Spectral theory, computation, and computability
Spectra, spectral measures, generalized eigenfunctions, optimal algorithms, and the limits of computation in infinite dimensions.
Matthew Colbrook
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.