Columbia University

What you'll study

The programme combines advanced coursework, qualifying examinations and original dissertation research. In the early years students normally take core and elective courses to build depth in analysis, computation and modelling, then move into focused research with a faculty advisor.

• Core topics: advanced real and functional analysis, partial differential equations, numerical analysis and scientific computing, numerical linear algebra.
• Electives and applied areas: stochastic processes and applied probability, optimisation and control, computational statistics and machine learning, computational biology, fluid dynamics, imaging and inverse problems, mathematical physics and dynamical systems.
• Computational skills: high-performance computing, numerical simulation, algorithm design and software development for scientific computing.
• Assessment: written and/or oral qualifying examinations to demonstrate mastery of core material, regular progress reviews, publication in peer-reviewed venues and a defended doctoral dissertation presenting original research.
• Teaching and professional development: many students gain teaching experience as instructors or teaching assistants and can access workshops on pedagogy, grant writing and career planning.

Entry requirements

Applicants are expected to demonstrate strong preparation in mathematics and related quantitative subjects. Typical qualifications include a bachelor’s degree in mathematics, applied mathematics, physics, engineering or a closely related discipline; many successful applicants hold a master’s degree or equivalent research experience.

• Academic transcripts demonstrating rigorous coursework in calculus, linear algebra, real analysis, differential equations and numerical methods.
• Research experience or evidence of mathematical maturity, such as independent projects, publications, or strong letters of recommendation from academics or research supervisors.
• A concise statement of purpose describing research interests and potential faculty mentors, a curriculum vitae and academic references.
• Standardised tests: policies may vary; applicants should consult the department for current guidance on GRE subject or general test submission. International applicants must satisfy English-language proficiency requirements where applicable.

Career prospects

Graduates from Columbia’s applied mathematics doctoral programme move into a broad range of careers. Many pursue academic appointments in mathematics, engineering and computational science departments. Other common destinations include research positions in national laboratories, technology companies and financial institutions, roles in data science, quantitative research, optimisation and modelling in industry, and leadership positions in R&D organisations.

The programme’s emphasis on both rigorous analysis and computational implementation equips graduates to tackle interdisciplinary problems in areas such as machine learning, computational biology, climate and geophysical modelling, medical imaging and computational finance.

Why study at Columbia University

Columbia offers access to an active research environment in New York City, with collaborative opportunities across departments and affiliated institutes in engineering, computer science, biology and finance. Faculty in the Department of Applied Physics and Applied Mathematics and related groups are engaged in cutting-edge research spanning theoretical analysis, numerical methods and data-driven modelling.

• Interdisciplinary research culture with connections to engineering schools, medical centres and industry partners in the city.
• Robust computing resources and support for high-performance numerical work and software development.
• Strong mentoring and professional-development resources tailored to academic and non-academic career paths.