Spectral theory, computation, and computability
Spectra, spectral measures, generalized eigenfunctions, optimal algorithms, and the limits of computation in infinite dimensions.
Matthew Colbrook
Associate Professor in DAMTP, University of Cambridge
I am an Associate Professor in the Department of Applied Mathematics and Theoretical Physics at the University of Cambridge. My research lies at the interface of analysis, numerical analysis, and data-driven computation.
A recurring question in my work is how to turn infinite-dimensional mathematical objects and finite data into algorithms with convergence guarantees and error control, and how computation can be used as part of mathematical proof. This connects spectral theory and computability, Koopman operator learning, with applications across PDEs and mathematical physics.
Across these areas, I also develop and apply mathematical and computational tools that combine rigorous foundations with reliable, broadly useful methods for scientific and engineering problems.
This last year, I have become interested in Geometric spectral theory. You can read more about the programme I attended here:
Isaac Newton Institute GST Programme
Research
Research areas
Rigorous mathematics and reliable algorithms for infinite-dimensional and data-driven problems.
Rigorous numerics and computer-assisted proofs
Verified algorithms, residual-based certification, explicit error control, and computation designed to support proof.
Koopman operators and data-driven dynamics
Convergent methods for DMD, spectral analysis, forecasting, system identification, and model verification.
Optimization, inverse problems, and learning
Stable and accurate neural networks, unrolled and first-order methods, compressed sensing, and reliable recovery.
Background
Academic path
My work spans pure and applied mathematics, computation, and the foundations of reliable algorithms.
Current
Associate Professor, University of Cambridge
Department of Applied Mathematics and Theoretical Physics.
Previously
Assistant Professor, University of Cambridge
Research and teaching across mathematics and scientific computing.
Earlier
Trinity College, Cambridge
Junior Research Fellow.
Paris
École Normale Supérieure
Fondation Sciences Mathématiques de Paris Fellow.
2020
PhD, University of Cambridge
The Foundations of Infinite-Dimensional Spectral Computations, completed in September 2020.
Education
St John’s College, Cambridge
Earlier degrees in mathematics at the University of Cambridge.