# Ph.D. Program in Mathematics

### Degree Requirements

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

### Coursework

Sample course schedules (Years 1 and 2) for students with a primary interest in:

Year I - Fall Term | Year I - Spring Term |
---|---|

Linear Algebra | Topology II |

Differential Geometry I | Differential Geometry II |

Real Variables | Ordinary Differential Equations |

Complex Variables | Functional Analysis I |

Year II - Fall Term | Year II - Spring Term |
---|---|

Advanced Topics in Geometry: Isometric Immersions Before and After Nash | Advanced Topics in Geometry: Randomness and Complexity |

Advanced Topics in Geometry: High Dimensional Expanders and Ramanujan Complexes | Advanced Topics in Geometry: Topics in Geometric Nonlinear Functional Analysis |

Harmonic Analysis | Advanced Topics in Geometry: Analysis and Geometry of Scalar Curvature |

Advanced Topics in PDE: Resonances in PDEs | Advanced Topics in PDE: Analytic Aspects of Harmonic Maps |

Applied Math (Math Biology, Scientific Computing, Physical Applied Math, etc.)

Year I - Fall Term | Year I - Spring Term |
---|---|

Linear Algebra | Applied Stochastic Processes |

PDE I | Asymptotic Analysis |

Fluid Mechanics | Continuum mechanics |

Numerical Methods I | Numerical Methods II |

Year II - Fall Term | Year II - Spring Term |
---|---|

Neurophysiology and Neuronal Networks | Data Analysis |

Complex fluids | Mathematical Physiology |

Real Variables | Geophysical Fluid Dynamics |

Computational Fluid Dynamics | Nonlinear Optimization |

**Additional information for students interested in studying applied math is available here.**

Year I - Fall Term | Year I - Spring Term |
---|---|

Stochastic Calculus | Probability: Limit Theorems II |

Probability: Limit Theorems I | Applied Stochastic Analysis |

Real Variables | Advanced Topics in Probability: Random Graphs |

Complex Variables | Advanced Topics in Math Biology:Stochastic Problems in Cellular Molecular and Neural Biology |

Year II - Fall Term | Year II - Spring Term |
---|---|

Advanced Topics in Probability: Ergodic Theory of Markov Processes | Advanced Topics in Geometry: Randomness and Complexity |

Advanced Topics in Probability: Motion in Random Media | Advanced Topics in Probability: Random Matrices |

Advanced Topics in Applied Math: Quantifying Uncertainty in Complex Turbulent Systems | Advanced Topics in Probability: Markov Chain Analysis |

Derivative Securities | Advanced Topics in Numerical Analysis: Monte Carlo Methods |

Year I - Fall Term | Year I - Spring Term |
---|---|

Linear Algebra | Topology II |

PDE I | Ordinary Differential Equations |

Real Variables | PDE II |

Complex Variables | Functional Analysis I |

Year II - Fall Term | Year II - Spring Term |
---|---|

Differential Geometry I | Algebra II |

Harmonic Analysis | Advanced Topics in PDE: Extreme Problems for Elliptic Eigenvalues |

Advanced Topics in Analysis: Calculus of Variations | Advanced Topics in Analysis: Dynamics of the Nonlinear Schroedinger Equation |

Probability: Limit Theorems I | Probability: Limit Theorems II |

### 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 page on the written comprehensive exams.

### 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.

### Visiting Doctoral Students

Information about spending a term at the Courant Institute's Department of Mathematics as a visiting doctoral student is available on the Visitor Programs page.