Mathematics

Graduate courses in all the principal branches of mathematics are offered regularly each year. The department is prepared to direct research in a variety of fields, including various branches of analysis, algebra, topology, number theory, applied analysis, and mathematical logic and foundations.

Admission Requirements

Applicants apply for admission to the program via the J. Jeffrey and Ann Marie Fox Graduate School application for admission. Requirements listed here are in addition to Graduate Council policies listed under GCAC-300 Admissions Policies.

Scores from the Graduate Record Examinations Aptitude Test (GRE), or from a comparable substitute examination accepted by the Mathematics graduate program, are not required for admission. Those scores can still be included if the applicant has them and they will then be considered by the admission committee.

To be admitted to the Ph.D. program without undergraduate deficiency, an applicant should have completed at least 18 credits in mathematics at the advanced undergraduate level (400 series or their equivalents). The undergraduate student is urged to take at least 6 credits in foundations of analysis (MATH 401), 6 in modern algebra (MATH 435 and MATH 436), and 3 in topology (MATH 429) or their equivalents. These courses are essential preparation for the graduate program, and if they are taken after admission, a maximum of 6 credits may be counted toward an advanced degree. Students are typically admitted directly to the Ph.D. program, though may on occasion be admitted first to the M.A. degree when not meeting all of the above requirements.

Students with a 3.00 junior/senior average and with appropriate course backgrounds will be considered for admission. The best-qualified applicants will be accepted up to the number of spaces that are available for new students.

The language of instruction at Penn State is English. English proficiency test scores (TOEFL/IELTS) may be required for international applicants. See GCAC-305 Admission Requirements for International Students for more information.

Degree Requirements

Master of Arts (M.A.)

Requirements listed here are in addition to Graduate Council policies listed under GCAC-600 Research Degree Policies.

For the M.A. degree the department offers two tracks:

1. the thesis track requires 12 credits of approved 500-series or 600-series courses in mathematics, 6 to 9 credits of thesis, sufficient credits in approved 400- or 500-series courses to make a total of 30 credits, and a final oral examination based on the thesis and general course material; and
2. the nonthesis track requires 18 credits of 500-series courses in mathematics, sufficient credits in approved 400- or 500-series courses to make a total of 30 credits, and a term paper on an approved topic in mathematics. No final examination is given in this tracks.
   
   For both tracks, a grade of A or B is required in all courses.

Doctor of Philosophy (Ph.D.)

Requirements listed here are in addition to Graduate Council policies listed under GCAC-600 Research Degree Policies.

All doctoral students are required to pass a single multi-part examination that is composed of three written qualifying examinations and the completion of one required course with a grade of A- or better. All qualifying exams and the course must be completed by the end of students' third semester (excluding summer sessions) of entry into the doctoral program.

-The real analysis qualifying exam is mandatory for all students.

-Students will choose one of the following qualifying exams:  linear algebra or abstract algebra.

-Students will choose one of the following qualifying exams:  functional analysis or topology

- Students will choose and pass with a grade of A- or better one of the following courses: Complex Analysis (MATH 502), Differential Manifolds (MATH 528), Mathematical Logic (MATH 557), Algebraic Number Theory I (MATH 567), Numerical Analysis I (MATH 523), PDE 2 (MATH 514). This list may be modified after approval by the graduate studies committee of the program. 

• Optional route: students may choose to take both the functional analysis and topology qualifying exams, and not be required to complete the course requirement above.

Qualifying examinations are offered twice a year: in May, after the end of the Spring semester and in December/January, after the end of the Fall semester. The corresponding semester-long graduate-level courses that prepare for these qualifying exams are offered every year: Linear algebra (MATH 535), Abstract algebra (MATH 536), Real analysis (MATH 501) in the Fall semesters, and Functional analysis (MATH 503) and Topology (MATH 527) in the Spring semesters.

The other courses are offered at least once every two years. They are required to have a final exam, and their final grade must be based at least for 60% on written tests (quizzes, midterm, final exam).

After passing all three qualifying exams and the course, students are expected to select a dissertation adviser and form a Ph.D. committee. The committee administers the comprehensive exam (no later than the end of the sixth semester of study) and offers counsel to the student as their research progresses.

In addition to the qualifying examination, Ph.D. degree requirements include successful completion of the following: English competence requirements, a comprehensive examination, and a final oral examination (the dissertation defense). To earn the Ph.D. degree, doctoral candidates must write a dissertation that is accepted by the Ph.D. committee, the head of the graduate program, and the Fox Graduate School.

Dual-Titles

Dual-Title M.A. and Ph.D. in Mathematics and Operations Research

Requirements listed here are in addition to requirements listed in GCAC-208 Dual-Title Graduate Degree Programs.

Admission Requirements

Students must apply and be admitted to the graduate program in Mathematics and The Graduate School before they can apply for admission to the dual-title degree program. After admission to their primary program, students must apply for admission to and meet the admissions requirements of the Operations Research dual-title program. Refer to the Admission Requirements section of the Operations Research Bulletin page. Doctoral students must be admitted into the dual-title degree program in Operations Research prior to taking the qualifying examination in their primary graduate program.

Degree Requirements

To qualify for the dual-title degree, students must satisfy the degree requirements for the degree they are enrolled in Mathematics. In addition, students must complete the degree requirements for the dual-title in Operations Research, listed on the Operations Research Bulletin page.

The qualifying examination committee for the dual-title Ph.D. degree will be composed of Graduate Faculty from Mathematics and must include at least one Graduate Faculty member from the Operations Research program. Faculty members who hold appointments in both programs’ Graduate Faculty may serve in a combined role. There will be a single qualifying examination, containing elements of both Mathematics and Operations Research. Dual-title graduate degree students may require an additional semester to fulfill requirements for both areas of study and, therefore, the qualifying examination may be delayed one semester beyond the normal period allowable.

In addition to the general Graduate Council requirements for Ph.D. committees, the Ph.D. committee of a Mathematics and Operations Research dual-title Ph.D. student must include at least one member of the Operations Research Graduate Faculty. Faculty members who hold appointments in both programs’ Graduate Faculty may serve in a combined role. If the chair of the Ph.D. committee is not also a member of the Graduate Faculty in Operations Research, the member of the committee representing Operations Research must be appointed as co-chair. The Operations Research representative on the student’s Ph.D. committee will develop questions for and participate in the evaluation of the comprehensive examination.

Students in the dual-title program are required to write and orally defend a dissertation on a topic that is approved in advance by their Ph.D. committee and reflects their original research and education in Mathematics and Operations Research. Upon completion of the doctoral dissertation, the candidate must pass a final oral examination (the dissertation defense) to earn the Ph.D. degree. The dissertation must be accepted by the Ph.D. committee, the head of the graduate program, and the Graduate School.

Minor

A graduate minor is available in any approved graduate major or dual-title program. The default requirements for a graduate minor are stated in Graduate Council policy GCAC-218 Minors.

Student Aid

Graduate assistantships available to students in this program and other forms of student aid are described in the Tuition & Funding section of the J. Jeffrey and Ann Marie Fox Graduate School’s website. Students on graduate assistantships must adhere to the course load limits set by the Fox Graduate School.

Courses

Graduate courses carry numbers from 500 to 699 and 800 to 899. Advanced undergraduate courses numbered between 400 and 499 may be used to meet some graduate degree requirements when taken by graduate students. Courses below the 400 level may not. A graduate student may register for or audit these courses in order to make up deficiencies or to fill in gaps in previous education but not to meet requirements for an advanced degree.

Mathematics (MATH) Course List

Learning Outcomes

Master of Arts (M.A.)

1. Understand advanced mathematical topics and methods and implement some of them.
2. Summarize and explain state-of-the-art research in Mathematics.
3. Communicate mathematical results and ideas in a pedagogical manner, both through written publications and oral presentations and in the context of teaching in a classroom.
4. Carry out fully rigorous mathematical arguments.
5. Conduct themselves in accordance with the highest ethical standards and values.

Doctor of Philosophy (Ph.D.)

1. Understand and implement advanced mathematical topics and methods.
2. Conduct research and advance mathematical knowledge at the highest level of excellence.
3. Communicate mathematical results and ideas in a pedagogical manner, both through written publications and oral presentations and in the context of teaching in a classroom.
4. Design and carry out fully rigorous mathematical arguments.
5. Conduct themselves in accordance with the highest ethical standards and values.

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