**Bryce McLeod**'s parents were John McLeod, an engineer, and Adeline Annie Bryce. John and Annie (the name she was known by) were married in the Rubislaw district of Aberdeen in 1928. Their son John Bryce McLeod, the subject of this biography, was born on 23 December 1929 but, as one would expect, his birth was not registered until January 1930. He was educated in his home city of Aberdeen, first attending primary school there. When he was ten years old, shortly after World War II broke out, Bryce was on the point of moving from his primary school to Aberdeen Grammar School. However, the Grammar School was only operating on half the days since it was being used to train soldiers on the other half. Bryce's parents, afraid that their son's education would suffer, sent him to be taught by his grandfather when the pupils were not at school. His grandfather was by this time retired but he had been Head of Mathematics at Aberdeen Grammar School. "Let's teach young Bryce some mathematics," he thought [4]:-

After this introduction to algebra from his grandfather, Bryce progressed rapidly in his schooling at Aberdeen Grammar School. He was an outstanding pupil and completed his secondary schooling much more rapidly than the norm. He graduated from the Grammar School in 1945 and, later in that year, he entered the University of Aberdeen at the age of sixteen. The University of Aberdeen Bursary Competition was an important event for those in their final year at high school and McLeod had sat the examination and had been awarded a top bursay. He graduated in 1950 with First Class Honours in Mathematics and Natural Philosophy. We note that this degree was an M.A. (not a B.A. as given in certain of the references below) since the M.A. was, and still is, the traditional first arts degree in the Scottish system. [The Scottish M.A. is equivalent in standard to the B.A.] McLeod was awarded a scholarship from Aberdeen to study at Oxford University so, in 1950, he matriculated at Christ Church, Oxford. At this time McLeod was aiming to become a school teacher.Apparently, this gentleman had lost track of what mathematics a10-year-old would have been exposed to, and he began the first lesson with algebra, completing linear equations in around15minutes, 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.

McLeod spent two years at Christ College, Oxford, working for his B.A. His tutor at the college was Theodore William Chaundy (1889-1966) who was also a University Reader in Mathematics. During the time McLeod was studying for his Oxford B.A., Chaundy taught the courses 'Elementary differential equations and Legendre's functions' and 'Partial differential equations. Elliptic equations. Parabolic equations' and McLeod became fascinated with the topic. The rest of his mathematical career was influenced by the courses and tutorials given by Chaundy on differential equations. McLeod was awarded a First Class B.A. by the University of Oxford in 1952.

After obtaining his B.A. from Oxford, McLeod spent the year 1952-53 in Vancouver funded by a Rotary Foundation Fellowship from Aberdeen. This fellowship would have funded his studies anywhere in the world and he had asked advice from Chaundy about the best place to go. Chaundy suggested British Columbia, more because he had a daughter there than because of its mathematical standing. However, although it wasn't a top place for mathematics, nevertheless McLeod had a great year there and made many good contacts. Returning to England, he did two years National Service which, at that time, was compulsory. These years, 1953-55, were spent as an Education Officer in the RAF, teaching physics and mechanics at the Royal Air Force Technical College at Henlow, Bedfordshire. He soon discovered, however, how little mathematics the men he was teaching knew but he found it a very worthwhile experience showing him that he had to adjust his teaching to suit his pupils. He also made some good friends in these two years.

He then returned to Oxford to undertake research for his D.Phil. advised by Edward Titchmarsh. During 1955-56 he was a Harmsworth Senior Scholar of Merton College, Oxford. In 1956 he married Eunice Martin Third; they had a daughter, Bridget, and three sons, Kevin, Callum and Patrick. While still undertaking research, McLeod was appointed as a Junior Lecturer in Mathematics at Oxford in 1956, a position he held for two years. He was awarded a D.Phil. by Oxford in 1958 for his thesis *Some Problems in the Theory of Eigenfunction Expansions*. This year of 1958 also marks the time that his first paper was published, namely *On a functional equation* which he wrote jointly was T W Chaundy who, as we have already noted, had been his tutor and had inspired him to work in this area. In 1959 he had three papers published: *On four inequalities in symmetric functions*; *Two expressions for the distribution of eigenvalues*; and (jointly with E C Titchmarsh) *On the asymptotic distribution of eigenvalues*. These papers mark the beginnings of a long publication list and we note that MathSciNet lists 156 published items for McLeod.

In 1958, after completing the work for his D.Phil., McLeod was appointed as a lecturer in Mathematics at the University of Edinburgh. He spent two years in Edinburgh before being appointed as a fellow of Wadham College, Oxford in 1960. At the same time he was appointed as a Lecturer in Mathematics at Oxford University. He was a Junior Proctor in 1963-64. He held the fellowship at Wadham until 1991 and the University Lectureship until 1988 but in 1987 he accepted the University of Pittsburgh's offer of a research professorship, a position that meant he became a tenured full professor. This was a move to a place that McLeod knew well since he had spent several summers and sabbaticals in the United States where he spent most time at the University of Wisconsin and the University of Pittsburgh. Among the reasons he chose to leave Oxford to take up the appointment at Pittsburgh were the fact that he realised that his area of applied analysis was more appreciated in the USA than in Britain. Another reason was that, by 1987, he realised that he faced mandatory retirement in a few years time in the UK but could continue teaching in the USA.

Sam Howison describes his approach to mathematics in [5] and [6]:-

What better way to describe Mcleod's achievements in mathematics than to quote the citation for the Naylor Prize and Lectureship in Applied Mathematics which he received from the London Mathematical Society in 2011:-Bryce considered himself a problem-solving mathematician rather than a builder of general theories. He liked to focus on a specific hard problem and to find something new to say about it that was at the same time rigorous, interesting and useful. He was, of course, fully au fait with modern techniques but he added to this a deep understanding in the style of the more classical tradition he had inherited from Chaundy, Titchmarsh and their predecessors. He solved problems with consummate skill across an extraordinary range of areas as diverse as fluid mechanics, general relativity, plasma physics, mathematical biology, superconductivity, Painlevé equations, coagulation processes, nonlinear diffusion and pantograph equations, among many others.

We note that the Naylor Prize and Lectureship in Applied Mathematics is awarded in even numbered years in memory of V D Naylor:-The Naylor Prize and Lectureship in Applied Mathematics is awarded to Professor J Bryce McLeod, FRS, of the University of Oxford, in recognition of his profound and versatile lifelong achievement in the analysis of nonlinear differential equations arising in mechanics, physics and biology, and of its lasting influence. Since the early1960s, McLeod has played an important role in this work. For example, in1962he devised a proof, far ahead of its time, that an infinite system of coagulation - fragmentation equations has non-trivial solutions; in1971, his seminal paper with Tosio Kato on the asymptotic behaviour of functional differential equations broke completely new ground in what was then a new area; in1977, with Paul Fife he established that solutions to reaction-diffusion equations converge to travelling waves(this now-classic paper has since been extended, in particular, by McLeod and others to an important integral equation from mathematical neuroscience); in1979, he devised an ingenious proof that Krasovskii's conjecture concerning the maximum slope of a water wave is false; in the1980s, his work with Stuart Hastings and others was among the first contributions to the rediscovery in modern times of the Painlevé transcendents; in fundamental work with Avner Friedman in1985he proved that for reaction-diffusion equations with certain nonlinearities, solutions blow up in finite time and in1995, with Gero Friesecke, he showed that dynamics provides a mechanism which prevents the solution to a model of phase transformations from generating an infinitely fine microstructure. The list could go on. Bryce McLeod is an applied analyst with a terrific ability in problem solving. Over an extended period, he has made crucial contributions over a very broad range of topics by solving hard, specific problems, most arising in applications. His work is marked by power and virtuosity, transparency and elegance, and over the decades has been a source of inspiration to many young researchers.

McLeod gave the Naylor Lecture at the London Mathematical Society Annual General Meeting on 16 November 2012. The topic of his lecture was 'The Wedge Entry Problem'.The grounds for the award may include work in, influence on, and contributions to applied mathematics and the applications of mathematics, and lecturing gifts.

McLeod has written one book, *Classical methods in ordinary differential equations. With applications to boundary value problems* jointly authored with Stuart P Hastings (the author of the obituary [4]). Here is an extract from Stephen Schecter's review of the book:-

The article [7] is a review by McLeod of a text onWhich excites you more, the problem you want to solve, or the method you envision using to solve it? For many of us, it's the method. We have techniques we love, and we like to use them, extend them, and show the world how great they are. This book is about the authors' favorite methods, which they call "classical", for solving nonlinear ODE boundary value problems. The methods include shooting(their most favorite), using estimates to bound the solutions, rewriting differential equations as integral equations(to set up an iteration scheme, and to give a rigorous approach to asymptotic expansions), and rewriting linear ODEs in self-adjoint form. Methods they don't like so much include fixed point arguments in function spaces, variational methods, unjustified asymptotic expansions, geometric singular perturbation theory, and Mel'nikov's method. The book covers a lot of problems that the authors treat using their favorite methods. For many of the problems they also discuss approaches using the methods they don't like, often giving introductions to these methods. In a number of cases their motivation for looking at a problem seems to be to see if a problem done another way can also be done their way, and to see if they have anything to add. ... The book is clearly and carefully written, with lots of complete proofs. Often, to improve clarity, the most general case is not attempted, as is appropriate in a book of this type. The authors say that their approach often gives more information than alternative approaches. They prove this by numerous examples. ... I would be happy to own this book. It is a summary by two distinguished mathematicians of the techniques they have found useful so that others can use them. More of us should write such books.

*Gamma and Beta Functions, Legendre Polynomials, Bessel Functions*. McLeod makes some general comments in this review which give an indication of his ideas about teaching these topics:-

To get an idea of McLeod's qualities as an undergraduate teacher, we cannot do better than to record some comments from University of Pittsburgh students:If it is desired to introduce the special functions at such an early stage, with all the limitations and inadequacies that this implies, then this book may have something to say. But I should have thought that intending mathematicians would be well-advised to do a bit more analysis first, while non-mathematicians, interested presumably in applications, might become a bit impatient with the elaborate care with which the more elementary properties are proved and with the complete absence of some of the more advanced(and more useful)ones.

- [McLeod is] very clear and always involves the class. His tests are very fair. ... If you do the homework and study that for the tests then you will do fine. Awesome and entertaining teacher! Take him by all means!
- Very organized, clear, and knowledgeable. His lectures are fast paced but always very interesting. Pay attention in class and do the homework and you will do well. Without a doubt, the best lecture class I've taken yet. He's a really funny guy too. McLeod is the MAN!
- One of the best maths teachers I ever had. Very clear and crisp lectures. But he expects you to know the basics thoroughly and he moves through the lectures fast.
- [It] seemed like the class was moving fast but never really covered any hard material which made the tests easy. He doesn't really care about doing homework at all.

In the interview [1] McLeod gives some wonderful insights into his love of mathematics. He said:-

McLeod was the thesis advisor for many students at both Oxford and at Pittsburgh. Quite a number have become professors of mathematics. In 1962 Ian Michael became McLeod's first research student. Ian describes his experiences with McLeod in [8]:-Mathematics is fun ... Whether it's delving into the mathematics itself or talking to other mathematicians about it ... it's fun. ... In my experience, that fun comes from not getting hold of one problem and spending your life digging deeper and deeper and deeper into that problem. It lies in keeping your mind open to what other people are doing. I never would have wanted to do anything else.

We have already mention that McLeod was awarded the Naylor Prize and Lectureship in Applied Mathematics by the London Mathematical Society in 2011. He had, however, received a number of highly prestigious awards before that time. He was elected a fellow of the Royal Society of Edinburgh in 1974 and a fellow of the Royal Society of London in 1992. He received the Whittaker Prize from the Edinburgh Mathematical Society in 1965 and the Keith Medal and Prize from the Royal Society of Edinburgh in 1987.I remember Bryce as a patient and humane supervisor. At one point in his Naylor lecture of2012Bryce emphasised the importance of asking questions. I hope that this has remained a feature of my later work as an Anglican priest, work which neither of us would have predicted in1962.

McLeod retired from Pittsburgh in 2007 and, the family having kept their home in Abingdon during the twenty years they were in Pittsburgh, they were able to take up residence near Oxford again. In fact they had returned to Abingdon most summers during their twenty years in the United States and McLeod had maintained strong contacts with the Oxford Centre for Industrial and Applied Mathematics and the Oxford Centre for Nonlinear PDEs. McLeod spent the last years of his career at the latter Oxford Centre.

Let us end with some tributes to McLeod. First one from Carson C Chow [3]:-

Sam Howison writes ([5] or [6]):-Bryce was an extraordinary mathematician and an even better human being. I had the fortune of being his colleague in the math department at the University of Pittsburgh. I will always remember how gracious and welcoming he was when I started. One of the highlights of my career was being invited to a conference in his honour in Oxford in2001. At the conference dinner, Bryce gave the most perfectly constructed speech I have ever heard. It was just like the way he did mathematics - elegantly and sublimely.

Juan J Manfredi, mathematics professor at Pittsburgh, said [2]:-Many, many people throughout the mathematical community remember Bryce with great fondness: for his kindness and support for students and colleagues alike; for his intensely amused laughter or his rapt concentration on an explanation; for his zest for life and mathematics.

He was a distinguished scientist and mathematician but he also was a great gentleman of mathematics. It was a pleasure to talk with him about mathematics. ... He was kind with everybody, even those who did not agree with him.

**Article by:** *J J O'Connor* and *E F Robertson*

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