# The Ph.D. Programs

# PhD Program

The Ph.D. degrees in Mathematics and in Atmosphere-Ocean Science and
Mathematics are open to students who wish to pursue a career in
academic research and teaching, as well as in the private and public
sectors. consistent with its scientific breadth, the Institute welcomes
applicants whose primary background is in quantitative fields such as
economics, engineering, physics, or biology, as well as mathematics.
Doctoral students take advanced courses in their areas of
specialization, followed by a period of research and the preparation
and defense of the doctoral thesis.

The Graduate Department of Mathematics at the Courant Institute
offers balanced training in mathematics and its applications in the
broadest sense. The Department occupies a leading position in
pure and applied mathematics, especially in ordinary and partial
differential equations, probability theory and stochastic processes,
differential geometry, numerical analysis and scientific computation,
mathematical physics, material science, fluid dynamics, math biology,
Atmosphere and Ocean science, and Computational Biology.

## The Ph.D. in Mathematics

### Degree Requirements

A candidate for the Ph.D. degree in mathematics must fulfill a number of different departmental requirements:

- » 72 points of credit;
- » a written comprehensive examination;
- » an oral preliminary examination;
- » an oral defense of the dissertation.

### Coursework

All students in the Ph.D. program must complete 72 points of credits (24 courses of 3 points each). It is possible, with departmental permission, to take courses, relevant to students' course of study, in other departments at NYU or at other universities. A base minimum of 32 points of credits must be completed at the Department of Mathematics. Information on potential transfer of credit is contained in the forthcoming section on GSAS and Departmental Policies and Procedures.

### The Written Comprehensive Examination

The examination tests the basic knowledge required for any serious mathematical study. It consists of the three following sections: Advanced Calculus, Complex Variables, and Linear Algebra. The examination is given on two consecutive days, twice a year, in early September and early January. Each section is allotted three hours and is written at the level of a good undergraduate course. Samples of previous examinations are available in the departmental office. Cooperative preparation is encouraged, as it is for all examinations. In the fall term, the Department offers a workshop, taught by an advanced Teaching Assistant, to help students prepare for the written examinations.

Entering students with a solid preparation are encouraged to consider taking the examination in their first year of full-time study. All students must take the examinations in order to be allowed to register for coursework beyond 36 points of credit; it is recommended that students attempt to take the examinations well before this deadline. Graduate Assistants are required to take the examinations during their first year of study.

For further details, consult the section on the Written Comprehensive Examinations.

### The Oral Preliminary Examination

This examination is usually (but not invariably) taken after two years of full-time study. The purpose of the examination is to determine if the candidate has acquired sufficient mathematical knowledge and maturity to commence a dissertation. The phrase "mathematical knowledge" is intended to convey rather broad acquaintance with the basic facts of mathematical life, with emphasis on a good understanding of the simplest interesting examples. In particular, highly technical or abstract material is inappropriate, as is the rote reproduction of information. What the examiners look for is something a little different and less easy to quantify. It is conveyed in part by the word "maturity." This means some idea of how mathematics hangs together; the ability to think a little on one's feet; some appreciation of what is natural and important, and what is artificial. The point is that the ability to do successful research depends on more than formal learning, and it is part of the examiners' task to assess these less tangible aspects of the candidate's preparation.

The orals are comprised of a general section and a special section, each lasting one hour, and are conducted by two different panels of three faculty members. The examination takes place three times a year: fall, mid-winter and late spring. Cooperative preparation of often helpful and is encouraged. The general section consists of five topics, one of which may be chosen freely. The other four topics are determined by field of interest, but often turn out to be standard: complex variables, real variables, ordinary differential equations, and partial differential equations. Here, the level of knowledge that is expected is equivalent to that of a one or two term course of the kind Courant normally presents. A brochure containing the most common questions on the general oral examination, edited by Courant students, is available at the Department Office.

The special section is usually devoted to a single topic at a more advanced level and extent of knowledge. The precise content is negotiated with he candidate's faculty advisor. Normally, the chosen topic will have a direct bearing on the candidate's Ph.D. dissertation.

All students must take the oral examinations in order to be allowed to register for coursework beyond 60 points of credit. It is recommended that students attempt the examinations well before this deadline.

### The Dissertation Defense

The oral defense is the final examination on the student's dissertation. The defense is conducted by a panel of five faculty members (including the student's advisor) and generally lasts one to two hours. The candidate presents his/her work to a mixed audience, some expert in the student's topic, some not. Often, this presentation is followed by a question-and-answer period and mutual discussion of related material and directions for future work.

### Summer Internships and Employment

The Department encourages Ph.D. students at any stage of their studies, including the very early stage, to seek summer employment opportunities at various government and industry facilities. In the past few years, Courant students have taken summer internships at the National Institute of Health, Los Alamos National Laboratory, Woods Hole Oceanographic Institution, Lawrence Livermore National Laboratory and NASA, as well as Wall Street firms. Such opportunities can greatly expand students' understanding of the mathematical sciences, offer them possible areas of interest for thesis research, and enhance their career options. The Director of Graduate Studies and members of the faculty (and in particular the students' academic advisors) can assist students in finding appropriate summer employment.

## The Ph.D. in Atmosphere Ocean Science and Mathematics

### Coursework

The curriculum for the program provides a balance between mathematics and physical science, and aims at preparing students for research and teaching in all aspects of applied mathematics. Please refer to the Center for Atmosphere Ocean Science Graduate Program for further information on coursework (including a detailed list of courses and sample distribution of courses by term), degree examinations and other aspects of the program.

## Visiting Doctoral Students

Advanced doctoral students at other universities who wish to spend a term at the Courant Institute's Department of Mathematics need to provide the following documentation:

- a letter from the student's advisor requesting that the student spend at term at the Courant Institute to conduct research under the supervision of a faculty mentor;

- confirmation from the Courant Institute faculty in question that he or she are willing to work with the visiting student;

- official documentation showing financial support of at least
$2,000 U.S. per month for the term of stay.

These students will have to submit an application for non-degree
studies to NYU's Graduate School of Arts and Science for the semeter
for their proposed visit. The application can be
downloaded from the Graduate Admission web page at
http://gsas.nyu.edu/page/grad.admissionsapplication, and sent in with
all transcripts, a statement of purpose and the application fee (GRE
tests or letters of recommendation are not required for non-degree
applications).

With this documentation, New York University will be able to sponsor a J-1 visa for a visiting student.

For any questions, please contact Tamar Arnon at the Courant Institute (arnon@cims.nyu.edu).