
Applied Mathematics Graduate Program

M.S. Program in Applied Physics / Concentration in Applied Mathematics

Ph.D. Program in Applied Physics / Concentration in Applied Mathematics
Columbia University offers an exceptional environment for graduate study in applied mathematics leading Master of Science (M.S.), Master of Philosophy (M.Phil.), Doctor of Philosophy (Ph.D.) and Doctor of Engineering Science (Eng.Sc.D. or DES) degrees. The applied mathematics faculty in the Department of Applied Physics and Applied Mathematics are expert in a wide variety of application areas and perform multidisciplinary research spanning atmospheric, oceanic, and climate science; geophysics and solidearth dynamics; waves in fluids, optics and quantum systems; imaging & image processing; inverse problems, inference and machine learning; bioinformatics & systems biology. Close collaborations exist with Columbia’s Lamont Doherty Earth Observatory and Columbia’s medical school.
"Applied mathematics deals with the use of mathematical concepts and techniques in various fields of science and engineering. Historically, mathematics was first applied with great success in astronomy and mechanics. Then it developed into a main tool of physics, other physical sciences, and engineering. It is now important in the biological, geological, and social sciences. With the coming of age of the computer, applied mathematics has transcended its traditional style and now assumes an even greater importance and a new vitality.
Compared with the pure mathematician, the applied mathematician is more interested in problems coming from other fields. Compared with the engineer and the physical scientist, he or she is more concerned with the formulation of problems and the nature of solutions. Compared with the computer scientist, he or she is more concerned with the accuracy of approximations and the interpretation of results. Needless to say, even in this age of specialization, the work of mathematicians, scientists, and engineers frequently overlaps. Applied mathematics, by its very nature, has occupied a central position in this interplay and has remained a field of fascination and excitement for active minds." (C.K. Chu, 19871988 SEAS Bulletin)
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M.S. Program in Applied Physics / Concentration in Applied Mathematics
This 30point program leads to a Master of Science degree. Students must complete five core courses and five electives. All degree requirements must be completed within five years. A candidate is required to maintain at least a 2.5 grade point average.
 APMA E4001: Principles of applied mathematics

APMA E4101: Introduction to dynamical systems
 APMA E4150: Applied functional analysis
 APMA E4200: Partial differential equations
 APMA E4204: Functions of a complex variable

APMA E4300: Introduction to numerical methods
 APMA E4301: Numerical methods for partial differential equations
 APMA E6301: Analytic methods for partial differential equations

APMA E6302: Numerical analysis for partial differential equations
 Students must also take a required Research Seminar course, APMA E6100 x or y
Elective Courses
A student must select five elective courses from those listed below (or any of those not used to satisfy the core requirements from the list above) for a total of 15 points of graduate credit. Additional courses not listed below can be applied toward the elective requirements, subject to the approval of the faculty adviser. *
* Absolutely no courses from the dept of economics, business school, sipa or quantitative courses offered by continuing education may be applied as electives toward the degree.
Computer science elective courses include:
 CS0R W4231: Analysis of algorithms, I
 C0MS W4236: Introduction to computational complexity
 C0MS W4241: Numerical algorithms and complexity
 C0MS W4252: Computational learning theory
* Industrial engineering/operations research elective courses include:
 IEOR E4003: Industrial economics
 IEOR E4004: Introduction to operations research: deterministic models
 IEOR E4007: Optimization: models and methods
 IEOR E4106: Stochastic models

STAT GU4001: Introduction to probability and statistics
 IEOR E4403: Advanced engineering and corporate economics
 IEOR E4407: Game theoretic models of operations
 STAT GU4606: Elementary stochastic processes
 IEOR E4700: Introduction to financial engineering
*Please check IEOR website for registration procedures required of nonIEOR students
http://ieor.columbia.edu/registrationguidelinesnonieorstudents
Other elective courses may be chosen from other departments in SEAS and Arts and Sciences, e.g., the Departments of Mechanical Engineering, Mathematics, and Statistics.
Ph.D. Program in Applied Physics / Concentration in Applied Mathematics
After completing the M.S. program in applied physics/concentration in applied mathematics, doctoral students specialize in one applied physics field (such as Applied Mathematics). Some specializations have specific course requirements for the doctorate; elective courses are determined in consultation with the program adviser. Successful completion of an approved 30point program of study is required in addition to successful completion of a written qualifying examination taken after two semesters of graduate study. An oral examination, taken within one year after the written qualifying examination, and a thesis proposal examination, taken within two years after the written qualifying examination, are required of all doctoral candidates.
Degree Requirements for Doctor of Philosophy (Ph.D.) in Applied Physics: Applied Mathematics

Complete requirements for the M.S. with a 3.0 Minimum GPA
(unless a Master’s Degree from another institution has already been earned, in which case students receive 30 points and 2 Residence Units of advanced standing)

Complete requirements for the Master of Philosophy (M.Phil.) Degree:

Written Qualifying Examination
Courses suggested for preparation at the level of the general and plasma physics parts of the written qualifying examination are listed in the qualifying examination memorandum.

30 points of courses and/or research (beyond M.S.) taken for a letter grade with minimum 3.0 GPA
Can be fulfilled with core and related courses of specialization not used for the MS degree as well as research points, but no more than 15 points of research can be applied to this 30 point requirement.

6 Residence Units
One per semester not including summer, takes 3 years without M.S. or 2 years with M.S.

Oral Exam (usually Spring of 2nd year)

Thesis proposal (usually Spring of 3rd year)

Ethics requirements
Online ethics course during Fall of 1st, year, attend departmental ethics seminar during Spring of 1st and 2nd years

Master of Philosophy Degree awarded

Written Qualifying Examination

Complete Dissertation

Successful Defense
Degree Requirements for Doctor of Engineering Science (Eng.Sc.D. or DES) in Applied Physics: Applied Mathematics

Complete requirements for the M.S. with a 3.0 Minimum GPA
(unless a Master’s Degree from another institution has already been earned, in which case student receives 30 points and 2 Residence Units of advanced standing)

Written Qualifying Examination
Courses suggested for preparation at the level of the general and plasma physics parts of the written qualifying examination are listed in the qualifying examination memorandum.

Ethics requirement
Online ethics course during Fall of 1st year, attend departmental ethics seminar during Spring of 1st and 2nd years

Oral Exam (usually Spring of 2nd year)

30 points of courses and/or research (beyond M.S. taken for a letter grade with 3.0 GPA
Can be fulfilled with core and related courses of specialization not used for the M.S. degree as well as research points, but no more than 15 points of research can be applied toward this 30 point requirement

12 points of APAM E9800: Doctoral Research Instruction

Thesis proposal (usually Spring of 3rd year)

Complete Dissertation

Successful Defense
Core Courses
 APMA E4001: Principles of applied mathematics

APMA E4101: Introduction to dynamical systems
 APMA E4150: Applied functional analysis
 APMA E4200: Partial differential equations
 APMA E4204: Functions of a complex variable

APMA E4300: Introduction to numerical methods
 APMA E4301: Numerical methods for partial differential equations

APMA E4400: Introduction to biophysical modeling

APMA E6301: Analytic methods for partial differential equations
 APMA E6302: Numerical analysis for partial differential equations
Related Courses of Specialization

AMCS E4302: Parallel scientific computing

APMA E6901: Special topics in applied mathematics

APMA E8308: Asymptotic methods in applied mathematics

APPH E4210: Geophysical fluid dynamics

APMA E9815: Geophysical fluid dynamics seminar

COMS W4205: Combinatorial theory

COMS W4241: Numerical algorithms and complexity

EESC GU4008: Introduction to atmospheric science

PHYS G4019: Mathematical methods in physics

MATH W4032: Fourier analysis

MATH G4151/4152: Analysis and probability/Analysis II

MATH G6351G6352: Applications of harmonic analysis
 MATH G6430G6431: Mathematical physics
Applied Mathematics Faculty
Daniel Bienstock, IEOR & APAM
Qiang Du, Applied Mathematics Program Committee Chair & Applied Mathematics Doctoral Committee CoChair
Eitan Grinspun, CS & APAM
Kyle Mandli
Lorenzo Polvani, APAM & EESC  Applied Mathematics Doctoral Committee CoChair
Donsub Rim
Christopher Scholz, EESC & APAM
Adam Sobel, APAM & EESC
Marc Spiegelman, APAM & EESC
Michael Tippett
Michael Weinstein, APAM and Department of Mathematics
Chris Wiggins
Drew Youngren
EESC: Department of Earth & Environmental Sciences
IEOR: Department of Industrial Engineering & Operations Research
Multidisciplinary and External Advisors
Admitted students may work with scientific advisors external to the applied mathematics faculty, as long as student secures funding from said faculty. Examples of other faculty in areas closely related to applied mathematics include:
Larry Abott, Neuroscience
Brian Cairns, NASA/GISS and Adjunct Assoc. Professor, APAM
Vittorio Canuto, NASA/GISS and Adjunct Professor, APAM
Barbara Carlson, NASA/GISS and Adjunct Professor, APAM
Panagiota Daskalopoulos, Mathematics
Anthony Del Genio, NASA/GISS and Adjunct Assoc. Professor, APAM
Julien Dubedat, Mathematics
Donald Goldfarb, IEOR
Timothy Hall, NASA/GISS and Adjunct Professor, APAM
Ron Miller, NASA/GISS and Adjunct Professor, APAM
Ben O'Shaughnessy, Chemical Engineering
Ovidiu Savin, Mathematics
Bruce Shaw, LDEO
Edward Spiegel, Astronomy