## Matthew Colbrook

You have 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 and contact details
(I'm always keen to work with new people). My CV can be found here.

From October 2020 I will be a Junior Research Fellow at Trinity College, Cambridge.

## 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