University of Michigan

What you'll study

The programme combines advanced coursework, qualifying examinations and an extended original research project leading to a doctoral thesis. Core study areas emphasise numerical analysis, scientific computing, and the theory and practice of algorithms for partial differential equations, optimisation, stochastic simulation, and high-performance computing.

• Core topics: numerical linear algebra, error analysis, stability and convergence theory, finite element and finite difference methods, spectral methods.
• Computational topics: scalable algorithms for large-scale PDEs, domain decomposition, multigrid methods, fast solvers, time-stepping schemes, and uncertainty quantification.
• Theoretical foundations: functional analysis, approximation theory, probability for computation, and numerical aspects of optimisation.
• Complementary areas: scientific machine learning, data assimilation, inverse problems, computational geometry, and software engineering for scientific codes.

Typical doctoral progression includes one to two years of coursework (tailored to each student's background), passing a written and/or oral qualifying examination, progressing to candidacy, and then concentrated research under the supervision of one or more faculty advisors. Students are expected to present research at seminars, participate in reading groups and workshops, and often collaborate with research groups across engineering, computer science, and applied sciences.

Entry requirements

Applicants are normally expected to hold a strong undergraduate degree in mathematics, applied mathematics, or a closely related discipline; many successful applicants will also hold a master’s degree. Strong preparation in real and complex analysis, linear algebra, differential equations, and numerical methods is expected. Programming experience and familiarity with scientific computing environments are advantageous.

• Academic record: evidence of high achievement in advanced mathematics courses and relevant computational subjects.
• Research potential: prior research experience, publications, project work or detailed research statements strengthen an application.
• References: at least two or three academic references who can comment on the applicant’s mathematical ability and research promise.
• Additional materials: a statement of purpose outlining research interests, transcripts, and where applicable, GRE scores or language proficiency evidence in line with the university’s requirements.

Career prospects

Graduates from the Computational Mathematics stream move into a wide range of careers that apply rigorous mathematical and computational skills. Many continue in academia as postdoctoral researchers and faculty in mathematics, applied mathematics, engineering and computer science. Others join research laboratories or industry roles that require advanced modelling and algorithmic expertise.

• Academic and research positions in universities and national laboratories.
• Quantitative and modelling roles in finance, energy, climate science and engineering firms.
• Data science, machine learning and algorithm development positions in technology companies.
• Software engineering and high-performance computing roles focused on scientific applications and simulation codes.

Why study at University of Michigan

The University of Michigan offers a strong, research-intensive environment with faculty working across numerical analysis, scientific computing and interdisciplinary applications. Students benefit from access to world-class computational resources, active seminar series, and collaboration opportunities with engineering, computer science, physics and applied sciences.

The department’s culture emphasises close mentorship, a broad choice of advanced courses and informal research groups, enabling doctoral students to tailor their training to theoretical, methodological or application-driven projects. Additionally, proximity to interdisciplinary research centres and industry partnerships provides ample opportunities for applied collaborations and professional development.