Applied Mathematics Program

Applied mathematics connects mathematical concepts and techniques to various fields of science and engineering. The objective of applied mathematical research is not only to intelligently apply existing mathematical tools and insights to solve scientific problems, but also to develop novel and useful mathematics inspired and driven by the applications. With the emergence of new computational technology, applied mathematics transcended its traditional style, and now assumes an even greater importance and a new vitality.

While the interdisciplinary nature of applied mathematics is vital to its success in modern science and technology, its unique mission makes it different from the areas it connects to. 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 is more concerned with the formulation of problems and the nature of solutions. Compared with the computer scientist, he 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.

Our faculty are leaders in the fields of applied mathematics and atmospheric science.

The core group of faculty members of the Applied Mathematics Program, many of whom have joint appointments over different departments in SEAS, have research expertise covering mathematical analysis, partial differential equations, numerical analysis, probability, dynamical systems, multiscale modeling, high performance scientific computation, and numerical optimization with applications in optics and photonics, material science, machine learning, data science, imaging science, biology, and climate modeling, to name some examples.

Daniel Bienstock: Applied mathematics, methodology and high-performance implementation of optimization algorithms, applications of optimization: preventing national-scale blackouts, emergency management, approximate solution of massively large optimization problems, higher-dimensional reformulation techniques for integer programming, robust optimization 

Liliana Borcea: Wave propagation in random media with applications to wave based imaging and free space optical communications; Inverse problems for hyperbolic, elliptic and parabolic partial differential equations; Data driven reduced order modeling and applications to inverse problems; Scientific computing.

Qiang Du: Numerical analysis, mathematical modeling and scientific computation with selected applications in physical, biological, materials, data and information sciences

Lorenzo M. Polvani: Atmospheric and climate dynamics, geophysical fluid dynamics, numerical methods for weather and climate modeling, planetary atmospheres

Kui Ren: Numerical analysis, scientific computation, applied analysis and partial differential equations, inverse problems and imaging, random graphs and networks, kinetic modeling and simulations

Adam H. Sobel: Atmospheric and climate dynamics, tropical meteorology, extreme weather

Marc W. Spiegelman: Advanced computation for multi-physics problems with applications to coupled fluid-solid mechanics in Earth Sciences (e.g. magma dynamics, carbon sequestration

Michael Tippett: Predictability and variability of the climate system, with emphasis on the application of statistical methods to data from observations and numerical models

Michael I. Weinstein: Applied and fundamental mathematics, partial differential equations, multi-scale analysis, dynamical systems; waves in nonlinear, inhomogeneous and random media; applications to optics and photonics, quantum and fluid systems

Chris H. Wiggins: Applied mathematics, mathematical biology, biopolymer dynamics, soft condensed matter, genetic networks and network inference, machine learning

Qihao Ye: Numerical Analysis, Scientific Computing, Applied and Computational Mathematics, Partial Differential Equations, Inverse Problems, Fast Algorithms, Machine Learning, Formal Reasoning, AI-Assisted Mathematical Research

Drew Youngren: Microlocal Analysis, Partial Differential Equations, Mathematics Education

Learn more about our faculty's cross-cutting research addresses key and emerging areas in society, such as energy, environment, and health:

1. Applied mathematics is the nexus of the quantitative sciences. Its powerful abstractions (algebraic systems, differential systems, geometrical descriptions, probability distributions, discrete structures, etc.) are fed by all of the other quantitative disciplines; its results, developed in one area, inform all of the other areas that share the same abstractions. Applied mathematicians can therefore choose to work anywhere along the spectrum from collaborative teamwork in the applications to autonomous research on the abstractions, and freely move back and forth on this spectrum.
2. Applied mathematicians and computational scientists are essential to advances in science and engineering from aerodynamics to biomedicine, from geophysics to materials science, from quantum chromodynamics to webpage ranking.
3. The applied mathematics curriculum at Columbia is highly customizable with elective chains or minors in diverse applications of the student’s choice.
4. The applied mathematics BSE degree is easily combined with an MS degree in five years at Columbia, or, with careful planning and advanced placement, in just four years.
5. Applied mathematics is a strategic major for professional school applications, since, when combined with the appropriate set of electives or minor, the degree communicates quantitative proficiency essential to meeting conceptual challenges in business, law, or medical studies.
6. Applied mathematics is a strategic major for PhD program applications throughout the sciences and engineering.
7. Applied mathematics provides a strong foundation for careers involving risk management, a growth area in government and industry.
8. Applied mathematics is an excellent major for immediate entry into the financial industry.
9. The career prospects of applied mathematicians are robust because they can work independently of major experimental devices and large research programs. Their primary tool, the computer, is continually becoming more powerful and less expensive, meaning that over their careers an ever increasing degree of scientific and engineering work will be simulated first, and demonstrated second.
10. Mathematicians of all kinds are in demand as teachers and can find satisfaction in professional and voluntary roles as instructors, tutors, and consultants, at all stages of life, and in cultures all around the globe regardless of the local language and economy.