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 solid-earth 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, 1987-1988 SEAS Bulletin)

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

M.S. Program Requirements

This 30-point 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 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
     
  • Any other 4000, 6000 or 8000-level APMA course not listed here


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:

  • C0MS W4236: Introduction to computational complexity


* Industrial engineering/operations research elective courses include:

  • IEOR E4004: Introduction to operations research: deterministic models
  • IEOR E4106: Introduction to operations research: stochastic models
  • SIE0 W4150: Introduction to probability and statistics
  • IEOR E4403: Advanced engineering and corporate economics
  • IEOR E4700: Introduction to financial engineering

*Please check IEOR website for registration procedures required of non-IEOR students   
http://ieor.columbia.edu/registration-guidelines-non-ieor-students


Other elective courses include:

  • MSAE E4215: Mechanical behavior of structural materials

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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 30-point 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

  • 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 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

 



Applied Mathematics Faculty

Guillaume Bal
Daniel Bienstock, IEOR & APAM
Mark Cane, EESC & APAM
Qiang Du, Applied Math Program Committee Chair & Applied Mathe Doctoral Committee Co-Chair
Eitan Grinspun, CS & APAM
Kyle Mandli
Lorenzo Polvani, APAM & EESC - Applied Math Doctoral Committee Co-Chair
Vincent Quenneville-Bélair
Christopher Scholz, EESC & APAM
Adam Sobel, APAM & EESC
Marc Spiegelman, APAM & EESC
Michael Tippett
Michael Weinstein, APAM and Department of Mathematics
Chris Wiggins

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

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