**Admissions and Placement**

Applicants to the mathematics graduate program are evaluated on the basis of their transcripts (courses taken and grades earned), letters of recommendation, GRE scores, TOEFL or TSE scores (for non-native English speakers), personal statement, and other material in their applications. The entire application package is considered, and no one portion guarantees admission (although weaknesses in one area should be compensated by strengths in another area). Applicants should have taken courses in advanced calculus/analysis and in abstract and linear algebra with proofs; and mathematics course grades should be mostly A or A- (or the equivalent). Letters of recommendation should be from mathematics professors who can evaluate the applicant's knowledge and ability, concerning mathematical material at an advanced level.

Scores on the Advanced Math Subject Test of the GRE should preferably be at least about 750, although applicants with lower scores may be admitted if the rest of their application is strong and provides evidence of mathematical initiative. The average GRE scores of the students who entered our Ph.D. program in the recent past were: Verbal: 597; Quantitative: 789; Advanced Math Subject Test: 820. Non-native English speakers should preferably have a TOEFL score of at least 100 on the ibt exam (and at least 20 on the spoken English portion of that exam), or at least 250 on the computer-based exam; scores on the TSE should be at least 50. Among those admitted to our program in recent years who took the TOEFL, the average score was 116 on the ibt exam.

Over the past few years, the average number of applications submitted to our graduate program per year was 200. We admitted about 20-30 students per year to our Ph.D. program. Of these, an average of 10-12 entered our program per year.

A majority of our Ph.D.'s take academic jobs in mathematics following their Ph.D. The remainder take other mathematically related jobs, in areas such as computer vision, genomics, medical imaging, mathematical finance, or artificial intelligence.

Graduating students receive help from the Mathematics Department in their employment search. This includes guidance about how to apply in an effective way; logistical help with submitting letters of recommendation; and help in providing information about the applicant's teaching and research to potential employers.

**Students and Support**

Currently there are 42 Ph.D. students in our graduate program. Of these, almost all are full-time. Almost half of our graduate students entered our program from colleges or universities outside the United States, from a wide variety of countries including Bulgaria, Chile, China, Germany, Ghana, Romania, Russia, South Korea, Switzerland, and Turkey. About one quarter of our Ph.D. students are women. Students typically receive their Ph.D.'s in five years, and our financial support package lasts for five years for students making good progress toward their Ph.D. This includes at least four semesters with no teaching required, as well as three summers of funding with no teaching required.

The current academic year stipend for Ph.D. students is $26,700 for ten months, in addition to a full-tuition scholarship and free health insurance. The summer stipend is an additional $4,400. (Students are expected to be in residence during the summer months when they are funded, unless they arrange otherwise with their advisor.) Mathematics graduate students also receive office space with a computer, and access to the Mathematics Reading Room and the Math-Physics-Astronomy Library.

### Courses, Requirements and Mentoring

The Penn Mathematics Graduate Program offers three degrees: the masters degree (A.M.), the M.Phil. degree (in between the masters and the Ph.D.), and the Ph.D. The Ph.D. is a research degree; the M.Phil. is a terminal professional degree; and the masters degree can be earned either as a professional degree or as a step in the Ph.D. program.

Ph.D. students take four courses each semester that they are on fellowship (including the first year), and three courses in other semesters, until they complete the course requirement. There are many courses to choose from, but all Ph.D. students must either take, or place out of, the following required courses:

- In geometry-topology: Math 600, 601, 618.
- In algebra: Math 602, 603.
- In analysis: Math 608, 609.

These courses are mostly taken in the first year of the program. Students who place out of these courses, by demonstrating their knowledge of the material to the instructor, may take other courses instead. In the second year, students typically take courses related to their areas of greatest interest, and in preparation for their oral exam (the Ph.D. Preliminary Exam). In the third year, students concentrate on their research, and typically complete their 20-course requirement by taking topics courses in the area of their dissertations, independent studies, seminars, etc. (Students can complete their course requirement sooner by transfering up to eight courses taken as a graduate student elsewhere.)

Ph.D. students can receive a masters degree on the way to the Ph.D. by completing the requirements for that degree (eight graduate courses, with at least one each in algebra, analysis, and geometry-topology; passing the Masters Preliminary Exam; and submitting and successfully defending a masters thesis. Apart from the masters thesis (which is optional for Ph.D. students), Ph.D. students are expected in their first year to complete the requirements for the masters degree, and to achieve a distinguished pass on the Masters Preliminary Exam. (By doing this and by doing well in their classes, students satisfy the Qualifications Evaluation.) Ph.D. students are expected to take their oral Ph.D. Preliminary Exam by the end of the second year, unless given an extension by the Graduate Group Chair.

Students receive mentoring from both their faculty advisor and the Graduate Chair. Students are assigned to a faculty advisor upon entering the program, and students can change advisors if they wish. Following passage of the oral exam, students select a Ph.D. thesis advisor, who becomes their main mentor. Students also receive mentoring concerning their teaching (as TAs) from the faculty and senior grad students who coordinate the TA training program. They also receive additional mentoring from their Ph.D. Advisory Committee, with whom they discuss the topic of their dissertation (Ph.D. thesis). These discussions are in lieu of a formal written dissertation proposal.

**More information** about Graduate Courses.

**More information** about requirements for the masters, M.Phil, and Ph.D. programs.