The PhD in Applied Mathematics at the University of Colorado Boulder is a research-focused doctorate training students to develop mathematical methods and computational tools for real-world problems across science, engineering and industry. It suits students with strong mathematical preparation who want to pursue original research in areas such as numerical analysis, partial differential equations, scientific computing, dynamical systems, stochastic processes and inverse problems.
What you'll study
The PhD programme combines advanced coursework, qualifying examinations, teaching experience and an extended original research project leading to a doctoral dissertation. Core areas of study include numerical analysis and scientific computing, theory and analysis of partial differential equations, dynamical systems and nonlinear analysis, probability and stochastic processes, optimisation and control, and inverse problems and data assimilation.
- Advanced coursework in real and functional analysis, numerical linear algebra, spectral methods and finite element methods.
- Specialist modules covering topics such as computational PDEs, multiscale methods, high-performance computing for scientific applications, uncertainty quantification, and statistical learning for physical systems.
- Seminars and reading courses aligned with faculty research—examples include mathematical biology, fluid dynamics, wave propagation, inverse problems in imaging and geophysics, and stochastic modelling of complex systems.
- Qualifying or preliminary examinations to assess mathematical breadth and readiness for research; students typically demonstrate competence in both theory and computation.
- Original research under the supervision of a faculty advisor, culminating in a written dissertation and public defence. Collaborative projects with departments such as physics, engineering, geosciences, computer science and national laboratories are common.
- Teaching assistantships or instructional experiences, which develop communication and pedagogical skills.
Entry requirements
Successful applicants typically hold a strong bachelor’s degree with honours in mathematics, applied mathematics or a closely related discipline; many entrants also hold a master’s degree. Prior coursework in advanced calculus, real analysis, linear algebra, differential equations and numerical methods is expected, along with demonstrated competency in programming and scientific computing.
- Academic transcripts showing strong performance in mathematics and quantitative courses.
- A statement of purpose outlining research interests and fit with prospective supervisors in the department.
- Two or three academic references who can speak to the applicant’s mathematical ability and research potential.
- A curriculum vitae listing relevant coursework, projects, publications or research experience.
- Some evidence of mathematical maturity such as independent research, advanced projects or published work is advantageous.
The department assesses applications holistically; specific requirements such as standardised tests or minimum grade thresholds may vary and applicants should consult the department’s admissions pages for current details.
Career prospects
Graduates of the PhD in Applied Mathematics pursue research and leadership roles across academia, government and industry. Typical career paths include:
- Academic positions in mathematics, applied mathematics and allied departments, involving research and teaching.
- Research scientist roles at national laboratories and research institutes where mathematical modelling, simulation and uncertainty quantification are central.
- Data scientist, quantitative analyst or machine learning engineer positions in technology, finance, energy and consulting sectors that require strong mathematical modelling and computational skills.
- R&D and computational modelling roles in aerospace, climate science, geophysics, biomedical engineering and imaging industries.
- Technical leadership and interdisciplinary collaboration roles, leveraging expertise in numerical methods, optimisation and large-scale computation.
Why study at University of Colorado Boulder
The Department of Applied Mathematics at the University of Colorado Boulder offers a collaborative, interdisciplinary environment with strong links to neighbouring research centres and national labs. Students benefit from access to high-performance computing resources, a broad spectrum of research expertise across theory and computation, and opportunities to work with colleagues in physics, engineering, geosciences and computer science.
- Proximity to national research institutions and laboratories provides avenues for collaborative projects and practical applications of mathematical research.
- A supportive doctoral training environment with seminars, specialised reading groups and regular colloquia featuring international researchers.
- Opportunities for funded assistantships and fellowships that combine research and teaching experience.
- Emphasis on transferable computational and quantitative skills that prepare graduates for diverse career paths in academia, government and industry.
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