# Degree Programs

## Mathematics

The graduate Mathematics Program at Harvard is designed for students who hope to become research mathematicians and show definite promise in this direction. Once the student has demonstrated a command of basic mathematical concepts by passing the qualifying examination, the emphasis is on getting to the frontiers of some field by independent reading, advanced courses, and seminars. The Cambridge area is one of the most active centers of mathematics in the world. Harvard, Brandeis, and Northeastern Universities and the Massachusetts Institute of Technology have an especially close association in mathematics, sharing several seminars and a weekly colloquium.

### The PhD Program

The degree of doctor of philosophy is awarded to students who have demonstrated their mastery of the basic techniques of mathematics and their ability to do independent research. The former is tested in the qualifying examination the latter in the dissertation. The dissertation, however, is the more important of the two.

The qualifying examination is given twice annually, and students are encouraged to take it as soon as possible so that they may begin work towards their dissertation research. Most students pass the exam during their first year, but if need be, the exam may be retaken a number of times. Students are expected to pass the examination by the end of the second year.

The PhD dissertation is an original treatment of a suitable subject leading to new results, usually written under the guidance of a faculty member. The final manuscript must conform to the requirements described in *The Form of the PhD Dissertation*, which is available in the Publications section on the GSAS website.

Many of the more advanced courses and seminars are designed to lead the student to areas of current research.

The University requires a minimum of two years of academic residence (16 half-courses). (See *The Graduate School of Arts and Sciences Handbook* for financial residence requirements.) On the other hand, the PhD usually takes four to five years.

A reading knowledge of two modern languages, French, Italian (if deemed related to the student’s field of study), German, or Russian, is required of PhD candidates. One language requirement must be passed by the end of the second year of graduate study and the other by the end of the third year. Each candidate must also write a "minor thesis." This is an original presentation of a standard subject about which the student is ignorant but wishes to learn. It is intended to give the student experience in assimilating and presenting unfamiliar material. The research and writing must be done during a three-week period, soon after the qualifying examination is passed. Each candidate must also participate in the Teaching Apprentice Program and have two semesters of classroom experience of teaching, usually as a teaching fellow.

Applications for transfer from other programs granting PhDs in mathematics are not ruled out, but are discouraged.

### AB-AM Degree

Candidates for the AB-AM degree in mathematics must meet both the academic and course requirements for the AB degree in mathematics and for the AM degree in mathematics. A given course can be counted for only one of the two degrees, i.e., one course cannot meet the requirement for the AB degree and then be counted again for the AM degree. See below for the AM requirements. Any undergraduate who wishes to apply for this degree must file an application form for the graduate program in mathematics just as any other student files for graduate work at Harvard. Only students with advanced standing are eligible to apply for this four-year program. Undergraduates taking graduate courses in their third year may bracket those courses they wish to apply toward their graduate degree. Candidates for the AB-AM degree must bracket courses by the beginning of the final term before graduation.

### Requirements for the AM Degree

The Master of Arts (AM) degree is not a prerequisite for the PhD, but is often obtained by students on their way to a doctorate. However, applicants are not accepted for the terminal AM in mathematics. The formal requirements are a minimum academic residence of one year, eight half-courses in mathematics at 100 or 200 level, with at least four at the 200 level; a reading knowledge of one of three languages—French, German, or Russian—is also required. (See *The Graduate School of Arts and Sciences Handbook* for financial residence requirements.)

### Financial Aid

All students in the Department of Mathematics receive substantial financial support during their graduate training. This support may be in the form of grants or teaching fellowships from Harvard, or fellowships and research assistantships from outside organizations such as the National Science Foundation. Students are strongly encouraged to apply for outside awards.

Sources from outside the University support a great many graduate fellowships. In particular, students who are US citizens should investigate the predoctoral fellowship opportunities provided by the Fannie and John Hertz Foundation (Box 5032, Livermore, CA 94551–5032), the National Defense Science and Engineering Graduate Fellowship Program (200 Park Drive, Suite 211, P.O. Box 13444, Research Triangle Park, NC 27709–3444), the National Physical Science Consortium for Minorities and Women (c/o New Mexico State University, O’Loughlin House, University Boulevard, Box 30001, Department 3 NPS, Las Cruces, NM 88003-8001), and the National Science Foundation (Washington, DC 20550). International students are encouraged to apply for the Fulbright IIE, Knox and Kennedy fellowships (applications for the Kennedy and Knox are available through the Harvard Committee on General Scholarships), and other private and government scholarships and fellowships available in their home country.

Students without outside support are required to teach as part of their financial aid package. Graduate students do not teach in their first year. Students begin as teaching fellows for one half-course (i.e., for a one-semester course) in their second through fourth years, and for two half-courses if they stay for a fifth year. Teaching fellows ordinarily teach their own sections of undergraduate calculus, but have a course assistant to help with grading and problem sections. There are a few upper-class tutorials taught by experienced teaching fellows. All students must participate in the Teaching Apprentice Program run by the department and demonstrate English proficiency before they may teach.

Applications for admission and for scholarships or nonteaching fellowships, together with information regarding admissions procedures, are available on the Graduate School of Arts and Sciences Prospective Students page on the GSAS website. Applicants may apply for admissions beginning in the fall until December 15. All applications and accompanying materials must be submitted online.

### Senior Faculty Research Interests

**Noam D. Elkies**, *Professor of Mathematics*. Number theory, computation, classical algebraic geometry, music.

**Dennis Gaitsgory,** *Professor of Mathematics*. Geometric aspects of representation theory.

**Robin Gottlieb,** *Professor in the Teaching of Mathematics.*

**Benedict H. Gross,** *George Vasmer Leverett Professor of Mathematics*. Algebraic number theory, Diophantine geometry, modular forms.

**Joseph Harris,*** Higgins Professor of Mathematics.* Algebraic geometry.

**Michael J. Hopkins**, *Professor of Mathematics.* Algebraic topology.

**Mark Kisin,** *Professor of Mathematics. *Number theory and arithmetic geometry.

**Peter Kronheimer,** *William Caspar Graustein Professor of Mathematics.* Topology, differential and algebraic geometry, and their applications.

**Jacob Lurie***, Professor of Mathematics.* Algebraic geometry, algebraic topology, and higher category theory.

**Barry Mazur,** *Gerhard Gade University Professor.* Number theory, automorphic forms and related issues in algebraic geometry.

**Curtis T. McMullen, ***Maria Moors Cabot Professor of the Natural Sciences.* Riemann surfaces, complex dynamics, hyperbolic geometry.

**Martin Nowak,** *Professor of Mathematics and Biology.* Mathematical biology, evolutionary dynamics, infectious diseases, cancer genetics, game theory, language.

**Wilfried Schmid,*** Dwight Parker Robinson Professor of Mathematics*. Lie groups, representation theory, complex differential geometry.

**Yum-Tong Siu**,* William Elwood Byerly Professor of Mathematics*. Several complex variables.

**Shlomo Sternberg**, *George Putnam Professor of Pure and Applied Mathematics. *Differential geometry, differential equations, Lie gro ups and algebras, mathematical physics.

**Clifford Taubes,** *William Petschek Professor of Mathematics*. Nonlinear partial differential equations and applications to topology, geometry, and mathematical physics.**W. Hugh Woodin,** *Professor of Philosophy and of Mathematics. *Set theory, determinacy, and strong axioms of infinity.**Horng-Tzer Yau**, *Professor of Mathematics*. Probability theory, quantum dynamics, differential equations, and nonequilibrium physics.**Shing-Tung Yau, ***William Caspar Graustein Professor of Mathematics.* Differential geometry, partial differential equations, topology, mathematical physics.

Junior and visiting faculty interests represent a diverse and important addition to the department. As these appointments vary in length from one term (on the part of visitors) to three-year appointments as a Benjamin Pierce Lecturer on Mathematics, Assistant Professor of Mathematics, they will be listed annually in the *Courses of Instruction* and in the *Graduate Guide to Mathematics* (available from the department). Generally there are from nine to 12 appointments in these categories.