Must be enrolled in one of the following Departments: CST:Mathematics.

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may not be repeated for additional credits.

**Pre-requisites:**

MATH 8041|Minimum Grade of B-|May not be taken concurrently.

**MATH 8051. Functions of a Complex Variable I. 3 Credit Hours.**

**Department Restrictions:** Must be enrolled in one of the following Departments: CST:Mathematics.

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may not be repeated for additional credits.

**MATH 8052. Functions of a Complex Variable II. 3 Credit Hours.**

**Department Restrictions:** Must be enrolled in one of the following Departments: CST:Mathematics.

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may not be repeated for additional credits.

**Pre-requisites:**

MATH 8051|Minimum Grade of B-|May not be taken concurrently.

**MATH 8061. Differential Geometry and Topology I. 3 Credit Hours.**

**Department Restrictions:** Must be enrolled in one of the following Departments: CST:Mathematics.

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may not be repeated for additional credits.

**MATH 8062. Differential Geometry and Topology II. 3 Credit Hours.**

**Department Restrictions:** Must be enrolled in one of the following Departments: CST:Mathematics.

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may not be repeated for additional credits.

**Pre-requisites:**

MATH 8061|Minimum Grade of B-|May not be taken concurrently.

**MATH 8107. Mathematical Modeling for Science, Engineering, and Industry. 3 Credit Hours.**

**Department Restrictions:** Must be enrolled in one of the following Departments: CST:Mathematics, Engineering: Engineering.

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may not be repeated for additional credits.

**Pre-requisites:**

(MATH 8007|Minimum Grade of B-|May not be taken concurrently)

AND (MATH 8008|Minimum Grade of B-|May not be taken concurrently)

**MATH 8141. Partial Differential Equations I. 3 Credit Hours.**

**Department Restrictions:** Must be enrolled in one of the following Departments: CST:Mathematics.

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may not be repeated for additional credits.

**MATH 8142. Partial Differential Equations II. 3 Credit Hours.**

**Department Restrictions:** Must be enrolled in one of the following Departments: CST:Mathematics.

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may not be repeated for additional credits.

**Pre-requisites:**

MATH 8141|Minimum Grade of B-|May not be taken concurrently.

**MATH 8161. Topology. 3 Credit Hours.**

**Department Restrictions:** Must be enrolled in one of the following Departments: CST:Mathematics.

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may not be repeated for additional credits.

**Pre-requisites:**

MATH 5041|Minimum Grade of B-|May not be taken concurrently.

**MATH 8200. Topics in Applied Mathematics. 3 Credit Hours.**

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may be repeated for additional credit.

**MATH 8210. Topics in Applied Mathematics II. 3 Credit Hours.**

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may be repeated for additional credit.

**MATH 8700. Topics Computer Program. 3 Credit Hours.**

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may be repeated for additional credit.

**MATH 8710. Topics Computer Program. 3 Credit Hours.**

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may be repeated for additional credit.

**MATH 8985. Teaching in Higher Education. 1 to 3 Credit Hour.**

**Department Restrictions:** Must be enrolled in one of the following Departments: CST:Mathematics.

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may be repeated for a total of 3 credit.

**MATH 9000. Topics in Number Theory I. 3 Credit Hours.**

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may be repeated for additional credit.

**MATH 9003. Modular Functions. 3 Credit Hours.**

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may not be repeated for additional credits.

**MATH 9004. Modular Functions. 3 Credit Hours.**

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may not be repeated for additional credits.

**MATH 9005. Combinatorial Mathematics. 3 Credit Hours.**

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may not be repeated for additional credits.

**MATH 9010. Topics in Number Theory II. 3 Credit Hours.**

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may be repeated for additional credit.

**MATH 9011. Homological Algebra. 3 Credit Hours.**

**Department Restrictions:** Must be enrolled in one of the following Departments: CST:Mathematics.

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may not be repeated for additional credits.

**Pre-requisites:**

(MATH 8011|Minimum Grade of B-|May not be taken concurrently)

AND (MATH 8012|Minimum Grade of B-|May not be taken concurrently)

**MATH 9012. Representation Theory I. 3 Credit Hours.**

**Department Restrictions:** Must be enrolled in one of the following Departments: CST:Mathematics.

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may not be repeated for additional credits.

**Pre-requisites:**

(MATH 8011|Minimum Grade of B-|May not be taken concurrently)

AND (MATH 8012|Minimum Grade of B-|May not be taken concurrently)

**MATH 9013. Representation Theory II. 3 Credit Hours.**

**Department Restrictions:** Must be enrolled in one of the following Departments: CST:Mathematics.

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may not be repeated for additional credits.

**Pre-requisites:**

MATH 9012|Minimum Grade of B-|May not be taken concurrently.

**MATH 9014. Commutative Algebra and Algebraic Geometry I. 3 Credit Hours.**

**Department Restrictions:** Must be enrolled in one of the following Departments: CST:Mathematics.

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may not be repeated for additional credits.

**Pre-requisites:**

(MATH 8011|Minimum Grade of B-|May not be taken concurrently)

AND (MATH 8012|Minimum Grade of B-|May not be taken concurrently)

**MATH 9015. Commutative Algebra and Algebraic Geometry II. 3 Credit Hours.**

**Department Restrictions:** Must be enrolled in one of the following Departments: CST:Mathematics.

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may not be repeated for additional credits.

**Pre-requisites:**

MATH 9014|Minimum Grade of B-|May not be taken concurrently.

**MATH 9021. Riemannian Geometry. 3 Credit Hours.**

**Department Restrictions:** Must be enrolled in one of the following Departments: CST:Mathematics.

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may not be repeated for additional credits.

**Pre-requisites:**

(MATH 8061|Minimum Grade of B-|May not be taken concurrently)

AND (MATH 8062|Minimum Grade of B-|May be taken concurrently)

**MATH 9023. Knot Theory and Low-Dimensional Topology I. 3 Credit Hours.**

**Department Restrictions:** Must be enrolled in one of the following Departments: CST:Mathematics.

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may not be repeated for additional credits.

**Pre-requisites:**

(MATH 8061|Minimum Grade of B-|May not be taken concurrently)

AND (MATH 8062|Minimum Grade of B-|May not be taken concurrently)

**MATH 9024. Knot Theory and Low-Dimensional Topology II. 3 Credit Hours.**

**Department Restrictions:** Must be enrolled in one of the following Departments: CST:Mathematics.

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may not be repeated for additional credits.

**Pre-requisites:**

MATH 9023|Minimum Grade of B-|May not be taken concurrently.

**MATH 9031. Advanced Probability Theory. 3 Credit Hours.**

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may not be repeated for additional credits.

**MATH 9041. Functional Analysis I. 3 Credit Hours.**

**Department Restrictions:** Must be enrolled in one of the following Departments: CST:Mathematics.

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may not be repeated for additional credits.

**Pre-requisites:**

(MATH 8041|Minimum Grade of B-|May not be taken concurrently)

AND (MATH 8042|Minimum Grade of B-|May not be taken concurrently)

AND (MATH 8161|Minimum Grade of B-|May not be taken concurrently)

**MATH 9042. Functional Analysis II. 3 Credit Hours.**

**Department Restrictions:** Must be enrolled in one of the following Departments: CST:Mathematics.

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may not be repeated for additional credits.

**Pre-requisites:**

MATH 9041|Minimum Grade of B-|May not be taken concurrently.

**MATH 9043. Calculus of Variations. 3 Credit Hours.**

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may not be repeated for additional credits.

**MATH 9044. Harmonic Analysis. 3 Credit Hours.**

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may not be repeated for additional credits.

**MATH 9051. Several Complex Variables I. 3 Credit Hours.**

**Department Restrictions:** Must be enrolled in one of the following Departments: CST:Mathematics.

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may not be repeated for additional credits.

**Pre-requisites:**

(MATH 8051|Minimum Grade of B-|May not be taken concurrently)

AND (MATH 8052|Minimum Grade of B-|May not be taken concurrently)

**MATH 9052. Several Complex Variables II. 3 Credit Hours.**

**Department Restrictions:** Must be enrolled in one of the following Departments: CST:Mathematics.

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may not be repeated for additional credits.

**Pre-requisites:**

MATH 9051|Minimum Grade of B-|May not be taken concurrently.

**MATH 9053. Harmonic Analysis. 3 Credit Hours.**

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may not be repeated for additional credits.

**MATH 9061. Lie Groups. 3 Credit Hours.**

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may not be repeated for additional credits.

**MATH 9062. Lie Groups. 3 Credit Hours.**

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may not be repeated for additional credits.

**MATH 9063. Riemann Surfaces. 3 Credit Hours.**

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may not be repeated for additional credits.

**MATH 9064. Riemann Surfaces. 3 Credit Hours.**

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may not be repeated for additional credits.

**MATH 9071. Differential Topology. 3 Credit Hours.**

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may not be repeated for additional credits.

**MATH 9072. Differential Topology. 3 Credit Hours.**

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may not be repeated for additional credits.

**MATH 9073. Geometric Group Theory. 3 Credit Hours.**

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may not be repeated for additional credits.

**Pre-requisites:**

(MATH 8061|Minimum Grade of B-|May not be taken concurrently)

AND (MATH 8062|Minimum Grade of B-|May not be taken concurrently)

**MATH 9082. Independent Study. 1 to 3 Credit Hour.**

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may be repeated for additional credit.

**MATH 9083. Independent Study. 1 to 3 Credit Hour.**

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may be repeated for additional credit.

**MATH 9100. Topics in Algebra. 3 Credit Hours.**

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may be repeated for additional credit.

**MATH 9110. Topics in Algebra. 3 Credit Hours.**

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may be repeated for additional credit.

**MATH 9120. Seminar in Algebra. 3 Credit Hours.**

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may be repeated for additional credit.

**MATH 9200. Topics in Numerical Analysis. 3 Credit Hours.**

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may be repeated for additional credit.

**MATH 9210. Topics in Numerical Analysis. 3 Credit Hours.**

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may be repeated for additional credit.

**MATH 9300. Seminar in Probability. 3 Credit Hours.**

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may be repeated for additional credit.

**MATH 9310. Seminar in Probability. 3 Credit Hours.**

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may be repeated for additional credit.

**MATH 9400. Topics in Analysis. 3 Credit Hours.**

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may be repeated for additional credit.

**MATH 9410. Topics in Functional Analysis. 3 Credit Hours.**

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may be repeated for additional credit.

**MATH 9420. Topics in Differential Equations II. 3 Credit Hours.**

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may be repeated for additional credit.

**MATH 9991. Master's Research Projects. 1 to 6 Credit Hour.**

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Degree Restrictions:** Must be enrolled in one of the following Degrees: Master of Arts, Master of Science, Prof Science Masters.

**Repeatability:** This course may be repeated for additional credit.

**MATH 9994. Preliminary Examination Preparation. 1 to 6 Credit Hour.**

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may be repeated for additional credit.

**MATH 9995. Capstone Project. 1 to 6 Credit Hour.**

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Degree Restrictions:** Must be enrolled in one of the following Degrees: Master of Arts, Master of Science, Prof Science Masters.

**Repeatability:** This course may be repeated for additional credit.

**MATH 9996. Master's Thesis Research. 1 to 6 Credit Hour.**

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may be repeated for additional credit.

**MATH 9998. Pre-Dissertation Research / Elevation to Candidacy. 1 to 6 Credit Hour.**

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Repeatability:** This course may be repeated for additional credit.

**MATH 9999. Dissertation Research. 1 to 6 Credit Hour.**

**Level Registration Restrictions:** Must be enrolled in one of the following Levels: Graduate.

**Student Attribute Restrictions:** Must be enrolled in one of the following Student Attributes: Dissertation Writing Student.

**Repeatability:** This course may be repeated for additional credit.

## Contacts

### Program Web Address:

### Department Information:

Dept. of Mathematics

638 Wachman Hall

1805 N. Broad Street

Philadelphia, PA 19122-6094

215-204-3928