University of Arizona

What you'll study

The PhD programme combines advanced coursework, qualifying examinations, and original research culminating in a doctoral dissertation. Core topics emphasise numerical analysis, scientific computing, and the mathematics of algorithms used to solve large-scale problems arising from physics, engineering, data science and computational biology.

• Core coursework typically includes advanced numerical analysis, numerical linear algebra, scientific computing, approximation theory, and the theory of partial differential equations (PDEs).
• Supporting topics often cover optimisation, probability and statistics for computation, scientific machine learning, high-performance computing, and computational aspects of dynamical systems.
• Seminars and reading courses provide exposure to current research, techniques for implementing algorithms, and opportunities to present work-in-progress.
• Research training includes supervised projects with faculty in computational mathematics, preparation of research proposals, participation in interdisciplinary teams, and use of modern software and parallel computing environments.
• Qualifying examinations and dissertation require demonstration of mastery in core areas and the completion of an original research dissertation defended before a faculty committee.

Entry requirements

Applicants are expected to hold a strong undergraduate degree in mathematics or a closely related discipline; many successful applicants also have a masters degree. Typical preparation includes rigorous coursework in real analysis, linear algebra, differential equations, and numerical methods, plus programming experience in languages commonly used for scientific computing (for example Python, C/C++, or MATLAB).

Application materials normally include academic transcripts, a statement of research interests, letters of recommendation, and a curriculum vitae. International applicants must demonstrate English proficiency according to the university's standard requirements. Admissions are competitive and assessed on the strength of prior training, research potential, and fit with faculty expertise.

Career prospects

Graduates of the programme pursue careers in a range of sectors where advanced computational and mathematical skills are required. Common pathways include:

• Academic positions (postdoctoral research and faculty) in mathematics, applied mathematics, and computational science.
• Research scientist roles at national laboratories and government research centres working on large-scale simulations and modelling.
• Industry positions in technology, engineering, energy, aerospace and finance focusing on algorithm development, numerical simulation, and data-driven modelling.
• Data science and machine learning roles that combine statistical insight with scalable computational methods.
• Software development for scientific computing, including high-performance and parallel computing applications.

Why study at University of Arizona

The University of Arizona hosts an active mathematics department with faculty working across numerical analysis, computational PDEs, optimisation and scientific machine learning, offering rich opportunities for collaboration. Strong ties with neighbouring departments such as computer science, engineering, earth sciences and astronomy enable interdisciplinary projects that connect mathematical theory with real-world applications.

Doctoral students benefit from access to campus computational resources and training in high-performance and parallel computing, regular research seminars and interdisciplinary centres, and opportunities for funding through research assistantships and teaching assistantships. The department's research environment and the university's location provide a supportive setting for developing the technical and professional skills needed for research and advanced careers in computational mathematics.