## Research Interests

Broadly speaking, I am interested 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 algebra 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

## 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