**John Lewis**'s parents were Trevor Lewis, a shipbroker, and his wife Tegwyn. Although born in Swansea, John was brought up in Cardiff where he attended Marlborough Road Elementary School for Boys. One of his teachers at this school was Thomas George Thomas who later became an MP and Speaker of the House of Commons. After studying at the elementary school, John attended Cardiff High School from 1942 to 1948. It was at this school that John received excellent teaching in mathematics from Arthur Davies. Arthur Davies had a son, Brian Davies, who became a mathematician at the University of Oxford and then professor at King's College. London. By a strange coincidence, many years later Lewis wrote a paper with Brian Davies

*An operational approach to quantum probability*(1970).

John's mother died when he was young and in 1948 the family moved to Belfast where John continued his secondary education at the Royal Belfast Academical Institution. He studied at this grammar school for boys from 1948 to 1949. Then in the autumn of 1949, after winning a Foundation Entrance Scholarship, he entered Queen's University, Belfast where his main subjects were mathematics and physics. He was awarded a B.Sc. degree in 1952 with First Class Honours in Applied Mathematics. For his outstanding performance in the honours applied mathematics classes he was awarded the William Blair Morton Prize in Applied Mathematics. This prize, founded in 1945 to commemorate William Blair Morton, Professor of Physics at Queen's University Belfast, was awarded to a student whose work in applied mathematics, particularly essay work, was the most outstanding.

While at Queen's University, Lewis met Maureen MacEntee, who was an organic chemist. They were married in September 1959 and had four children: Caitríona, who became a software consultant; Michael, who became a quantity surveyor; Roisín, who became an artist; and Ciarán, who became a barrister. Lewis was a keen rower and during his time at Queen's University he was captain of the boat club, and a very successful captain too for they won the national senior championship for eights in 1953.

David Robert Bates had been appointed as Professor of Applied Mathematics at Queen's University, Belfast, in 1951. Alex Dalgarno (1928-2015) had worked with Bates at University College London before joining him in Belfast. When Lewis began undertaking research at Queen's University, Belfast, in 1952 his official Ph.D. advisor was Bates but he was greatly assisted by Dalgarno. After two years, Dalgarno became his official Ph.D. advisor for his third year of study. Lewis was awarded his doctorate in 1955 for his thesis *Quantal Calculations Relating to Certain Rate Processes. *The authors of [1] write:-

Lewis's first publicationIn the final year of his graduate studies John worked with Alex Dalgarno on the use of variational methods in quantum mechanical perturbation theory. Emerging from their studies was a procedure for the evaluation of infinite summations of matrix elements which is widely applicable. It has come to be known as the Dalgarno-Lewis sum rule or method. They also explored the structure of the series representation of atomic interactions in inverse powers of the interatomic distance and demonstrated that the series were unique and asymptotically divergent.

*Inelastic heavy particle collisions involving the crossing of potential energy curves*, written jointly with David Bates, appeared in 1955 and in the same year his first joint publication with Alex Dalgarno was published, namely

*The exact calculation of long-range forces between atoms by perturbation theory*. Two further papers by Lewis appeared in 1955, the single authored paper

*Ionic configuration interaction in some excited states of the hydrogen molecule*and the paper

*Properties of the hydrogen molecular ion*written jointly with Martin R C McDowell, and Benno L Moiseiwitsch.

After the award of his doctorate, Lewis went to the University of Oxford where he was a postdoctoral student working with Charles Coulson, the Rouse Ball Professor of Mathematics. Lewis and Coulson, together with A D Buckingham, published *The quadrupole moment of a hydrogen atom in a uniform field* (1956). After a year as a postdoctoral student, Lewis spent sixteen further years at Oxford. The first two of these, 1957-59, were spent as a Research Lecturer at Christ Church, Oxford, during which time he also held an appointment as a University Junior Lecturer in Mathematics. He continued with his passion for rowing and rowed for Christ Church reaching the final of an event at the Henley Royal Regatta in 1957.

In 1959 he moved to Brasenose College, Oxford where he held the position of Hulme Lecturer in Mathematics for one year before being appointed as a fellow of Brasenose College in 1960, a position he continued to hold until 1972. However, in addition to these roles, he was Dean of Brasenose College from 1964 to 1967 and a University of Oxford Lecturer in Mathematics from 1966 to 1972. All four of John and Maureen Lewis's children were born during the years they spent at Oxford. Towards the end of his time in Oxford, Lewis took sabbatical leave, spending 1969 at the Institute for Advanced Study in Princeton and 1970 as a Visiting Scientist at Rockefeller University In New York. We note that Rockefeller University was only given this name five years before Lewis was a visitor there; before that it was part of the State University of New York. The invitation to Rockefeller University came from Mark Kac who Lewis had met in Oxford in the spring of 1969.

During the years from 1956 to 1972 Lewis's interests broadened considerably. He became expert in the representation theory of groups and during the academic year 1966-67, attended a series of lectures on group representations and their applications at Oxford given by George Whitelaw Mackey who was George Eastman visiting professor. Lewis then wrote two papers with Christopher Martin Edwards, who was a fellow in mathematics at the Queen's College, Oxford, on *Twisted group algebras*. They write:-

Martin Edwards had been one of Lewis's doctoral students, obtaining his doctorate from Oxford in 1966. Lewis also wrote a paper with David Joseph Judge on the commutation relation in quantum mechanics. While at Rockefeller University Lewis had worked with Mark Kac who interested him in quantum probability, dissipation in quantum mechanics and Bose-Einstein condensation. Their collaboration led to Kac spending further time at Oxford later in 1970 after Lewis's visit to the United States. At Rockefeller University, Lewis had also collaborated with George William Ford (known as Bill), to whom he had been introduced by Kac. Ford, from the University of Michigan, was also visiting Rockefeller University at that time. Lewis also met Bob O'Connell from Louisiana State University and Ford and O'Connell became long term collaborators with whom Lewis wrote many papers. We have counted 13 Ford-Lewis-O'Connell three-author papers, and 5 further papers of Lewis with Ford, some with additional authors.Projective representations of locally compact groups arise naturally in quantum mechanics in several ways. Groups of automorphisms of the algebra of bounded operators on a Hilbert space H give rise to projective representations on H, and so the study of symmetry groups acting on the observables leads to a study of their projective representations. Weyl's form of the canonical commutation relations define a projective representation of the vector spaceR^{2n}.

In 1972 Lewis left Oxford and returned to Ireland where he was appointed to a Senior Professorship in the School of Theoretical Physics of the Dublin Institute for Advanced Studies. He had been a regular speaker at Symposia at that School since 1962. The position had become vacant due to the retirement of John Lighton Synge who had built the School into a major international one. At the time of Lewis's appointment, Lochlainn O'Raifeartaigh was the Director of the School of Theoretical Physics. In January 1975 Lewis was appointed as Director of the School when O'Raifeartaigh made an extended visit to the Institut des hautes études scientifiques at Bures-sur-Yyvette near Paris. Lewis continued to be the Director for the rest of his career [3]:-

The Dublin Institute for Advanced Studies did not give Lewis the opportunity to teach undergraduates, but he offered his services as a teacher to the Department of Mathematics at Trinity College, Dublin [3]:-It was characteristic of John's generosity that one of his first acts as Director in Dublin Institute for Advanced Studies was to introduce an open access policy to its facilities, enabling scientists from all over Ireland to further their research.

In 1977, he published the monographJohn was a gifted teacher with a rare ability to present clearly the essential principles of a difficult subject without compromising its complexity. In Dublin, although his position was free of teaching duties, his passion for teaching mathematics led him to offer his talents to the School of Mathematics in Trinity College. For24years, he taught statistical mechanics and probability theory to undergraduates. He constantly refined his lectures, striving to present the latest insights from his research.

*Dilations of Irreversible Evolutions in Algebraic Quantum Theory*which he had written in a collaboration with David E Evans who had come from Oxford to join Lewis at the Dublin Institute for Advanced Studies in 1975. Richard Kadison reviewed their monograph and wrote [2]:-

In 1987 the Irish government seriously considered closing the Dublin Institute for Advanced Studies and they reduced the grant to the Institute for 1988 by 35%. In March 1988 a deputation from the Institute, including Lewis, met with the Department of Education. By the end of 1988 the financial position in Ireland had improved somewhat and the government made a normal financial settlement for 1989. However, they refused to allow James McConnell's Senior Professorship to be filled; he had retired in December 1987. Lewis and the other School directors made presentations to the Secretary of State making a strong case to continue funding the Institute. By November 1989 when the Secretary of State visited the Institute, he declared that the government was satisfied with the presentations but told members of the Institute that they would need to continue to justify all further funding. The Institute was saved from closure [5]:-This tract is an extended, somewhat expository account of recent work in the theory of dynamical semigroups, with emphasis on the theoretical foundations of quantum statistical mechanics. It is presented in the C*-algebra framework. Although the authors give brief accounts of material needed from various mathematical disciplines, a potential reader ought to be well versed in the basics of functional analysis.

Lewis was keen to collaborate with other mathematicians as can be clearly seen from his publication list. Only twelve of his 105 papers are single authored, and most of these twelve are papers derived from addresses he gave at various conferences around the world in places such as London (1974), Kyoto (1975), Esztergom (1979), Villa Mondragone (1982), Swansea (1983), Bielefeld (1984), Groningen (1985), London (1987), Swansea (1986), Amsterdam (1985-1987), and Leuven (1988). He invited many foreign scientists to the Dublin Institute, and he was particularly keen to try to obtain the necessary permissions for scientists from countries in the Eastern block to visit Dublin. In 1989 the Steklov Mathematics Institute invited Lewis to visit Moscow and Kiev as a Guest of the USSR Academy of Sciences. In Moscow, Lewis [3]:-There are many people who can claim a share in reversing this decision but John's part was certainly very important and significant.

Although the Moscow visit had given him ideas, there was a practical reason why Lewis moved his research in this direction. We explained above that when the funding for the Dublin Institute for Advanced Studies was agreed in 1989 it was on condition that they continued to justify that funding. Noel Lindsay, who was then secretary of the Department of Education, had made it clear when he visited the Institute that continued funding would be dependent on the Institute showing that its work was relevant on a practical scale. Lewis certainly took up this challenge with his new programme as he explained [6]:-... met leading researchers in the Institute for Problems in Information Transmission. They supported their research in pure mathematics with applications to telecommunications. Inspired by this, he launched a programme investigating the applications of Large Deviation Theory to the Internet. After a revolutionary insight on how to measure Internet traffic, John persuaded visionaries from the Computer Laboratory in Cambridge and the Swedish operator Telia to join him in a three-year research contract funded by the European Commission. Starting in1996, the project supported nearly a dozen students in pursuit of higher degrees in mathematics.

In 1999 the company Measure Technology Ireland was set up. Lewis explained [6]:-That's when I began the activity in the school of theoretical physics to find an area in which the skills I had developed in basic research could be applied. This was pre-Internet. At that time, they were developing big computers - called switches - for broadband networks. The switches stored transmitted data in queues, read the addresses and routed it, accordingly. I hit on studying problems associated with queueing and it turned out that the mathematics I had developed for my particular interest - in statistical mechanics - was just what was needed for tackling these problems. Instead of making elaborate models of tele-traffic on broadband and calculating the rate function, I decided to measure it.

Lewis always declared that he was self-taught in pure mathematics. He explained his position in relation to pure and applied mathematics which was quoted in [6]:-The idea of the project is to use mathematics in the way we did in the earlier project - in telecommunications, as before, but also in multi-media computer operating systems and in parallel processing computer systems. In all these areas, there are problems of allocating scarce resources. We will use the statistical properties of these systems to achieve the optimal utilisation of resources. Our aim is to produce a research group which will have an international reputation and train graduates in research in this area.

Lewis received notable honours for his contributions. He was elected to the Royal Irish Academy in 1977, serving on its council from 1985 to 1989 and again from 1997 to 1999. He was Vice President of the Academy from 1999 to 2001. In 1999 he was awarded an honorary doctorate by the Dublin Institute of Technology and, in the same year, an honorary professorship at Trinity College, Dublin. He also held honorary professorships at the Universities of Swansea (1992-1995) and Cardiff (1998-2004). Lewis retired from his position at the Dublin Institute for Advanced Studies in 2001.I get huge satisfaction from pure mathematics, but I find it enormously satisfying to see some abstract mathematical idea having an application. I will be thrilled if the company[Measure Technology Ireland]develops a successful product which embodied the original idea.

Outside his family and research interests, Lewis was fond of wine, leading to family holidays visiting vineyards, and fond of reading, particularly biographies. We hope that he enjoyed reading biographies from our MacTutor website.

Let us end this biography with a quote from the authors of [1]:-

John made significant contributions to theoretical chemistry, quantum measurement, quantum stochastic processes, the Ising model, boson condensation and telecommunications. ... John's impressive scientific work is a monument to our memory of him. His friends and collaborators will recall him for his ability, enthusiasm, insight, encouragement and eagerness to help. Many mathematicians would not have considered academic life without John's influence. While considering John's contribution to science, we should not forget the contributions that he made to many of our lives. An old friend of John's from his time at Oxford put it well when he said: "John was a lovely man and a perfect fit to my ideal of an academic: a man of great learning, lightly worn, and no conceit of himself."

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

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