I grew up in Duffield near Derby, in the midlands of England. In 1957 a new secondary school opened there and I, with 73 other 11-year olds was one of the first pupils. With that small number, each teacher had to cover several areas and for a couple of years Mathematics was taught by the French master. Eventually, as the school grew, they employed a dedicated mathematician and I began to be attracted to the subject.
I was accepted in 1964 to study Mathematics at Jesus College in Oxford but, as was common then, I left school after the Entrance Examination and in my case got a temporary job in the Engineering Computing Department of Rolls-Royce in Derby. There I was surrounded by mathematics graduates and I absorbed several notions which were absent from my school curriculum. I was also given some interesting problems which required lateral thinking.
As an undergraduate in Oxford my mathematical interests were more on the pure side and in 1968 I became a graduate student with the topologist Brian Steer as supervisor. Michael Atiyah at that time had moved from Oxford to be a permanent member at the Institute for Advanced Study in Princeton but he returned each summer term, and one year while my supervisor was on sabbatical I had the benefit of being supervised by him. This extended my horizons enormously and broadened my interests by looking at questions which involved algebraic geometry and topology as well as differential geometry. This mixture of topics was formative for my future work.
In 1971 I moved to Princeton as Atiyah’s assistant. This was an eye-opening experience for me, exchanging ideas with young postdocs, learning from senior visitors and being invited to give talks at various US universities. It was there that I met my wife Nedda, who was visiting her cousin, one of the other mathematical members. We married in 1973 and then spent a year in New York. At New York University I began reading the papers of Roger Penrose on zero rest-mass field equations in relativity.
When I returned to Oxford as a postdoc the following year Penrose had recently been appointed to a Chair and I began to learn that, through his newly-developed twistor theory, the Riemannian geometry I was interested in and the geometry of relativity were both put on the same footing. It meant that questions about Einstein’s equations which were occupying me at the time made sense in this new setting. This was perhaps the first occasion I realized that there was an interface between my own interests and physics which I could exploit.
We had many senior visitors in Oxford then, and in 1977 Isadore Singer came with some new ideas from his physics colleagues at MIT. These were called instantons — Euclidean versions of the Yang-Mills equations of particle physics. The formalism however fitted perfectly with my earlier studies. Moreover Richard Ward, a student of Penrose, had just shown how twistor methods could be applied to these equations. Week by week we introduced new results in a seminar devoted to this subject and finally, combining recent work in differential geometry and algebraic geometry, Atiyah and I (and independently Drinfeld and Manin in Moscow who had also been following this development) gave a complete solution to the problem. As a consequence I travelled a great deal at this time giving talks (unfortunately for my wife around the time of the birth of our first child) and in 1979 I was eventually given a permanent Lectureship in Oxford, with a Fellowship at St Catherine’s College.
I followed up the instanton work by attacking a related concept, magnetic monopoles. Then in 1983–84 I spent a sabbatical at Stony Brook — I had been approached to join their strong group in differential geometry. Instead I found myself discussing an idea of Martin Rocek in the Theoretical Physics group. This resulted in a framework within hyperkähler geometry which explained a number of previous facts and pointed to many more. Although we each described this in a different language it was clear that we were doing the same thing. In the end, however, despite an appeal from C N Yang, I returned to Oxford.
It was soon afterwards that the setting this work provided suggested a new gauge-theoretic concept which could naturally be applied to the classical area of Riemann surfaces. It yielded an entry point into a whole range of geometrical problems, providing a link between algebraic geometry and the representation theory of surface groups. Almost incidentally, the algebraic geometry also gave a vast generalization of integrable systems that had been studied piecemeal for decades. The consequences of this work were gradually elucidated by myself and various graduate students and ultimately came to the attention of string theorists who demonstrated a link with the geometric Langlands programme.
In the meantime I had left Oxford to become a Professor at the University of Warwick. Then in 1994 I was appointed to the Rouse Ball Chair in Cambridge (which brought me into closer contact with theoretical physicists) but three years later I accepted the Savilian Professorship of Geometry back in Oxford.
Subsequent work, as then, was often guided by the intuition of physicists, which differs from that of most mathematicians. I have found repeatedly that when the two worlds interact, fertile mathematical ideas emerge.
27 September 2016 Hong Kong
