Case Western Reserve University

What you'll study

The programme combines advanced coursework, qualifying examinations and an extended original research project leading to a doctoral dissertation. Early-stage coursework typically covers core areas of applied mathematics such as numerical analysis, partial differential equations, applied functional analysis, scientific computing and optimization. Students choose elective modules or seminars tailored to research interests — examples include computational fluid dynamics, inverse problems and imaging, stochastic processes and uncertainty quantification, mathematical biology and bioinformatics, control theory, machine learning for scientific computing, and high-performance computing.

Training emphasises both theory and computation: students learn to formulate mathematical models, analyse qualitative behaviour, design and analyse numerical schemes, and implement scalable algorithms on modern computing architectures. The programme includes a sequence of research seminars and reading courses that expose students to current problems and methods. Typical elements of progression are a qualifying or preliminary examination to demonstrate readiness for research, formation of a dissertation committee, an oral proposal defence, and periodic dissertation progress reviews prior to the final defence.

Teaching experience is part of the doctoral training. Most students serve as teaching assistants or instructors for undergraduate mathematics courses, gaining experience in communication and pedagogy. Students are encouraged to present at internal seminars and national conferences and to publish in peer-reviewed journals.

Entry requirements

• Academic background: A strong bachelor's degree in mathematics, applied mathematics, or a closely related quantitative discipline is normally required; many successful applicants hold a master's degree or have substantial graduate-level coursework in analysis, differential equations, numerical methods and linear algebra.
• Transcripts and academic performance: Evidence of strong numerical and theoretical preparation in undergraduate and any graduate coursework. Applicants should highlight rigorous courses in real/complex analysis, ODEs/PDEs, numerical analysis, probability/statistics and linear algebra.
• Letters of recommendation: At least three academic references that can speak to research potential, mathematical maturity and readiness for a PhD programme.
• Statement of purpose: A clear research statement explaining interests, relevant experience (such as research projects, computational experience or publications) and fit with faculty research areas.
• Supporting evidence: Relevant research experience, publications, or strong programming/computational skills strengthen an application. Familiarity with scientific computing environments and languages (e.g. MATLAB, Python, C/C++, MPI) is advantageous.
• English language proficiency: Required for applicants whose first language is not English; standardised test scores or institutional equivalents are accepted according to university policy.
• Standardised tests: Applicants should consult the department for current policy on GRE general or subject tests; requirements may vary over time.

Career prospects

Graduates of the PhD in Applied Mathematics pursue diverse careers in academia, industry and the public sector. Common academic paths include postdoctoral appointments and faculty positions in mathematics, engineering, physics and computational biology departments.

Industry opportunities are broad: quantitative research and modelling roles in finance and risk management; data science and machine learning positions in technology companies; algorithm and simulation development in engineering, aerospace and energy firms; and computational work in biomedical companies and medical imaging. Graduates also work in national and international laboratories, government agencies, and start-ups that require advanced modelling, optimisation and uncertainty quantification skills.

Why study at Case Western Reserve University

Case Western Reserve University offers an applied mathematics PhD within a research-intensive environment that encourages interdisciplinary collaboration. The Department of Mathematics maintains active links with the Case School of Engineering, the medical school and local research institutions, enabling projects that span imaging and inverse problems, biomedical modelling, data-driven science and computational engineering. Proximity to clinical and industrial partners in Cleveland supports translational research, particularly in biomedical applications and imaging.

Students benefit from working with faculty whose expertise spans numerical analysis, PDEs, stochastic modelling, optimisation and scientific computing, and from access to institutional resources such as high-performance computing facilities and specialised research centres. The department fosters a collegial atmosphere with regular seminars, reading groups and opportunities to collaborate on funded research. Financial support packages, including teaching and research assistantships, are typically available to qualified doctoral students, allowing them to focus on research and professional development.