University of North Carolina at Chapel Hill

What you'll study

The programme combines advanced coursework with sustained original research. Early study typically covers foundational topics such as real and complex analysis, advanced linear algebra, and probability, alongside core computational courses in numerical analysis, numerical linear algebra, and numerical methods for partial differential equations. Students then move into specialised modules and seminars that reflect active faculty research areas, for example:

• Numerical methods for PDEs and finite element/finite volume techniques
• Scientific computing and high‑performance numerical algorithms
• Iterative methods and preconditioning for large linear and nonlinear systems
• Computational optimisation and inverse problems
• Uncertainty quantification, stochastic simulation and probabilistic numerical methods
• Data‑driven computation, numerical methods for machine learning and scientific data assimilation

Programme structure normally includes a period of structured coursework and seminars (commonly in the first two years), written and/or oral qualifying examinations, selection of an adviser and research group, a proposal or preliminary examination, and completion of a doctoral dissertation based on original research. Many students gain experience implementing algorithms on parallel and GPU architectures and engage with interdisciplinary projects in applied sciences, engineering and data science.

Entry requirements

Applicants are expected to hold a strong bachelor’s degree in mathematics, applied mathematics, statistics, computer science, engineering or a closely related discipline; many applicants present a master’s degree. Essential preparation includes rigorous training in real analysis and linear algebra and substantial experience with undergraduate or graduate numerical analysis and programming. Typical application materials include:

• Official academic transcripts demonstrating advanced coursework in mathematics and computation
• A statement of purpose outlining research interests and fit with faculty expertise
• Curriculum vitae or résumé
• Letters of recommendation from academic or professional referees able to attest to mathematical ability and research potential
• Evidence of programming experience and any prior research or project work

Admissions committees evaluate applicants holistically, with emphasis on mathematical maturity and research promise. International applicants must meet the University’s English language proficiency requirements.

Career prospects

Graduates of a computational mathematics PhD are prepared for careers in academia, national laboratories and research institutes, and in industry sectors that depend on large‑scale modelling and computation. Common career paths include:

• Research and teaching positions in universities and colleges
• Research scientist or applied mathematician roles in national laboratories and government research centres
• Quantitative and modelling roles in finance, energy and engineering firms
• Data science, machine learning and algorithm development positions in technology companies
• High‑performance computing, software development and scientific consulting

Many graduates combine theoretical skills with software development and parallel computing expertise, making them competitive for interdisciplinary teams tackling large computational challenges.

Why study at University of North Carolina at Chapel Hill

UNC Chapel Hill’s Department of Mathematics has a strong tradition in both pure and applied areas, and a critical mass of faculty working in computational mathematics and scientific computing. Students benefit from close mentorship, a collaborative graduate community and access to research groups spanning applied mathematics, statistics, computer science and engineering. The department fosters interdisciplinary collaboration with neighbouring institutions and research organisations in the Research Triangle, offering opportunities to work on practical problems and to access shared high‑performance computing resources.

Doctoral students typically have opportunities to teach undergraduate courses, participate in specialised seminars and workshops, and present work at national and international conferences. The department’s emphasis on rigorous analysis together with practical computation prepares graduates to pursue independent research careers or to take leadership roles in industry and government research organisations.