University of Colorado Boulder

What you'll study

The PhD emphasises advanced coursework and original research in computational mathematics and scientific computing. Core topics typically include numerical analysis, numerical linear algebra, partial differential equations (PDEs) and their discretisations, numerical optimisation, uncertainty quantification, and high-performance computing. Students also study supporting theoretical areas such as real analysis, functional analysis, and probability or statistics as required for their research.

• Coursework and seminars: Advanced numerical analysis, methods for PDEs, computational linear algebra, optimisation and inverse problems, statistical computing, and graduate-level analysis courses. Departmental and research-group seminars expose students to current computational mathematics research.
• Qualifying and preliminary examinations: Students normally complete written and/or oral examinations to demonstrate breadth in mathematical foundations and readiness for dissertation research.
• Research and dissertation: After passing exams, students undertake supervised research culminating in a dissertation. Research topics often address algorithm design, rigorous error and stability analysis, scalable software for scientific computing, uncertainty quantification, and data-driven modelling.
• Interdisciplinary collaboration: Many students collaborate with researchers in physics, engineering, computer science, atmospheric and geosciences, or national laboratories on computational problems, leveraging campus and regional research centres.
• Teaching experience: Graduate students typically gain teaching experience through assistantships, developing communication and instruction skills.

Entry requirements

Competitive applicants hold a strong bachelor’s or master’s degree in mathematics, applied mathematics, or a closely related discipline (for example, physics, engineering or computer science) with substantial coursework in advanced calculus/analysis, linear algebra, differential equations and numerical methods. Demonstrated programming experience and familiarity with scientific computing are highly beneficial.

• Academic record: A strong academic transcript showing solid preparation in undergraduate and, if applicable, graduate mathematics.
• Research potential: Evidence of research experience or technical projects—such as a masters thesis, publications, or substantial computational project—strengthens an application.
• Supporting materials: Statement of purpose describing research interests, curriculum vitae, and strong letters of recommendation from academic referees.
• English language proficiency: Required for applicants whose first language is not English; standard institutional tests are accepted according to university policy.
• Direct entry considerations: Applicants with only a bachelor’s degree may be admitted if they show exceptional preparation; many entrants hold a master’s degree in a relevant field.

Career prospects

Graduates of the PhD in Computational Mathematics pursue careers across academia, national laboratories and industry. Typical paths include tenure-track or research faculty positions, postdoctoral research, and technical or research scientist roles at government and national labs.

• Academic research: Positions in mathematics, applied mathematics and computational science departments.
• National and government labs: Research scientist roles at national laboratories and agencies that require advanced numerical modelling and simulation expertise.
• Industry: Roles in data science, quantitative modelling, computational engineering, software development for scientific computing, finance, energy, aerospace and technology companies that rely on large-scale simulation and algorithm development.
• Interdisciplinary teams: Opportunities within interdisciplinary research centres, translating mathematical methods into applications in climate science, materials, imaging, and machine learning.

Why study at University of Colorado Boulder

The Department of Mathematics at the University of Colorado Boulder offers a collaborative research environment with easy access to interdisciplinary partners and regional research organisations. Students benefit from proximity to centres of computational research and national institutes as well as campus high-performance computing resources.

• Active research groups: Faculty work in numerical analysis, scientific computing, optimisation, uncertainty quantification and data-driven modelling, providing a broad range of potential advisors and projects.
• Interdisciplinary connections: Strong ties with physics, engineering, computer science, earth and atmospheric sciences, and nearby national labs support cross-disciplinary research and applied computational projects.
• Resources and training: Access to seminars, workshops, and institutional computing infrastructure helps students develop skills in algorithm development, parallel computing and scientific software engineering.
• Professional development: Teaching opportunities, conference travel support and a research-focused curriculum prepare graduates for academic and non-academic careers that require deep computational and mathematical expertise.