After graduating with his first degree, Roach joined the Royal Air Force. He worked in the Education Branch between 1955 and 1958 being a Flying Officer and attaining the rank of Flight Lieutenant. In 1958 he took a job as a Research Mathematician with the British Petroleum Company but during the three years he worked for BP he also studied part-time for a Master's Degree from Birkbeck College, London. He was awarded an M.Sc. with distinction in 1960. During the years when he was working for BP and also studying for his M.Sc., Roach met and married Isabella Grace Willins Nicol, a young girl from Fife, Scotland. In 1961 he resigned his post at BP when offered a position of Lecturer in Mathematics at the University of Manchester Institute of Science and Technology (UMIST). While working as a lecturer at UMIST, he also registered as a research student and studied for his doctorate. He was awarded a Ph.D. for his thesis entitled Dynamical Theory of Viscous Tides in Close Binary Systems in 1964. He was awarded a Fellowship of the Royal Astronomical Society and, leaving UMIST in 1966, he spent the academic year 1966-67 in Canada as a Visiting Professor at the University of British Columbia. While there he worked with Robert A Adams, an analyst who joined the Mathematics Department of the University of British Columbia in 1966 after completing a Ph.D. in Mathematics at the University of Toronto.
Back in the UK, Roach was appointed as a lecturer at Strathclyde University in Glasgow, Scotland. He and his wife Isabel lived in Fintry, a village in Stirlinshire about 20 km north of Glasgow across the Campsie Hills. He worked at Strathclyde University for the rest of his career. He was promoted to Senior Lecturer in 1970, to Reader in 1971, and to Professor in 1979. In 1982 he was honoured with being appointed to the 1825 Chair of Mathematics in succession to Donald Pack. He held this prestigious professorship until he retired in 1996 when he was made Professor Emeritus. He served Strathclyde University in a number of roles throughout his career: as Head of the Department of Mathematics (1980-1982), and as the first Dean of the new expanded Faculty of Science (1982-1985). He had periods of service on both the Senate and on the University Court. The Universities of Glasgow and Strathclyde Military Education Committee was established in April 1908 with oversight of all three Service Units in the two universities. The War Office had stipulated that any University establishing a contingent of the Officers Training Corps must have a Committee responsible for Military Education established by the host University. Roach was vice-convener of the committee from 1992 to 1996 and convener in 1996-97.
Roach's research was mainly in the area of applied analysis. In fact he lists his research interests as operator theory, nonlinear evolutionary processes, scattering theory, scattering processes in nonhomogeneous, anisotropic media, inverse scattering problems, and applications of mathematics to industry. His early papers were: On the approximate solution of elliptic, self adjoint boundary value problems (1967); Fundamental solutions and surface distributions (1968); Approximate Green's functions and the solution of related integral equations (1970); and (jointly with Robert A Adams) An intrinsic approach to radiation conditions (1972). These were the first of over 150 papers published by Roach. His first book Green's functions. Introductory theory with applications was published in 1970. D O Banks writes in a review:-
As indicated by the title, this book is intended to give a self-contained and systematic introduction to the theory of Green's functions and the general ideas involved in their application to boundary value problems associated with ordinary and partial differential equations. It is the author's hope that such an account will enable mathematical scientists to acquire enough confidence in the technique of solving boundary value problems so that they are able to use more advanced and comprehensive expositions and apply this technique to specific examples. This hope is based on the assumption that mathematical scientists frequently do not use a given mathematical tool because of their lack of understanding of the mathematical ideas involved. To the extent that this is true, the author has fulfilled a need. For, true to his aim, the author has written a mathematically self-contained book. ... Generally, this book should fill the need of those who want an introduction to the theory of Green's functions but lack the mathematical background to understand more advanced accounts.In 1982 Roach published a second edition of the book which he had updated and added two additional chapters. He published four further books, three of which were written after he retired. The one which was published one year before he retired was An introduction to linear and nonlinear scattering theory (1995). Vladimir Georgescu writes in a review:-
The author's main purpose is to provide "a reasonably self-contained introduction to linear and nonlinear scattering theory for those who might wish to begin working in the area". The book is quite different from the existing textbooks on scattering theory; the emphasis is on presenting the fundamental concepts of the theory in the natural mathematical framework while most of the technical details are skipped. This makes it especially useful for "applied scientists" by making accessible to them various "abstract" techniques, but also to students and mathematicians interested in the spectral and asymptotic analysis of wave propagation phenomena. The text should indeed be accessible to a very large audience. The required preliminaries do not exceed an elementary knowledge of the initial-boundary value problem for the wave equation. ... I think that the author succeeded in his aim and has offered us a very attractive introduction to scattering theory and its applications to some problems in mathematical physics and applied sciences.Roach's last three books are: (1) Wave scattering by time-dependent perturbations. An introduction (2007); (2) An introduction to echo analysis. Scattering theory and wave propagation (2008); and (3) (with I G Stratis and A N Yannacopoulos) Mathematical analysis of deterministic and stochastic problems in complex media electromagnetics (2012). We quote short extracts from three reviews of these books, by Leon S Farhy, Hideo Soga, and Vilmos Komornik respectively:-
- The book offers a comprehensive introductory text in acoustic wave propagation and scattering by time-dependent perturbations which occur in a broad range of applications, including radar, sonar, engineering diagnostics, geophysical prospecting, ultrasonic medicine, etc. Even though the focus is placed more on the concepts and development of constructive methods rather than on the detailed proofs, the monograph is presented in a way that should appeal to both the theoretical and applied scientist working in the field of modern scattering theory and its applications.
- This monograph deals with the scattering of waves in media with some disturbance. The main aim is to give a self-contained and introductory explanation of the mathematical framework and the results for scattering, which were initiated and developed by T Ikebe, P D Lax and R S Phillips, C H Wilcox, and many other researchers.
- The purpose of this book is to provide a user-friendly introduction to many new results obtained in the last 20-30 years in the field of complex media, motivated by the recent apparition of artificial materials with astonishing properties. Interesting historical outlines, detailed physical and mathematical explanations and rigorous mathematical treatments are characteristic of the text. ... This monograph is of a very high standard, allowing the reader to learn many facets of the rapidly growing field of complex media and to get up-to-date information on a number of open research problems.
... radio controlled model planes. At flying displays, he acted as air traffic control, ensuring that no two planes were using the same radio frequency.Other interests are detailed in :-
He held office in a range of charitable organisations. In 1997, he was Deacon of the Incorporation of Bonnetmakers and Dyers, one of the 14 Incorporated Crafts of the Trades House of Glasgow. In this role he played an active part in the City's affairs for that year, with an emphasis on charitable work and education. From 1996 he was a member of The Trades House of Glasgow School Craft Project Committee, visiting schools and taking a keen interest in what the pupils produced. At his instigation the Committee introduced an award (The Deacon Convener's Achievement Award) to recognise outstanding work. In 1983 Gary joined The Nomads Club. This body, formed in 1895, exists as a forum for discussion and debate, and Gary presented an Annual Paper to the Club, on a matter chosen by the President of the year. This gave Gary considerable pleasure, owing to the subject usually having absolutely nothing to do with Science or Mathematics! He was President in Session 1998-99 and was made an Honorary Member in 2010.Roach received many honours including being elected a fellow of the Royal Society of Edinburgh (1977), awarded a D.Sc. by the University of Manchester (1991), and awarded an honorary Sc.D. by the University of Łódź, Poland, in 1993. He was president of the Edinburgh Mathematical Society (1981-82). He served as Convener of the Conference of Professors of Applied Mathematics and of the University and Colleges Admissions Service (Scotland) Coordinating Committee. He was also a fellow of the Royal Astronomical Society, of the Institute of Mathematics and its Applications, and of the Royal Society for Encouragement of Arts, Manufacture and Commerce.
We note that Roach had a long list of letters after his name: OStJ, FRSE, FRSA, FRAS, BSc, MSc, PhD, DSc, ScD, CMath. Now BSc, MSc, PhD, DSc, ScD are academic degrees we have already noted. OStJ is Officer of the Most Venerable Order of the Hospital of Saint John of Jerusalem. FRSE is fellow of the Royal Society of Edinburgh. FRSA is fellow of the Royal Society of Arts, a shortened form of the Royal Society for Encouragement of Arts, Manufacture and Commerce. FRAS is fellow of the Royal Astronomical Society. CMath is Chartered Mathematician, a professional qualification from the Institute of Mathematics and its Applications.
The authors of  write about Roach's health in the final years of his life:-
His sporting exploits probably contributed to the arthritic condition that took its toll on him in recent years curtailing his travel and attendance at research conferences, but which he bore with the greatest of humour. He remained cheerful and charming, and always seemed to have a smile on his face as he spoke with his gently Welsh lilt.The obituaries referenced below all note that he died at Bannockburn Hospital on Saturday, 17 March, 2012, having :-
... watched his beloved Welsh rugby team defeat France to win the Grand Slam.A Memorial service was held on Saturday, 31 March 2012 in Fintry Parish Church.
Article by: J J O'Connor and E F Robertson