University of Hawaii at Manoa

What you'll study

The Computational Mathematics PhD curriculum blends advanced coursework, research seminars and a sustained original research project leading to a dissertation. Core topics include numerical linear algebra, numerical solution of partial differential equations, scientific computing, numerical optimization, and probabilistic and statistical methods for computation. Students also study supporting areas such as functional analysis, approximation theory, numerical methods for dynamical systems, and high-performance computing techniques.

• Core coursework: advanced numerical analysis, numerical linear algebra, and numerical PDEs.
• Computational tools: parallel computing, GPU programming, scientific software development and numerical libraries.
• Applied and interdisciplinary modules: computational fluid dynamics, inverse problems and optimization, data assimilation, machine learning for numerical applications.
• Theoretical foundations: real and functional analysis, approximation theory and stability/error analysis of algorithms.
• Research and seminars: regular departmental seminars, computational mathematics reading groups and a required dissertation based on original research.

Students typically complete a combination of required and elective courses in the first years, pass qualifying or comprehensive examinations as set by the department, and then focus on dissertation research under the supervision of a faculty advisor. Collaborative projects with other departments (for example, oceanography, physics, engineering, or computer science) and access to university or regional high-performance computing resources are common components of research training.

Entry requirements

Applicants are normally expected to hold a strong bachelor’s or master’s degree in mathematics, applied mathematics, computational science, engineering, physics, or a closely related discipline. Successful candidates typically demonstrate substantial undergraduate preparation in real analysis, linear algebra, differential equations and numerical analysis. Programming experience and familiarity with numerical software or high-performance computing are highly desirable.

• Academic transcripts: evidence of high academic achievement in relevant coursework.
• Letters of recommendation: typically two or three academic references who can speak to research potential.
• Statement of purpose: a clear research statement describing interests in computational mathematics and potential faculty mentors.
• Preparation: coursework in analysis and numerical methods; prior research experience or a master’s thesis strengthens applications.
• English language proficiency: required for international applicants in accordance with university policy.

Admission decisions also consider fit with departmental research strengths and availability of supervision and funding. Prospective applicants are encouraged to contact potential faculty supervisors before applying to discuss research alignment.

Career prospects

Graduates of this PhD programme pursue careers across academia, industry and government research labs. Typical roles include university faculty positions, postdoctoral researcher appointments, research scientist roles in national laboratories, and specialist positions in industry developing numerical methods and scientific software.

• Academia: tenure-track and teaching-focused roles, postdoctoral research in applied and computational mathematics.
• Industry: quantitative scientist, algorithm developer, data scientist, and software engineer roles in sectors such as aerospace, energy, finance, and technology.
• Government and national labs: computational modelling, simulation and analysis for environmental science, defence, and engineering projects.
• Interdisciplinary opportunities: collaborations with oceanography, meteorology, materials science and bioinformatics where high-fidelity simulations and data-driven computation are needed.

Why study at University of Hawaii at Manoa

The Department of Mathematics at the University of Hawaii at Manoa has active research groups in numerical analysis, scientific computing and applied mathematics, and offers opportunities for interdisciplinary collaboration across campus centres and institutes. Its location provides unique opportunities for computational research related to oceanography, geosciences and atmospheric science, with faculty who engage in regionally relevant modelling and large-scale simulation projects.

• Research environment: mentorship from faculty with expertise in computational methods and access to collaborative projects across science and engineering.
• Computational resources: access to university and state-supported computing facilities and partnerships for high-performance computing needs.
• Interdisciplinary collaborations: strong links with applied departments and research centres addressing environmental and computational challenges specific to the Pacific region.
• Supportive setting: a department that emphasises student research development, seminar culture and opportunities to present at conferences and workshops.

Prospective students should review faculty research interests and contact potential advisors to ensure alignment of research priorities and funding possibilities prior to applying.