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Matthew Colbrook

You've reached the webpage of Matthew Colbrook, a final year PhD student at the University of Cambridge (UK). Here you can find info on things like interests, publications, numerical code and contact details (I'm always keen to work with new people).


Research Interests

Broadly speaking, I am intersted in mathematical and numerical analysis. My work particularly concerns the existence and development of algorithms related to spectral theory, solutions of PDEs and inverse problems/optimisation.

More specifically:

  • Spectral theory and the existence (or non-existence) of algorithms to solve computational problems in infinite dimensions. Infinite dimensional spectral problems are notoriously difficult. Part of my PhD is the development of a new resolvent based family of algorithms to tackle these problems.
  • The Solvability Complexity Index Hierarchy. I am one of the main contributors to the development of the SCI hierarchy, a classification of the difficulty of numerical problems. Part of my PhD is the extension of this hierarchy and developing new techniques to study it.
  • The quest for stability in deep learning and neural networks. Recent results show neural networks are in general unstable. This is not just a problem in image classification but also in image denoising and reconstruction (where stable methods already exist), and is attracting an increasing amount of interest/awareness. I seek to fix this in the context of noisy subsampling.
  • Spectral/pseudospectral methods and linear algbra in infinite dimensions.
  • Scattering problems and other problems related to computational fluid mechanics.


Main Academic Awards

IMA Lighthill-Thwaites (2nd) Prize 2019
Biannual international prize given by IMA to top early career (Ph.D./postdoc) applied mathematicians
(see here for details).

Smith-Knight/Rayleigh-Knight Prize 2018
Grade I (top grade). Best first-year graduate piece of work in Mathematics Department of University of Cambridge.
(see here for details).

Mayhew Prize 2016
Best masters degree in Department of Applied Mathematics and Theoretical Physics, University of Cambridge
(see here for details).

Caltech SURF Scholarship 2015
(see here for details).


Cambridge-Imperial Computational PhD Seminar

Introducing the first Cambridge-Imperial Computational PhD Seminar Series. This will occur twice per term featuring a host of student talks and guest plenary speakers. Details can be found here. If you want to give a talk, please get in touch!


Contact Details

Email: m[dot]colbrook[at]damtp.cam.ac.uk

Address: Department of Applied Mathematics and Theoretical Physics (DAMTP), University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK