## PROGRAM DESCRIPTION

The Department of Mathematics of the University of Pennsylvania offers a full Graduate Program in Mathematics, conferring the degrees of Master of Arts (A.M.), Master of Philosophy (M.Phil.), and Doctor of Philosophy (Ph.D.) . The educational aim of this program is to provide well-rounded mathematical training for a career of research and teaching in mathematics, as well as for other careers which use advanced mathematics in a substantial way.

The purpose of this document is to describe the graduate program in Mathematics at Penn. This program covers a variety of areas, including analysis, geometry-topology, algebra, mathematical physics, combinatorics, logic, and probability. See our Graduate Mathematics Page and our Math Department home page for information about other aspects of this program and of the Penn Math Department as a whole, including its colloquia, seminars and lecture series, and the research interests of the faculty. In addition, Penn offers an interdisciplinary graduate program in Applied Mathematics and Computational Science, for which there is separate application process.

Supplementary information about the program is available here, concerning graduate fellowships, the financial support package for Ph.D. students, and the Ph.D. thesis.

### ADMISSIONS AND APPLICATIONS

The Penn Mathematics Graduate Program welcomes applications from interested and dedicated students who have done substantial mathematical work of superior quality as undergraduates. Note that it is not necessary to have a masters degree to enter the Ph.D. program. Applications may be submitted by students in the last year of their bachelors degree program, with the understanding that they will receive the bachelors degree prior to starting the Ph.D. program.

Applicants for admission are expected to have a mathematical background that is comparable to a strong undergraduate mathematics major. In particular, applicants should be familiar with the material on our Graduate Graduate Preliminary Exam, which our graduate students take upon entering our program, and which in particular serves as a placement exam. Applicants' transcripts should include courses in advanced calculus and in abstract and linear algebra that involved proofs, and preferably advanced mathematical electives as well. Mathematics course grades should mostly be A or A- (or the equivalent).

**Applications** should be submitted electronically. This can be done beginning in October. Applications must be submitted by December 31, when the online admission system closes. Letters of recommendation can be uploaded by the recommenders, once the application is submitted with the recommenders' names indicated. Those letters can be uploaded after December 31, but it is best that this is done soon afterwards so that they will be seen by the admissions committee during its review process. Supporting application materials that cannot be submitted through this website can be sent by email to the Mathematics Graduate Administrator, Reshma Tanna.

**More information** about the application procedure is available here. Those applying should be careful to indicate whether they are interested in being considered for the Graduate Mathematics Program (discussed below) or in Penn's AMCS program in Applied Mathematics and Computational Science.

Those who are admitted will be notified by email and letter. Those who are rejected will have their status indicated on the online application website. While some decisions will be made in February or early March, others will not be made until close to the uniform reply date of April 15, since our admissions committee must wait for replies from those admitted in the first round before being able to make offers to those on our waitlist. All applicants must take the Graduate Record Examination (GRE): both the General test and the Advanced Mathematics Subject Test. (Institution code is 2926 and Department code is 0703.)

Applicants whose native language is not English must take the TOEFL (Test of English as a Foreign Language)(Institution code is 2926 and Department code is 72). This TOEFL requirement is waived for applicants who were undergraduates in the US, or Canada (or in a country whose primary language is English), for their entire four year bachelors program. If you are uncertain whether you are required to take the TOEFL exam, you should contact us to ask, since the lack of a required TOEFL score would disqualify your application.

The GRE and English language exams should be taken early enough for the scores to be received at Penn in time to be considered by the Admissions Committee. Note that it can take a month or two after taking the exam before the score is reported, and that advanced registration is required for these tests. Also, the GRE subject test is given only a few times each year, so advanced planning is needed.

Applicants are considered on the basis of their courses, grades, letters of recommendation, test scores (basic and advanced GRE and TOEFL), and other supporting material. Letters of recommendation should preferably be from mathematics professors who know the applicant well, and who can comment on the applicant's abilities, accomplishments, and potential.

Applicants for the graduate program in mathematics are ordinarily admitted for the Fall semester each year, with the deadline for applications being December 31 of the previous calendar year. It is important that the application be submitted by December 31, and that all related materials, including letters of recommendation and test scores, arrive by the end of December (or at latest by mid-January). It is therefore important to take the Advanced Mathematics GRE no later than October. Applications for which the related materials are late may be considered on the basis of available space and financial aid.

On the first page of the application, applicants should indicate the final degree that they hope to be awarded from the Penn Mathematics Graduate Program (either A.M., M.Phil., or Ph.D.). Students who intend to obtain an A.M. on the way to an M.Phil. or Ph.D. at Penn should indicate the latter degree.

The Mathematics Department also offers a submatriculation program for Penn undergraduates, which gives students the opportunity to work toward a Bachelors and Masters degree at the same time. Penn students interested in submatriculation should contact the department's Undergraduate Chair for additional information and guidance.

### FELLOWSHIPS AND FINANCIAL AID

Full-time students admitted to our Ph.D. program are offered a financial support package combining fellowships and teaching assistantships, including a fellowship (with no teaching) in the first year and two additional semesters (often in the fourth year). Ph.D. students who are working on funded research projects can receive additional fellowships during which they would not teach. There is also funding provided without teaching responsibilities during three summers. This package, which covers tuition and fees and health insurance and also provides a stipend (See Tax structure), continues for five years for Ph.D. students in good standing who are making timely progress.

The academic year stipend will be $32,255 during 2021-22, and the summer 2021 stipend will be $5,085. These amounts are expected to rise in future years, as they have in the past.

Teaching assistants most often run recitations of calculus, meeting with up to four small groups of undergraduates once a week to discuss homework. Other possible TA assignments can include running problem sessions in more advanced courses such as advanced calculus or algebra, or running review sessions or grading for courses in which there are no problem sessions. Counting contact hours, office hours, grading, record keeping, etc., the TA responsibilities take up to twelve hours per week (and replace one of the graduate student's own courses). Before beginning a teaching assistantship, graduate students go through a two-day TA training program run by the Math Department. There is also mentoring for TA's during the semester.

Graduate students often teach courses over the summer, for an additional stipend. (Doing this is optional, but is good experience.) Unlike TA responsibilities during the academic year, grad students who teach during the summer have full responsibility for the course, giving the lectures and assigning homework and grades.

Students in the masters or M.Phil. program may also be considered for teaching assistantships, which provide a stipend. These are awarded based on merit and availablity, and are decided on a semester-by-semester basis.

### FIRST YEAR CLASS SCHEDULE

All entering graduate students take the Graduate Preliminary Examination before the Fall semester's classes begin. This is a five hour exam which covers essential undergraduate material in mathematics, including basic algebra, linear algebra and advanced calculus. It serves in part as a placement exam, to help determine each student's schedule of classes during the first year.

After taking this exam, students meet individually with the Graduate Group Chair to design their first year program of courses. They also meet with their faculty advisors. (Incoming students are assigned advisors by the Graduate Group Chair, until such time as they choose advisors for themselves.) Most often, first-year students are advised to take the three main introductory Ph.D.-level courses in Geometry-Topology (Math 600, 601), Algebra (Math 602, 603), and Analysis (Math 608, 609), plus a fourth courses of their choosing. But those who demonstrate to the instructors that they already have a mastery of the material in these introductory courses can instead take more advanced courses. On the other hand, those whose performance on the Masters Prelim showed significant gaps in training may be advised to take one or more introductory masters-level courses during the first year in Geometry-Topology (Math 500), Algebra (Math 502, 503), or Analysis (Math 508, 509).

One of the first year graduate courses, the "Proseminar" (Math 504, 505), attempts to bridge the gap between undergraduate and graduate mathematics by treating the topics covered in the Graduate Preliminary Exam more intensively than is typical in undergraduate courses, with an emphasis on problem solving and presenting proofs. Graduate students take this course during the first year of study here if they do not pass the Graduate Preliminary Exam upon arrival.

The Graduate Preliminary Exam is given for a second time at the end of the spring semester. Passing it by the end of the first year in the graduate program is a requirement for remaining in the graduate program. (The passing level is ordinarily higher for the Ph.D. program than for the masters.) The University's Qualifications Evaluation requirement is met by passing this exam and doing well in the first-year courses.

Syllabus of first year courses

Year-by-year sequence for graduate students

### THE MASTERS DEGREE

The Masters of Arts program in mathematics is intended for students who wish to go into professional careers in mathematics, and who have a background corresponding to an undergraduate mathematics major. The masters degree is typically earned in approximately two years, though it is possible to earn it in less time.

Students in the masters program take at least eight graduate mathematical courses numbered 500 and above, including at least one each in algebra, analysis, and geometry-topology. The other courses can be in either pure or applied mathematics, and some can be taken in other departments. There is also a requirement to pass the Graduate Preliminary Exam and to submit and defend a masters thesis.

The masters degree can also be earned by Penn Math Ph.D. students, on the way to their Ph.D. In addition, Penn Ph.D. students in allied fields of study can also apply to enter the masters program in mathematics, which can allow them to earn both degrees simultaneously.

Detailed requirements:

- Admission to candidacy: Admission to Masters candidacy is achieved by passing the Graduate Preliminary Exam by the end of the first year in the graduate program, and by satisfactory performance in the first year's courses.
- Course requirements: A minimum of eight units of graduate courses at the University of Pennsylvania, numbered 500 and above, is required for the Masters degree, including at least four taken at the Penn Mathematics Department. Among these courses, every student must take at least a semester of graduate courses in each of algebra, analysis and geometry-topology.
- Language requirement: There is no foreign language requirement for the Masters degree.
- Masters thesis: Each student must write, under the supervision of a Mathematics Department faculty member, a satisfactory Masters thesis, of modest length, which is typically expository in nature, but may also be a research paper. The preparation of this thesis should involve the mastery of some area of mathematics beyond the curriculum of the courses that the student has taken, and should ordinarily take less than one semester to complete.
- The Masters General Examination: The student will take a General Examination for the Masters degree. This exam consists of a presentation by the student of the content of the thesis, followed by questions from the faculty on the topic presented and related areas of mathermatics. The student must deliver a copy of the masters thesis to the graduate coordinator at least one week before this General Examination, so that it will be available for the reading by the mathematics faculty.
- Some important administrative details:

- Administration of the Graduate Program: The Graduate Group Chair directs and oversees the administration of the graduate program, advises the graduate students in this program, and assesses their progress on a continuing basis. The Graduate Group Chair is aided in this by the Graduate Preliminary Exam Committee, the Graduate Advising Committee, and the First Year Course Committee.
- Timing: The program leading to the Masters degree should ordinarily be completed within two years of full time study, and in any case must be completed within three years of full time study. Part time study, when allowed, will have a corresponding timetable set in each case by the Graduate Group Chair.
- Course schedules: Each semester, the student's course schedule must be approved in advance by the Graduate Group Chair and the student's advisor. Students are reminded that, while the course requirements for the masters degree include at least a semester of graduate courses in each of algebra, analysis and geometry-topology, the course requirements for the Ph.D. include a full year in each of these subjects at the 600 level, and may want to take this into account when planning their course schedules.
- Course requirement: Approval of the Graduate Group Chair is required for courses taken outside the Penn Mathematics Department to be counted toward the course requirement for the Masters Degree in Mathematics.
- The Masters General Examination Committee is appointed by the Graduate Group Chair after consultation with the student. It will consist of two or more faculty members, at least one of whom must be tenured, and one of whom will ordinarily be the supervising faculty member. The Graduate Group Chair will notify the entire mathematics faculty of the exam; all faculty are explicitly invited to attend, and those who do are full voting members of the Exam Committee.
- If the Masters General Examination is not passed on the first try, it may be taken just once more, and this must occur before the end of the following semester. Passing the Masters General Examination on the second try, at the latest, is a requirement for remaining in the graduate program.
- Satisfactory performance in the graduate program is determined by the Graduate Group Chair, with the aid of the Graduate Advising Committee and (for first year courses) the First Year Course Committee. Satisfactory performance, which includes maintaining a B average every academic year and meeting requirements in a timely manner, is a requirement for remaining in the graduate program.

### THE M.Phil. DEGREE

The Master of Philosophy in mathematics is a terminal professional degree that is designed for students who want to go beyond the usual masters degree. Entering students should have previously completed either an undergraduate or a regular masters program in mathematics. The degree is typically earned one or two years after beginning to take courses that go beyond the regular master of arts degree.

Students who already have a regular masters degree take six additional graduate courses numbered 600 or above to earn an M.Phil. degree. Students who enter the program with just an undergraduate degree will fulfill the usual masters course requirements plus an additional six (for a total of 14 courses). Students in this program also need to pass the Graduate Preliminary Exam and an oral exam in one area of mathematics, and to submit and defend an M.Phil. thesis (which is somewhat longer and more in depth than a thesis in the usual masters program).

Detailed requirements:

- Admission to candidacy: A Bachelors degree (but not a masters degree) is required for entrance to the program. Admission to M.Phil. candidacy is achieved by passing the Preliminary Exam by the end of the first year in the graduate program, and the oral M.Phil. Preliminary Exam (described below). If the student so requests before the exam, the Masters General Examination can serve as the M.Phil. Preliminary Exam. In this case, the exam will test the depth and breadth of knowledge expected of entering M.Phil. candidates.
- M.Phil. Preliminary Exam: This is an oral exam in the subfield of mathematics which is the student's intended area of specialization. The student should demonstrate both depth and breadth of knowledge, as well as real mastery of basic material and the ability to apply it to specific examples. Typically, the content of this exam will be somewhat more than that of a one-semester graduate course at or above the 610 level.
- Course requirements: A minimum of six units of graduate courses (beyond the course requirements for the Masters) at the University of Pennsylvania, numbered 600 and above, is required for the M.Phil. Degree, including at least four taken in the Mathematics Department. These courses cannot include any that are counted towards a Masters degree. Students who do not have a Masters degree must in addition satisfy the course requirement of a Masters degree (while those who already have a Masters degree need take just these six). Among these six courses, every student must take at least a semester of graduate courses in two of algebra, analysis and geometry-topology.
- Language requirement: There is no foreign language requirement for the M.Phil. Degree.
- Seminar requirement: The student must satisfactorily present a lecture in one of the department's graduate or research seminars, and answer questions about the material. The intention of this requirement is to help the student gain experience in digesting and presenting advanced material and fielding questions about it before an audience of mathematicians. Guidance in the preparation of these lectures is provided by faculty members in the department.
- M.Phil. Thesis: Each student must write, under the supervision of a Mathematics Department faculty member, a satisfactory M.Phil. thesis, which is typically expository in nature, but may also be a research paper. The preparation of this thesis should involve the mastery of some area of mathematics beyond the curriculum of the courses that the student has taken. (To obtain an M.Phil. degree, it is not necessary to have previously written a masters thesis. But a student in the M.Phil. program who previously submitted a masters thesis cannot resubmit that thesis toward the M.Phil.)
- The M.Phil. Comprehensive Examination: Following submission of the M.Phil. thesis, the student will take a Comprehensive Examination for the M.Phil. degree. This exam will consist of a presentation by the student of the content of the thesis, followed by questions from the faculty on the topic presented and related areas of mathematics.
- Some important administrative details:

- The administrative details listed earlier in connection with the Masters Degree program apply to the M.Phil. program as well. In addition, we have the following:
- When the student asks that the Masters Gen'l Exam serve as the M.Phil Prelim exam, there are two possible positive outcomes: Pass at the M.Phil level, which requires demonstration of the required depth and breadth in the subject matter; Pass at the M.A. level (described in the requirements for that degree). If the student does not pass at either of these levels, he may retake the exam one more time; if on the initial exam, the student passes at a level below the desired outcome then he may also retake the exam one more time. In either case, the single retake of the exam must occur before the end of the next semester.
- Timing: The program leading to the M.Phil. must be completed within two years of full time study (after completion of the masters course-work). Part time study, when allowed, will have a corresponding timetable set in each case by the Graduate Group Chair.
- Course requirement: Approval of the Graduate Group Chair is required for courses taken outside the Penn Mathematics Department to be counted toward the course requirement for the M.Phil. Degree in Mathematics.
- The M.Phil. Comprehensive Examination Committee is appointed by the Graduate Group Chair after consultation with the student. It will consist of two or more faculty members, at least one of whom must be tenured, and one of whom will ordinarily be the supervising faculty member. The Graduate Group Chair will notify the entire mathematics faculty of the exam; all faculty are explicitly invited to attend, and those who do are full voting members of the Exam Committee.
- If the M.Phil. Comprehensive Examination is not passed on the first try, it may be taken just once more, and this must occur before the end of the following semester.

### THE Ph.D. DEGREE

Information on the Ph.D. thesis.

The degree of Doctor of Philosophy in Mathematics is an advanced degree designed for those who wish to pursue a career involving mathematical research. It is conferred in recognition of marked ability and high attainment in advanced mathematics, including the successful completion of an original research project of merit. The Ph.D. program is designed to guide students, year-by-year, toward becoming research mathematicians. Typically the program takes five years to complete, including the dissertation (although it can be completed in less time, depending on the student).

There are several stages to the Ph.D. program. The first, which is centered on the course requirements and the graduate preliminary exam, is designed to help the student acquire a broad background in mathematics. The second stage includes the oral Ph.D. preliminary exam on two topics of the student's choosing, and on participation in seminars. Breadth is one of the goals of this second stage, but the main objective at that point is to assist the student in choosing a field of specialization and in obtaining sufficient knowledge of this specialized field, including recent research developments. Ph.D. students also have an opportunity to earn a masters degree at this stage. The third and key stage of the Ph.D. program is the dissertation (or "Ph.D. thesis"), in which the student will make an original contribution to mathematics. The entire Ph.D. program is designed to help students move toward taking this significant step in creating new mathematics. Along the way the student is required to acquire some teaching experience.

The program leading to this degree is described below, and may include work completed at the University of Pennsylvania for a Masters degree. (Up to eight courses taken at other universities, while a candidate for a graduate degree, can also be counted toward the Ph.D. requirements.)

Detailed requirements:

- Admission to candidacy: Admission to Ph.D. candidacy is achieved by passing the Ph.D. Preliminary Examination. Students must previously have passed the Graduate Preliminary Exam in their first year, and have taken at least six graduate mathematics courses (including those taken elsewhere). (See the third item of section 10 below.)
- The Ph.D. Preliminary Examination: This is an oral exam in two distinct subfields of mathematics, one of which should be the student's intended area of specialization. The student should demonstrate both depth and breadth of knowledge, as well as real mastery of basic material and the ability to apply it to specific examples. Typically, the content of each of the two parts of this exam will be somewhat more than that of a one-semester graduate course at or above the 610 level (though sometimes the material in an advanced 500-level course may be appropriate).
- Course Requirements: Twenty units of graduate courses, numbered 500 and above (or the equivalent), are required for the Ph.D. degree, including at least twelve courses taken at the University of Pennsylvania and, amongst these, at least eight taken at the Penn Mathematics Department. Of these courses, at least sixteen, including those transferred from other institutions, must be in mathematics. (Independent study courses at Penn may be counted toward the twenty course requirement.) Among the courses, every student must take at least two semesters of graduate courses at the 600 level in each of geometry-topology, algebra, and analysis (Math 600, 601; 602, 603; 608, 609). The approval of the Graduate Group Chair is required for any course taken outside the Penn Mathematics Department to be counted toward the Ph.D.
- Language Requirement: There is no foreign language requirement for the Ph.D. degree. Depending on the nature of their research, however, students may in practice find it necessary to learn to read mathematical papers that have been written in a language other than English.
- Seminar Requirement: It is expected that all advanced graduate students will regularly attend and participate in at least one seminar series each semester. Students must also satisfactorily present a lecture in one of the department's graduate or research seminars, and answer questions about the material, by the end of the third year. Although one lecture is the formal minimum, it is expected that students will give a number of such talks to audiences of students and faculty. Guidance in the preparation of these lectures is provided by faculty members in the department. The intention is for the student to gain experience in digesting and presenting advanced material and in fielding questions about it before an audience of mathematicians, as well as actively participating in research interactions and being a part of the mathematical community of graduate students in the Mathematics Department.
- Teaching requirement: In order to gain experience in classroom teaching, the student is required to perform satisfactorily as a teaching assistant or instructor for at least two semesters. Teaching for more than two semesters is encouraged, especially for those students who plan to teach after their Ph.D. Graduate students participate in a TA training program before they begin their teaching.
- The Dissertation: The dissertation, also known as the "Ph.D. thesis", is the heart of the Ph.D. program. It must be a substantial original investigation in a field of mathematics, done under the supervision of a faculty advisor.
- The Ph.D. Advisory Committee: This committee is appointed by the Graduate Group Chair after consultation with the student, when the student begins to work on a thesis. It consists of three faculty members, including the thesis advisor, and meets at least once a semester with the student to discuss progress and to offer helpful comments.
- Dissertation Examination: When the dissertation is complete, it must be defended in a Dissertation Exam, at which the student will be expected to give a short public exposition of the results of the thesis, and to satisfactorily answer questions about the thesis and related areas.
- Some important administrative details:

- The administrative details listed earlier in connection with the Masters degree program apply to the Ph.D. program as well. In addition, we have the following:
- Combining the Masters General Examination with the Ph.D. Preliminary Examination: The oral defense of the Masters thesis may be used as the examination on one of the two subjects of the Ph.D. Preliminary Exam, provided that it occurs simultaneously with the other part of the Ph.D. exam, and that the requisite depth and breadth of knowledge are demonstrated.
- The Ph.D. Preliminary Exam Committee: This committee is appointed by the Graduate Group Chair after consultation with the student. It consists of three or more faculty members, at least two of whom are tenured members of the department. The syllabi for the exam are agreed upon by the committee and the student, with the approval of the Graduate Group Chair. The Graduate Group Chair notifies the mathematics faculty of the exam (date, time, place, committee members, reading lists). All faculty are to be explicitly invited, and each faculty member who attends the exam is a full voting member of the Exam Committee.
- Scheduling the Ph.D. Preliminary Exam: To take this exam, the student should have passed the Graduate Preliminary Exam and be in good standing. The student also should have taken at least six graduate mathematics courses (including those taken elsewhere). The student discusses in advance the topics, syllabi and the composition of the oral exam committee with the Graduate Group Chair, whose written approval is needed. The exam is to be taken by the end of the student's second academic year in the program. A one or two semester extension may be requested from the Graduate Group Chair in the case of students who took mostly 500-level courses in their first year.
- If the Ph.D. Preliminary Exam is not passed on the first try, it may be taken just once more, and this must occur before the end of the following semester. Passing the Ph.D. Preliminary Exam on the second try, at the latest, is a requirement for remaining in the Ph.D. program.
- Course Requirements: The Graduate Group Chair may, in exceptional cases, modify the requirement that at least 16 of the 20 graduate courses for the Ph.D. be in mathematics. A maximum of 6 of these 20 courses may be reading courses (independent studies). Permission of the instructor is needed in order to sign up for an independent study course. First and second year students also require permission of the graduate chair, as do third year students who wish to take more than one such course in a given semester.
- Timing for satisfying the Seminar Requirement: This requirement must be satisfied no later than one year after admission to Ph.D. candidacy.
- The Dissertation Exam Committee: This committee is appointed by the Graduate Group Chair after consultation with the student, and consists of three or more faculty members, at least one of whom is a tenured member of the Penn Mathematics Department, and at least one of whom is outside the area of specialization of the thesis. All mathematics faculty are to be explicitly invited to attend the exam, and those who do so are full voting members of the Exam Committee.
- The student must deliver a finished copy of the Ph.D. thesis to the graduate coordinator at least two weeks before the Dissertation Exam, so that it will be available for reading by the mathematics faculty.
- Timing for the Dissertation Exam: This must be successfully completed no later than six years after entering the graduate program. It should occur no later than the end of the third academic year after admission to Ph.D. candidacy. In exceptional situations, the Graduate Group Chair may, after consultation with the Graduate Advising Committee, provide an extension.
- Satisfaction of Requirements: When not otherwise specified, this is determined by the Graduate Group Chair, in consultation with the Graduate Advising Committee and involved faculty members and (in the case of the teaching requirement) the Undergraduate Chair.