# Ph.D. Program Content

The graduate program in the field of mathematics at Cornell leads to the Ph.D. degree, which takes most students five to six years of graduate study to complete. One feature that makes the program at Cornell particularly attractive is the broad range of interests of the faculty. In addition to the usual areas of algebra, analysis and geometry, the department has outstanding groups in the areas of algebraic geometry, combinatorics, dynamical systems, logic, Lie groups, and partial differential equations, including their numerical treatment. The field also maintains close ties with distinguished graduate programs in the fields of applied mathematics, computer science, operations research, and statistics.

### Special Committee

The emphasis in the Graduate School at Cornell is on individualized instruction and training for independent investigation. There are very few formal requirements and each student develops a program in conjunction with his or her special committee, which consists of three faculty members, some of which may be chosen from outside the field of mathematics. An entering student is assigned a special committee, but over time the committee changes its membership to reflect the student's interests, and is finally chaired by the student's thesis adviser.

### First-Year Courses

A normal course load for a beginning graduate student is three courses per term. A typical first-year program includes four of the following six basic courses:

- MATH 6110,
*Real Analysis*(syllabus) - MATH 6120,
*Complex Analysis*(syllabus) At the level of Rudin,*Real and Complex Analysis.*--> - MATH 6310,
*Algebra*(syllabus) - MATH 6320,
*Algebra*(syllabus) At the level of Lang,*Algebra,*or Jacobson,*Lectures in Abstract Algebra.*--> - MATH 6510,
*Introductory Algebraic Topology*(syllabus) At a level similar to that of Massey,*Algebraic Topology,*and Eilenberg-Steenrod,*Foundations of Algebraic Topology.*--> - MATH 6520,
*Differentiable Manifolds I*(syllabus) At the level of Lee,*Introduction to Smooth Manifolds*, or Conlon,*Differentiable Manifolds*-->

There are no qualifying exams, but the program requires that all students take four basic courses to be selected from the above six core courses by the time they are ready to take the A exam. They are to be distributed among three main areas: analysis, algebra and topology/geometry. A student must take at least one course from each group. All entering graduate students are encouraged to eventually take all six core courses with the option of an S/U grade for two of them. Students who are not ready to take some of these courses may take MATH 4130-4140, *Introduction to Analysis*, and/or MATH 4330-4340, *Introduction to Algebra*, which are the honors versions of our basic undergraduate courses. A first-year student is also able to explore other areas, such as differential equations, mathematical logic, and probability theory, in filling out the normal course load.

### Admission to Candidacy

To be admitted formally to candidacy for the Ph.D. degree, the student must pass the oral admission to candidacy examination or **A exam.** This must be completed before the beginning of the student's fourth year. At this stage the committee is formed of faculty who are experts in the proposed area of research, and the chair is also the thesis advisor. The admission to candidacy examination is given to determine if the student is “ready to begin work on a thesis.” The content and methods of examination are agreed on by the student and his/her committee before the examination. The student must be prepared to answer questions on the proposed area of research, and to pass the exam, he/she must demonstrate expertise beyond just mastery of basic mathematics covered in the standard first-year graduate courses. Upon passing the A exam, the student will be awarded (at his/her request) an M.S. degree without thesis.

### Residence

To receive an advanced degree a student must fulfill the residence requirements of the Graduate School. One unit of residence is granted for successful completion of one semester of full-time study, as judged by the chair of the special committee. The Ph.D. program requires a minimum of six residence units. This is not a difficult requirement to satisfy since the program generally takes five to six years to complete. A student who has done graduate work at another institution may petition to transfer residence credit but may not receive more than two such credits.

### Language Requirement

Every doctoral student must pass a test of mathematical reading ability in a foreign language at the time of their A exam. The allowed languages are French, German, Russian, Chinese, Japanese, Spanish, and Portuguese. The choice of language must be approved by the student’s special committee chair.

More about the language requirement

### Thesis

The candidate must write a thesis that represents creative work and contains original results in that area. The research is carried on independently by the candidate under the supervision of the chairperson of the special committee. By the time of the oral admission to candidacy examination, the candidate should have selected as chairperson of the committee the faculty member who will supervise the research. When the thesis is completed, the student presents his/her results at the thesis defense or **B Exam**.

### Master's Degree in the Minor Field

Ph.D. students in the field of mathematics may earn a master's degree in certain minor fields. The available options are Master of Science in Education, Master of Arts in Teaching in Education, and Master of Science in Computer Science. More options may be made available in the future. Interested students must apply to the Graduate School using a form available for this purpose. To be eligible for one of these degrees, the student must have a member representing the minor field on the special committee and pass the A-exam in the major field. The rules and the specific requirements for each master's program are laid down here.

Cornell will award at most one master's degree to any student. In particular, a student awarded a master's degree in a minor field will not be eligible for a master's degree in the major field.

### Graduate Minor in Mathematics

In order to get a minor in mathematics, any graduate student in any field must request a member of the field of mathematics to serve on his/her special committee representing mathematics. This committee member, after consulting with the graduate student, will set course requirements. The number of courses required will typically be four or more, but can be less in special circumstances. The committee member will attend (and ask questions on) the student's qualifying, admission to the Ph.D. candidacy and final exams. In addition, he/she will read at least the mathematical aspects of the student's thesis and be responsible (together with the other committee members) for giving final approval to the thesis.