# Applied and Computational Math

## Overview

The Program in Applied and Computational Mathematics offers a select group of highly qualified students the opportunity to obtain a thorough knowledge of branches of mathematics indispensable to science and engineering applications, including numerical analysis and other computational methods.

## Applying

## Ph.D.

Students enroll in courses based on research topics that they choose in consultation with faculty. Typically, students take regular or reading courses with their advisers in the three topic areas of their choice, completing the regular exams and course work for these courses.

Students must choose three areas in which to take courses and be examined out of a list of six possibilities specified below. The student should choose their specific topics by the end of October. The director of graduate studies, in consultation with the student, appoints a set of advisers from among the faculty and associated faculty. The adviser in each topic meets regularly with the student, monitors progress and assigns additional reading material. Advisers are usually program or associated faculty, but members of the faculty from other departments may serve as advisers with approval. In consultation with the topic advisers, the first-year student should choose three topics from among the following six applied mathematics categories:

- Asymptotics, analysis, numerical analysis and signal processing;
- Discrete mathematics, combinatorics, algorithms, computational geometry and graphics;
- Mechanics and field theories (including computational physics/chemistry/biology);
- Optimization (including linear and nonlinear programming and control theory);
- Partial differential equations and ordinary differential equations (including dynamical systems);
- Stochastic modeling, probability, statistics and information theory.

Additional topics may be considered with prior approval by the director of graduate studies.

At the end of the first year, students will also take a preliminary exam, consisting of a joint interview by their three first-year topic advisers. Each student should decide with their first-year advisers which courses are relevant for their examination areas.

Students should assess their level of preparation for the preliminary examination by reviewing homework and examinations from the previous year’s work. Students who fail the preliminary examination may, with the support of the first-year advisers, take the examination a second time.

Before being admitted to a third year of study, students must pass the general examination. The general examination, or generals, is designed as a sequence of interviews with assigned professors that covers three areas of applied mathematics. The generals culminate in a seminar on a research topic, usually delivered toward the end of the fourth term. A student who completes all program requirements (coursework, preliminary exams, with no incompletes) but fails the general examination may take it a second time. Students who fail the general examination a second time will have their degree candidacy terminated.

The Master of Arts degree is normally an incidental degree on the way to full Ph.D. candidacy, but may also be awarded to students who for various reasons leave the Ph.D. program. Students who have satisfactorily passed all courses undertaken during their graduate study, satisfactorily resolved all incompletes and have passed the preliminary exam, may be awarded an M.A. degree. Upon learning the program’s determination of their candidacy to receive the M.A., students apply for the master's degree online through the advanced degree application system.

The doctoral dissertation must consist of either a mathematical contribution to some field of science or engineering, or the development or analysis of mathematical or computational methods useful for, inspired by, or relevant to science or engineering.

The Ph.D. is awarded after the candidate’s doctoral dissertation has been accepted and the final public oral examination sustained.

## Faculty

#### Director

Peter Constantin

#### Executive Committee

Emmanuel A. Abbe

Noga M. Alon

René A. Carmona

Maria Chudnovsky

Peter Constantin

Weinan E

Paul Seymour

Amit Singer

Howard A. Stone

Jeroen Tromp

Ramon van Handel

## Courses

Permanent Courses

Courses listed below are graduate-level courses that have been approved by the program’s faculty as well as the Curriculum Subcommittee of the Faculty Committee on the Graduate School as permanent course offerings. Permanent courses may be offered by the department or program on an ongoing basis, depending on curricular needs, scheduling requirements, and student interest. Not listed below are undergraduate courses and one-time-only graduate courses, which may be found for a specific term through the Registrar’s website. Also not listed are graduate-level independent reading and research courses, which may be approved by the Graduate School for individual students.