# Research

My main interests are in approaching mathematical physics from a rigorous standpoint, but I aim to keep a broad mathematical reading. My arXiv page is here, and my ORCid page is here.

## Classical Field Theory

My main area of research so far has been in using the modern theory of partial differential equations to investigate the theory of interacting electric charges, using the Schrödinger–Maxwell equations. The overall aim of the project is to understand the effects of the process known as charge shielding, which explains that the observed charge of a charged particle is different from its actual charge, and dependent on the distance separating it from the observer.

I have proven a considerable extension of a known existence result for compact subsets of three-dimensional space, generalising it to any compact manifold. More recently, I have been studying some of the solutions of the low-dimensional non-compact cases, and in particular, the existence of what could be called a one-dimensional hydrogen atom. In this situation, I have proven that there is a bound state for any potential satisfying certain reasonable integrability conditions.

## Quantum Field Theory

My Part III essay was on $$H \to \gamma\gamma$$, a discussion of methods of regularisation in the one-loop calculation of a standard-model Higgs boson decay into two photons. It may be found here.

I also more recently considered further some results that I noticed while writing this, in order to prove important asymptotic results about the behaviour of the regularised integrals in the technique of Loop Regularisation.

## History of Mathematics

### Robert Woodhouse

In the summer of 2012, supervised by Dr Piers Bursill-Hall, I studied the influence of Robert Woodhouse (Lucasian Professor 1820–2, Plumian Professor 1822–7) on the adoption of Continental-style analysis and reasoning in Cambridge (and hence England) in the early nineteenth century.

To place this in the context of people of whom you have heard, his successors in this respect were Babbage, Peacock, de Morgan and the Analytical Society, whose work swiftly resulted in Eulerian calculus being adopted into the Tripos. (You observe that the Tripos being woefully out of date is by no means a new phenomenon.) Woodhouse very likely had contact with these people early in their careers in Cambridge, and had written the first book on the subject in English: The Priciples of Analytical Calculation (published 1803). He also wrote a famous Calculus of Variations text using European notation called A Treatise on Isoperimetrical Problems and the Calculus of Variations, noted for its historical summary and comparison of dots and “d”s in the introduction.

The essay I wrote may be found here.

### G. H. Hardy

My interest in Hardy is catalogued on my personal website.

### Riemann

For several years I have given at least one talk on the life and mathematics of Riemann, one of the most interesting, profound, original, and misunderstood mathematicians of the nineteenth century, if not all time.

### Groups before Group Theory

More precisely, why did groups become and important thing to study abstractly? The answer is profoundly peculiar, and disappointingly little-known, mainly because it considers topics that are now unfashionable, and very few nonspecialists ask the right questions. While Galois is significant in the story, he is far from the only figure involved, and he did not simply “discover group theory”, as the popular accounts often suggest.

## Integration

I have for some years been interested in both definite integrals and integration in general. I must quote G.H. Hardy, in a letter of November 1926 to D. Coxeter, detailing the evaluation of some integrals the latter requested in the Mathematical Gazette:

I tried very hard not to spend time on your integrals, but to me the challenge of a definite integral is irresistible.

I now have the beginnings of a page on integration here.