University of Colorado Boulder

What you'll study

This programme emphasises numerical methods, algorithm development and mathematical modelling for large-scale computational problems. Core study areas typically include numerical analysis, scientific computing, computational linear algebra, numerical solutions of differential equations, optimisation and uncertainty quantification. Coursework also covers related foundations such as real analysis, probability and statistics as needed for computational work.

• Numerical Analysis and Scientific Computing: finite-difference and finite-element methods, error analysis, stability and convergence.
• Computational Linear Algebra: direct and iterative solvers, preconditioning, sparse matrix methods for large systems.
• Numerical PDEs and Modelling: techniques for time-dependent and steady-state partial differential equations used in physics and engineering models.
• Optimisation and Inverse Problems: algorithms for constrained and unconstrained optimisation, parameter estimation and regularisation techniques.
• High-performance and Parallel Computing: practical implementation on multi-core CPUs and GPUs, code profiling, parallel algorithms and use of scientific computing libraries.
• Data-driven Methods: numerical aspects of machine learning, statistical modelling, Bayesian computation and uncertainty quantification.

Students typically combine coursework with substantial programming and project work using languages and environments such as Python, MATLAB, C/C++ and parallel computing frameworks. The programme often offers both thesis and non-thesis (project) options, allowing students to pursue independent research under faculty supervision or to complete a substantial applied computational project.

Entry requirements

Applicants should hold a good bachelor’s degree in mathematics, applied mathematics, computer science, engineering or a closely related quantitative discipline. The department looks for evidence of strong quantitative preparation, including coursework in calculus, linear algebra, differential equations and mathematical analysis. Prior exposure to numerical methods and programming is strongly recommended.

• Academic record: a competitive undergraduate GPA in a quantitative subject.
• Mathematical background: coursework in advanced calculus/analysis, linear algebra and differential equations; numerical analysis desirable.
• Programming skills: experience with scientific programming (Python, MATLAB, C/C++, or similar) is expected.
• Supporting documents: statement of purpose outlining research or career goals, academic transcripts and letters of recommendation. International applicants will need to demonstrate English language proficiency as required by the university.

Some applicants may be advised to take preparatory coursework if their background is less directly aligned to computational mathematics. Admissions are competitive and evaluated holistically.

Career prospects

Graduates are prepared for technically demanding roles that require strong mathematical modelling and computational skills. Typical career paths include:

• Computational scientist/engineer: developing and validating simulation codes for industry, government laboratories or research institutes.
• Data scientist/machine learning engineer: applying numerical methods and statistical modelling to large-scale data problems.
• Quantitative analyst or risk modeller: positions in finance and insurance that require numerical optimisation and stochastic modelling.
• Software developer for scientific applications: building high-performance software tools and libraries for computational research.
• Academic and research careers: continuation to PhD study in applied mathematics, computational science or related disciplines.

Graduates benefit from Boulder’s active technology and research ecosystem, which supports internships and collaborative projects with industry and national laboratories.

Why study at University of Colorado Boulder

The Department of Mathematics at the University of Colorado Boulder has strong research groups in numerical analysis, scientific computing, mathematical modelling and applied analysis. Students gain access to interdisciplinary collaborations across engineering, computer science, physics and earth sciences, enabling applied projects that address real-world problems.

• Research and facilities: opportunities to work with faculty on computational research and to use campus high-performance computing resources and scientific software stacks.
• Interdisciplinary links: close collaborations with engineering departments, computer science, environmental sciences and nearby national research centres, facilitating applied projects and internships.
• Supportive academic environment: small class sizes in advanced courses, dedicated supervision for thesis or project work, and seminars that connect students with current research and industry speakers.
• Location and networks: Boulder’s active tech and research community offers networking and employment opportunities in scientific computing and data-driven industries.

These features make CU Boulder a strong choice for students seeking rigorous training in computational mathematics and pathways into research, industry or further graduate study.