University of Colorado Boulder

What you'll study

The programme blends theoretical foundations with computational practice, emphasising the formulation and solution of problems from physics, engineering, biology and data science. Students typically follow a mix of core and elective courses and may choose a thesis or non‑thesis (project/coursework) track.

• Core topics: advanced calculus and real analysis, linear and functional analysis, ordinary and partial differential equations, numerical analysis and scientific computing.
• Computational and applied modules: numerical linear algebra, finite element and finite difference methods, computational fluid dynamics, optimisation and control, stochastic modelling and Monte Carlo methods.
• Interdisciplinary electives: data science and machine learning, mathematical biology, dynamical systems and chaos, statistical modelling, inverse problems, and computational physics.
• Research/project work: thesis students undertake supervised research culminating in a master’s thesis; non‑thesis students complete a substantial project or additional coursework demonstrating applied competence.
• Seminars and colloquia: access to departmental seminars and interdisciplinary colloquia exposes students to current research in applied analysis, computation and modelling.

Entry requirements

Applicants are expected to hold a bachelor’s degree in mathematics, applied mathematics, physics, engineering, computer science or a closely related discipline, with substantial coursework in calculus, linear algebra, differential equations and probability or statistics.

• Academic background: strong performance in undergraduate mathematics courses; courses in real analysis and numerical methods are advantageous.
• Preparation: familiarity with programming (Python, MATLAB, or C/C++) and experience with mathematical software is recommended.
• Supporting documents: official transcripts, statement of purpose outlining research or career goals, and letters of recommendation.
• English language: applicants whose first language is not English must demonstrate proficiency through recognised tests, unless otherwise exempted by the university.
• Standardised tests: the programme’s requirements for tests such as the GRE vary; check the department guidance and consider contacting the admissions office if unsure.

Career prospects

Graduates move into a wide range of roles that require strong quantitative and computational skills. Common destinations include:

• Data scientist, quantitative analyst or machine learning engineer in finance, technology and start‑ups.
• Modeller or computational scientist in engineering, aerospace and energy sectors.
• Research or technical positions at national laboratories and government agencies, including environmental and atmospheric modelling organisations.
• Roles in scientific software development, simulation and high‑performance computing.
• Continuation to doctoral study in applied mathematics, computational science or related fields for careers in academia and long‑term research.

Why study at University of Colorado Boulder

CU Boulder has a well‑regarded mathematics department with active research groups in numerical analysis, partial differential equations, dynamical systems and computational mathematics. The university’s collaborative environment encourages cross‑disciplinary projects with engineering, physics, geosciences and computer science.

• Research connections: proximity to national research facilities and labs fosters opportunities for applied projects, internships and collaborations on real‑world problems.
• Computational resources: students have access to modern computing infrastructure and software used in scientific computing and data analysis.
• Industry and regional ecosystem: Boulder’s technology and start‑up ecosystem creates networking and employment opportunities for graduates with strong quantitative skills.
• Support for students: departmental mentoring, seminars and career services help students build research profiles, prepare for industry roles, or apply to PhD programmes.