University of Chicago

What you'll study

The PhD programme emphasises a blend of theoretical foundations and practical computation. Early years centre on coursework to build depth in analysis and breadth in computational topics, followed by qualifying examinations and a transition to independent research leading to a dissertation. Students typically work closely with a faculty advisor to define a research programme in areas such as numerical analysis, scientific computing, numerical linear algebra, computational partial differential equations, optimisation and inverse problems, uncertainty quantification, and algorithms for high-performance computing.

Programme structure

• Coursework in core subjects (real and functional analysis, advanced linear algebra, numerical analysis) and elective courses in computational disciplines.
• Qualifying examinations or equivalent milestones to demonstrate readiness for research.
• Reading courses and seminars aligned with the chosen research area.
• Independent research under a faculty advisor, culminating in a written dissertation and oral defence.
• Opportunities for collaboration with neighbouring centres and labs on computational projects, internships and joint appointments.

Typical modules and topics

• Real analysis and functional analysis for PDEs and operator theory
• Numerical linear algebra and iterative methods
• Finite element, finite difference and spectral methods for PDEs
• Scientific computing and high-performance computing techniques
• Numerical optimisation and convex/non-convex methods
• Uncertainty quantification and stochastic numerical methods
• Inverse problems and regularisation techniques
• Computational statistics and machine learning for large-scale problems

Entry requirements

Applicants are expected to have a strong undergraduate degree (or equivalent) in mathematics, applied mathematics, or a closely related quantitative discipline. Many successful applicants hold a master’s degree, but a bachelor’s degree with exceptional preparation may also be sufficient. Adequate preparation typically includes coursework in:

• Real analysis and advanced calculus
• Linear algebra and differential equations
• Numerical analysis or scientific computing
• Probability and/or mathematical statistics (recommended for certain specialisations)

Applications normally require academic transcripts, a personal statement outlining research interests, three or more letters of recommendation from academic referees, and a curriculum vitae. International applicants may need to demonstrate English proficiency through recognised tests where English is not the primary language of instruction. The department assesses applicants on mathematical potential, research preparedness, and fit with faculty research interests.

Career prospects

Graduates of the Computational Mathematics track pursue careers across academia, national laboratories, and industry. Typical paths include:

• Academic positions in mathematics, applied mathematics and computational science departments as postdoctoral researchers and faculty.
• Research scientist and computational researcher roles in national laboratories and government research facilities, often working on large-scale simulation and modelling problems.
• Industry roles in quantitative finance, data science, machine learning engineering, optimisation and algorithm development for technology companies.
• Positions in engineering firms, energy, aerospace and bioinformatics where modelling, simulation and high-performance computing are central.
• Entrepreneurial and applied research roles in start-ups focused on numerical software, scientific platforms and computational tools.

Why study at University of Chicago

The University of Chicago Department of Mathematics is known for rigorous training and close mentorship, which is particularly valuable for computational mathematicians who need a strong analytical foundation. Students benefit from interdisciplinary connections across the university, including collaborative links with the Computation Institute and partnerships with national laboratories that support joint projects and access to advanced computing facilities.

Research-active faculty cover a range of computational topics and supervise doctoral work across theory, algorithms and large-scale implementation. The programme’s environment emphasises seminar culture, regular colloquia, and opportunities to participate in cross-department research initiatives, preparing graduates for both theoretical and applied careers in computational science.