## Summary

Almost all the properties of everyday solids – their chemistry, electrical conductivity, strength, hardness, ductility, melting point, dielectric constant, refractive index, magnetic moment, and so on – depend on quantum mechanics. One can measure the density of a brick without using quantum theory, of course, but why does a brick have the density it does? The density depends on the sizes of the atoms, and the sizes of the atoms depend on the sizes of the electron orbits around those atoms. These orbits can only be described using quantum mechanical wavefunctions satisfying the Schrödinger equation. Most questions about molecules and solids work like this: as soon as you say "why" more than once or twice, you find yourself face-to-face with quantum theory.

The Schrödinger equation can only be solved by hand in the very simplest systems, so most of my work is highly computational. The technique with which I am most closely associated is the quantum Monte Carlo method, in which massively parallel computers are used to carry out huge numbers of idealised "experiments", averages of the results of which yields the results of real experiments almost exactly. In large systems, in particular, quantum Monte Carlo methods are far more accurate than any other available approach. My group was among the first to apply quantum Monte Carlo techniques to real solids and remains one of the leading groups in the field. The main drawback of quantum Monte Carlo simulations is that they are very hard to do and the range of tractable physical problems is correspondingly small. Since I am interested in physics as well as computational methods, I also use simpler approaches such as density-functional theory when appropriate.

Recent quantum Monte Carlo work has included studies of warm dense matter (in collaboration with Michael Bonitz of Kiel University and James Shepherd of the University of Iowa) and the first attempt to learn the wavefunctions of real molecules using deep neural networks (in collaboration with Alex Matthews, David Pfau and James Spencer of Deepmind Ltd.). During the last few years, working with Andrew Horsfield of the Department of Materials, I have also become interested in extending the quantum mechanical approaches used to study solids beyond the Born-Oppenheimer approximation, taking proper account of the exchange of energy between spins, electrons, and ions.

When not researching or answering email, most of my working life is spent teaching. In recognition of the quality of my lecture courses on quantum physics and differential equations, I was recently given an Imperial College teaching award.

## Publications

### Journals

Fagerholm E, Gallero-Salas Y, Foulkes W, et al. , 2021, Neural systems under change of scale, *Frontiers in Computational Neuroscience*, ISSN:1662-5188

Pfau D, Spencer JS, Matthews AGDG, et al. , 2020, Ab-initio solution of the many-electron Schrödinger equation with deepneural networks, *Physical Review Research*, Vol:2, ISSN:2643-1564

Fagerholm ED, Foulkes W, Gallero-Salas Y, et al. , 2020, Conservation laws by virtue of scale symmetries in neural systems, *Plos Computational Biology*, Vol:16, ISSN:1553-734X

Azadi S, Foulkes WMC, 2019, Efficient method for grand-canonical twist averaging in quantum Monte Carlo calculations, *Physical Review B: Condensed Matter and Materials Physics*, Vol:100, ISSN:1098-0121