Columbia University

What you'll study

The programme emphasises both theoretical and applied aspects of computational mathematics. Core study typically covers advanced real and complex analysis, numerical analysis, scientific computing, and the theory of partial differential equations. Students build practical skills in algorithm design, high-performance computing, numerical linear algebra, optimisation and computational statistics.

• Advanced real analysis and functional analysis to provide rigorous foundations for numerical methods
• Numerical linear algebra, eigenvalue problems and large-scale matrix computations
• Numerical methods for ordinary and partial differential equations, finite element and spectral methods
• Scientific computing, parallel algorithms and use of high-performance computing environments
• Computational optimisation and numerical stability/accuracy analysis
• Computational statistics, machine learning methods and data-driven modelling
• Electives and seminars drawn from applied mathematics, computer science, statistics and engineering for interdisciplinary breadth
• Capstone options including a supervised research project, master's thesis or applied computational practicum

Students can often tailor their programme by choosing electives in areas such as mathematical finance, imaging and inverse problems, computational biology, or scientific machine learning, and by collaborating with faculty across departments.

Entry requirements

Applicants are expected to hold a bachelor's degree or its equivalent, typically in mathematics, applied mathematics, computer science, engineering, physics or a related quantitative discipline. A strong background in undergraduate analysis, linear algebra and programming is normally required.

• Official transcripts from all post‑secondary institutions attended
• Academic references (usually two or three letters) that can speak to mathematical ability and research potential
• A statement of purpose outlining academic interests, relevant experience and intended goals
• A curriculum vitae or résumé detailing research, coursework and programming experience
• Proof of English language proficiency for applicants whose first language is not English (international examinations or equivalent evidence)

Standardised tests such as the GRE may be optional or considered on a case-by-case basis; candidates with strong quantitative coursework, research experience or relevant industry experience are competitive. Applicants with gaps in course preparation may be advised to take additional undergraduate courses before or during the programme.

Career prospects

Graduates with a computational mathematics master's from Columbia pursue a wide range of careers that leverage mathematical modelling, numerical methods and computing skills. Typical paths include:

• Quantitative roles in finance and risk management, including modelling and algorithmic trading
• Data scientist and machine learning engineer positions in technology and start-ups
• Research and development roles in engineering firms, national laboratories and simulation-driven industries (energy, aerospace, materials)
• Computational scientist or software developer for high-performance computing applications
• Academic careers and further study: many graduates continue to PhD programmes in applied mathematics, computational science or related fields

The programme’s location and Columbia’s connections to industry, research centres and finance provide regular opportunities for internships, collaborative projects and networking with employers.

Why study at Columbia University

Columbia offers a strong mathematics department with faculty active across pure and applied areas, including experts in numerical analysis, PDEs, scientific computing and data-driven modelling. Students benefit from interdisciplinary collaborations with the Data Science Institute, engineering and statistics departments, and access to computational resources and research centres.

• Exposure to cutting‑edge research and regular seminar series linking theory and applications
• Opportunities to work with faculty on funded research projects and cross‑departmental initiatives
• Proximity to New York City's financial, technology and research industries for internships and employment
• Supportive graduate community and resources, including computing clusters and library collections tailored to quantitative research

These features make Columbia a compelling choice for students seeking rigorous training in computational mathematics combined with practical experience and strong industry and academic links.