# J Bryce McLeod

### Independent obituary

Obituaries Index

#### Mathematician and leader in the discipline of applied analysis

When J Bryce McLeod was 10 his home city of Aberdeen was under threat of German bombs. As a result his schooling was partially interrupted so his parents sent him to his grandfather, former Head of Mathematics at Aberdeen Grammar School, for instruction.

Apparently, this gentleman had lost track of what maths a 10-year-old would have been exposed to, and he began the first lesson with algebra, completing linear equations in around 15 minutes, and then delving into the quadratic equation.

Young Bryce, having seen nothing beyond arithmetic before, had, he later recalled, no idea what these $x$s and $y$s were about, but was too in awe of his grandfather to admit it. He went home with an assignment, and agonised for hours trying to determine what was going on. But when he returned the next day he was able to solve every quadratic equation his grandfather gave him.

His grandfather did not live to see the results of his efforts, which led to a distinguished undergraduate career at Aberdeen University and Oxford, an Oxford DPhil, directed by the renowned EC Titchmarsh, and a position as perhaps the leading British researcher of the 1960s and '70s in what is now termed "applied analysis", a subject primarily devoted to the rigorous mathematical study of differential equations arising in the sciences and engineering.

In the 1940s and '50s, major UK figures in this area included Titchmarsh, ML Cartwright, and JE Littlewood, but then the subject fell in stature compared to more abstract and "pure" types of mathematics, such as ("modern") algebra and topology. McLeod became a Fellow at Wadham College, Oxford, in 1960, and a lecturer a few years later, but his work was being recognised more in the US than at home.

In fact, after Titchmarsh no specialist in differential equations held a Chair at Oxford until John Ball in 1996. By that time McLeod had was a University of Pittsburgh professor. As well as the lack of enthusiasm for his subject in Britain, he was motivated by looming mandatory retirement. He stayed in Pittsburgh for 20 productive years, though he and his wife Eunice kept their home in Abingdon and returned in the summers. He received an inquiry from a senior mathematician at Cambridge inviting him to apply for a Chair there. He had to reply that he was beyond the mandatory retirement age, something the writer had clearly overlooked.

McLeod's influence did much to resuscitate applied analysis in the UK. One indication of this was his FRS (1992). Others around Britain, including John Ball, were encouraged in their interest in differential equations by his work. His Oxford graduate students gained professorships in Britain and abroad.

These students particularly appreciated his optimistic approach to mathematical research. During a meeting held upon his retirement from Pittsburgh in 2007, each of his Oxford students in attendance expressed the same view: working with Bryce was a pleasure, especially compared with the ordeal they saw many of their fellow students enduring in that period.

McLeod s most cited paper, written with the American mathematician Paul Fife, explains mathematically the development of nerve impulses in an axon. Another important insight led to major theoretical advances in "inverse scattering", which plays a pivotal role in wave propagation, whether in water or other media. He collaborated widely, working with at least 40 co-authors from around the world. His paper on wave propagation in a neural network, written with the leading mathematical biologist, and Pittsburgh colleague, G Bard Ermentrout, has been widely influential.

One cannot hope to summarise all of McLeod's more than 150 research papers, but his landmark study in 1962 of the principal mathematical model of coagulation should also be mentioned. This paper is still of active interest, and a large number of publications have cited either it or its "offspring".

McLeod did not attempt to develop the sorts of elaborate theoretical structures that fascinate more abstract mathematicians. Instead he was a problem-solver of genius. His collaborations usually developed when another mathematician had a problem from an applied area which he could not solve. Very often the result would be a simple way of looking at the problem which led to an ingenious solution.

Those mathematicians who were stumped were not run-of-the-mill. One of the most eminent was Tosio Kato, Professor of Mathematics at Berkeley. The problem he and McLeod worked on was about wave motion in the overhead supply line to an electrified railway system. When Kato received McLeod's solution he wrote back,  How ever did you think of that? Many of his collaborators over the years had the same question.

It is indicative of the revival of applied analysis in the UK in the last 30 years that the 2011 Naylor Prize and Lectureship of the London Mathematical Society was awarded to McLeod,  in recognition of his important and versatile achievements in the analysis of nonlinear equations arising in applications to mechanics, physics, and biology.

Stuart Hastings

John Bryce McLeod, mathematician: born 23 December 1929; married 1956 Eunice Martin Third (one daughter, three sons); died 20 August 2014.