1.1. The Junior Whitehead Prize.

The London Mathematical Society awards the Whitehead Prize every year. It is to honour J H C Whitehead, a former President of the Society. Only those resident in the United Kingdom and under the age of 40 are eligible.

1.2. The 1981 Prize Committee.

The 1981 Junior Whitehead Prize Committee consisted of C T C Wall and P M Neumann.

1.3. Announcement of 1981 Prize.

Junior Whitehead Prizes are awarded to D F Holt of the Mathematics Institute, Warwick, for his work in Group Theory, and to N J Hitchin of St Catherine's College, Oxford, for his work in Differential Geometry.

1.4. Citation for Nigel James Hitchin.

Nigel Hitchin is awarded a Junior Whitehead Prize for his many important contributions to differential geometry, particularly in directions related to modern work in theoretical physics. Starting with his early paper on "Harmonic spinors" (Adv. in Math., 14 (1974), 1-55) he has since studied the Yang-Mills equations jointly with M F Atiyah and I M Singer ("Self-duality in four-dimensional Riemannian geometry", Proc. Roy. Soc. A. 362 (1978), 425-461), and contributed to the general Penrose twistor programme. His most remarkable paper entitled "Polygons and gravitons" (Proc. Camb. Phil. Soc. 85 (1979), 465-476) relates in a beautiful way the gravitational instantons of Gibbons and Hawking to the extensive earlier work on rational double points of algebraic surfaces, which in turn related to Lie groups and the Platonic solids.

2.1. The Senior Berwick Prize.

The Senior Berwick Prize is named after Professor W E H Berwick, a former Vice-President of the Society, and is awarded in even-numbered years. The Senior Berwick Prize for year X can only be awarded to a mathematician who is a member of the Society on 1st January of year X; it is awarded in recognition of an outstanding piece of mathematical research actually published by the Society during the eight years ending on 31st December of year X-1; and it may not be awarded to any person who has previously received the De Morgan Medal, Polya Prize, Senior Whitehead Prize or Naylor Prize.

2.2. Announcement of the 1990 Senior Berwick Prize.

The Senior Berwick Prize is awarded to N J Hitchin for his paper 'The self-duality equations on a Riemann surface', Proc. London Mathematical Society (3) 55 (1987) 59-126.

2.3. Citation for Nigel James Hitchin.

The Senior Berwick Prize is awarded to Professor Nigel Hitchin of Warwick University for his paper 'The self-duality equations over a Riemann surface', Proc. London Math. Soc. (3) 55 (1987) 59-126. In this paper Hitchin introduced a new partial differential field equation over a Riemann surface, which he obtained in the first instance by dimension reduction of the self-dual Yang-Mills equation in four dimensions. On the one hand, Hitchin's equation fits into the general framework of 'Yang-Mills-Higgs' theories, involving coupling between a connection and an auxiliary field. On the other hand, Hitchin showed that the solutions to his equation, and particularly the moduli space of all solutions, had remarkable special properties, some related to well-established areas in Riemann surface theory, and some quite new.

A central feature of Hitchin's paper is the development of different descriptions of the moduli space. The equivalence of these descriptions is non-trivial and rests upon general analytical existence theorems which make up a substantial part of the paper. From an abstract point of view these different descriptions are an illustration in infinite dimensions of a general principle involving 'moment maps' for group actions, a principle which Hitchin has developed systematically over the past few years and applied to the differential geometry of moduli spaces. One of the descriptions identifies the moduli space with the space of conjugacy classes of irreducible representations of the fundamental group of the surface in $PSL(2, \mathbb{C})$. It thus contains in a natural way both the space of conjugacy classes of representations in $SO(3)$, which can be regarded as a moduli space of rank 2 holomorphic vector bundles, and also the representations into $PSL(2, U)$, one component of which is the Teichmuller space of the surface. Hitchin's paper thus brings together the moduli theories for Riemann surfaces and for holomorphic vector bundles. He obtained, as a special case of his existence theorems, a new proof of the uniformisation theorem for Riemann surfaces and also generalised some important classical facts about Teichmuller space to the other components of $PSL(2, U)$ representations. On the other hand, working from another description of the moduli space, Hitchin showed that it has a hyperkahler Riemannian metric and a natural circle action which he exploited to compute the homology.

Hitchin's paper is a delight to read and a gold-mine of new ideas and results. Many of these ideas are being developed vigorously at present, notably in connection with conformal field theories in quantum physics. It seems certain that this paper will be an important landmark for many years to come.

3.1. The Royal Society's Sylvester Medal.

Professor James Joseph Sylvester, F.R.S. was Savilian Professor of Geometry in Oxford in the 1880s. Soon after his death in 1897, a number of his friends considered the advisability of funding some suitable honour of his name and life-work. The suggestion met with a ready response from all parts of the world, and a representative International Committee was formed.

A sum of nearly £900 was subscribed, and it was decided to found a medal and prize for the encouragement and reward of working mathematicians throughout the world. The Council of the Royal Society undertook the trust on condition that the medal should be awarded without regard to nationality. The Sylvester Medal, first awarded in 1901, is a bronze medal awarded by the Royal Society for an outstanding researcher in the field of mathematics.

It was a triannual award up to 2010, then biannual up to 2018, and from then on it became an annual award.

3.2. The 2000 Sylvester Medal.

Nigel Hitchin was awarded the Sylvester Medal in 2002 by the Royal Society of London:-

... for his important contributions to many parts of differential geometry combining this with complex geometry, integrable systems and mathematical physics interweaving the most modern ideas with the classical literature.

He also received a gift of £1,000.

4.1. The Pólya Prize.

The Pólya Prize is awarded in memory of Professor G Pólya, who was a Member (and later Honorary Member) of the Society for about 60 years. It is awarded in those years, not numbered by a multiple of three, in which the De Morgan Medal is not available for award. The Pólya Prize is awarded in recognition of outstanding creativity in, imaginative exposition of, or distinguished contribution to, mathematics within the United Kingdom. It may not be awarded to any person who has previously received the De Morgan Medal.

4.2. Pólya Prize Citation.

Professor Nigel Hitchin FRS of Oxford University is awarded the Pólya Prize for his fundamental and enormously influential contributions to geometry, as well as for his wider contributions to the development of mathematics and mathematical physics.

Nigel Hitchin is one of the world's foremost geometers, and beyond that he has exercised a major influence on the shape of theoretical physics. For over 20 years Nigel Hitchin has been producing a remarkable success ion of highly original and influential papers in differential geometry. The range of topics is extensive: minimal surfaces, integrable systems, moduli spaces, symplectic and Kähler geometry, Einstein mercies, hyper-Kähler manifolds. He has used a broad range of techniques, many involving novel interactions with complex algebraic geometry, and has produced definitive results.

One masterpiece is his work on the self-duality equations and his identification of the "Hitchin moduli space" of holomorphic bundles equipped with a Higgs field. This has provided the basis for many developments in the theory of integrable systems. The space has been central in recent years in the work of Beilinson and Drinfeld on Langlands duality in representation theory. In mathematical physics, Hitchin's work on mono poles and spectral curves have been essential to implementing Manton's ideas about slow-motion scattering. His research continues at full pace.

Hitchin was President of the London Mathematical Society from 1994 to 1996. He has, with quiet efficiency, played a significant role in ensuring the healthy development of mathematics in this country and abroad.

5.1. The Shaw Prize.

Mr Run Run Shaw, a visionary philanthropist, held a steadfast conviction in the power of knowledge. He recognised the critical role that scientists play in illuminating the intricate mysteries of nature, and understood that their tireless efforts are fundamental to the advancement of civilisation.

In 2002, under the auspice of Mr Shaw, the Shaw Prize Foundation was established. The inaugural Shaw Prize was presented two years later in 2004. The Shaw Prize consists of three annual awards, namely the Prize in Astronomy, the Prize in Life Science and Medicine, and the Prize in Mathematical Sciences. Each prize carries a monetary award, which has been set at of one million two hundred thousand US dollars since 2016.

The Shaw Prize honours individuals, regardless of race, nationality, gender, and religious belief, who are currently active in their respective fields and who have recently achieved distinguished and significant advances, who have made outstanding contributions in academic and scientific research or applications, or who in other domains have achieved excellence. The Shaw Prize is dedicated to furthering societal progress, enhancing quality of life, and enriching humanity's spiritual civilisation.

Since 2004, the Shaw Prize has recognised over a hundred exceptional individuals who have made ground-breaking contributions to their respective fields, many of whom have gone on to receive other prestigious international awards. The Shaw Prize Foundation has also taken a proactive role in advancing scientific literacy through a range of engaging activities, including lectures, public forums, exhibitions, and other outreach programmes, in partnership with esteemed local and international universities and institutions.

The Shaw Prize Foundation, committed to upholding Mr Shaw's vision, remains dedicated to the promotion of excellence and innovation, and aspires to serve as a beacon of inspiration for those who seek to make a positive impact on society. Through its unwavering commitment to this mission, the Shaw Prize Foundation is poised to encourage and elevate the next generation of scientists and innovators, for the benefit of humankind.

5.2. The 2016 Shaw Prize in Mathematical Sciences.

Nigel J Hitchin was awarded the 2016 Shaw Prize:-

... for his far-reaching contributions to geometry, representation theory and theoretical physics. The fundamental and elegant concepts and techniques that he has introduced have had wide impact and are of lasting importance.

5.3. Contribution of Nigel J Hitchin.

The Shaw Prize in Mathematical Sciences 2016 is awarded to Nigel J Hitchin, Savilian Professor of Geometry at the University of Oxford, UK, for his far-reaching contributions to geometry, representation theory and theoretical physics. The fundamental and elegant concepts and techniques that he has introduced have had wide impact and are of lasting importance.

Geometry has long been at the core of mathematics. It has close connections to other parts of mathematics such as representation theory, which is related to the study of symmetry, to differential equations, to number theory and more recently to theoretical high energy physics.

Hitchin is of one of the most influential geometers of our times. The impact of his work on geometry and on these allied subjects is deep and lasting. On numerous occasions, he has discovered elegant and natural facets of geometry that have proven to be of central importance. His ideas have stimulated work in areas far removed from the context in which they arose.

His work on Higgs bundles over Riemann surfaces provided algebraically integrable systems, called Hitchin fibrations, that are important examples of hyperkahler manifolds and have become fundamental objects in geometry. In addition, they have been an impetus to progress in the modern branch of representation theory called 'geometric Langlands', helping to guide the development of that subject. Hitchin fibrations are a foundational ingredient in Ngo's recent Fields Medal winning work in the theory of automorphic forms and number theory. In a related work, Hitchin used this theory to construct projectively flat connections over the moduli spaces of Riemann surfaces, which had been predicted by Witten's analysis of a 3-dimensional topological quantum field theory.

Hitchin's work with Atiyah, Drinfeld and Manin on a description of the moduli space of instantons on four-space in terms of linear algebra, even now thirty years later, forms a basis for much work in both mathematics and theoretical physics. His formulation of the Kobayashi-Hitchin conjecture relating algebro-geometric stability and solutions to the instantons on complex algebraic surfaces has created a large new area of non-linear partial differential equations.

By exploring ignored corners of geometry, Hitchin has repeatedly uncovered jewels that have changed the course of developments in geometry and related areas and with it the way mathematicians think about these subjects.

5.4. An Essay on the Prize.

From antiquity the subject of geometry has been at the centre of mathematics. The ancient Greeks were fascinated by the subject and studied it extensively, giving us Euclidean geometry. The modern view of geometry dates to the middle of the 19th century, when Gauss introduced and developed the theory of curved surfaces. He was followed by Riemann who constructed the theory of higher-dimensional curved spaces, now called Riemannian geometry. Their work began a period of flowering of geometry, and our present-day understanding of the subject emanates directly from their work.

Nigel Hitchin is one of the most influential geometers of our time. The impact of his work on geometry and on many of the allied areas of mathematics and physics is deep and lasting. On numerous occasions, he has discovered elegant and natural facets of geometry that have proven to be of central importance. His ideas have turned out to be crucial in areas of mathematics and physics, far removed from the context in which he first developed them. By exploring ignored corners of geometry, Hitchin has repeatedly uncovered jewels, many more than described below, that have changed the course of developments in geometry and related areas, and changed the way mathematicians think about these subjects.

In the modern period geometry has flourished and expanded greatly. With the development of topology (which may be described as the study of shapes, allowing deformation but not tearing) as an independent discipline in mathematics around 1900, the purview of geometry expanded to include auxiliary objects associated with spaces. A prime example is gauge theory, which is the study of the geometry of certain curved structures over spaces known as fibre bundles.

Geometry has close connections to other parts of mathematics such as representation theory, which is related to the study of symmetry, differential equations and dynamics, and number theory. It has had a profound impact on topology. Its connections to physics are long-standing as well: Einstein formulated general relativity in terms of the geometry of curved four-dimensional spacetime, essentially in the form that Riemann had introduced. More recently, the standard model of particle physics is formulated using the geometry of gauge theories, and much of theoretical high-energy physics beyond the standard model is formulated geometrically.

While Hitchin has introduced many important concepts and techniques in geometry, one of his most influential works is his study of Higgs bundles over a Riemann surface. The parameter space of all Higgs bundles over a fixed surface is itself a fibre bundle over that surface. On these fibre bundles Hitchin defined a natural function that produces an algebraically completely integrable Hamiltonian system on the space. The result is what is now called the Hitchin fibration of the space of Higgs bundles. One aspect of these spaces is that they give fundamental examples of important objects known as hyperkahler manifolds. But the impact of Hitchin fibrations is not limited to Higgs bundles and hyperkahler manifolds. Hitchin fibrations and his quantization of them are cornerstones of the construction of a modern branch of representation theory called 'the geometric Langlands programme'. In addition, they are a foundational ingredient in Ngo's recent Fields-Medal-winning work in the theory of automorphic forms and number theory. In a related work, Hitchin used this theory to construct projectively flat connections over the moduli spaces of Riemann surfaces, which had been predicted by Witten's analysis of Chern-Simons theory, a three-dimensional topological quantum field theory. Hitchin fibrations have also served as a point of departure for physicists in their studies of certain four- and six-dimensional quantum field theories. The story of Hitchin fibrations is a perfect example of Hitchin's ability to find appealing and natural questions that had been previously overlooked but whose answers are of fundamental importance across a wide spectrum of mathematics and physics.

Hitchin's work with Atiyah, Drinfeld and Manin brought mathematics to bear on a question of fundamental importance in physics, namely the description of the moduli space of instantons on four-dimensional Euclidean space. Their approach using linear algebra, even now, thirty years later, forms a basis for much work on the study of instantons in both mathematics and theoretical physics, and serves as an important bridge between mathematics and physics.

The Kobayashi-Hitchin conjecture relating algebro-geometric stability and solutions to the partial differential equations describing instantons on complex algebraic surfaces was one of the first examples of a close relationship between the notion of stability in algebraic geometry and solutions of non-linear partial differential equations. This fundamental principle has reoccurred in various forms in many areas of mathematics.

The influence of Hitchin's work across the wide sweep of geometry and its applications to allied fields will be felt far into the future. His many achievements richly merit the award of the Shaw Prize in Mathematical Sciences.

5.5. Autobiography of Nigel J Hitchin.

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 realised 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 generalisation 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.

5.6. Simon Donaldson on Nigel Hitchin's Shaw Prize award.

The American Mathematical Society Notices asked Simon Donaldson of the Simons Center for Geometry and Physics at Stony Brook University to comment on the work of Hitchin. Donaldson responded:-

Nigel Hitchin's work, over four decades, has introduced some of the most important ideas in modern differential geometry; in particular, his work has played a central part in many of the spectacular interactions with theoretical physics over this period. In the 1970s, Hitchin was a leader in developing Penrose's twistor theory in the Riemannian setting and was one of the four discoverers of the ADHM construction of Yang- Mills instantons. The part of his work that has perhaps been most influential goes back to the mid-1980s, with his introduction of Higgs bundles over Riemann surfaces, a notion that has proved to be extraordinarily fruitful and has led to major new lines of research extending into integrable systems and representation theory. But there is much more that one could mention, such as his contribution to the hyperkähler quotient construction, his invention of generalised complex structures, and his treatment of various exceptional differential-geometric phenomena through the algebra of exterior forms in low dimensions. Hitchin's work is characterised by exceptional elegance, and his novel ideas appear, in retrospect, natural and inevitable. Many of his papers fit his new ideas into historical contexts within differential and algebraic geometry; on the other hand, new developments springing from his work are sure to continue far into the future.

5.7. Marta Mazzocco writes in Mathematics Today on Nigel Hitchin.

This year's Shaw Prize in Mathematics was awarded to Professor Nigel Hitchin FRS (Savilian Professor of Geometry, University of Oxford) 'for his far reaching contributions to geometry, representation theory and theoretical physics. The fundamental and elegant concepts and techniques that he has introduced have had wide impact and are of lasting importance.'

Indeed, Hitchin's profound and imaginative work has had tremendous impact in many different areas of Mathematics, including algebraic geometry, differential geometry, complex analysis, topology, integrable systems, mathematical and theoretical physics.

So much so, that to choose which of his many spectacular results should be explained in this article is a daunting task. Last week at a conference in Moscow, I realised that the definition of Nigel's most influential contribution depends on the field of research of your interlocutor.

          "Surely it must be the Hitchin's connection!"
          "Hang on a second, what about his notion of generalised complex manifold?"
          "Nonsense, the self-duality equations are the most important."
          "You're forgetting the Hitchin's integrable systems."
          "…And the monopoles."

I changed subject, mathematicians can be fiercely opinionated at times!

Hitchin derived the self-duality equations in the context of the self-dual Yang-Mills equations that describe the behaviour of elementary particles in the Euclidean 4-dimensional space. In [1], he discovered that despite the fact that these equations lose all physically relevant solutions when restricted to the plane, they actually produce a surprisingly rich space of solutions when defined on a compact Riemann surface, i.e. a surface that can be covered by a finite collection of planes. In the same paper he proved rigorously that solutions of the self-duality equations define what he called stable pairs in differential geometry.

This is a typical feature of Hitchin's ground-breaking work: he proves that certain objects arising in theoretical physics (as for example the Higgs fields responsible for the famous Higgs boson) define some new concepts in algebraic or differential geometry. He then develops a rigorous pure mathematical theory of these concepts to deduce very powerful and elegant results that keep the physicists busy for twenty years or so.

Another example of this feature is given by his rigorous approach to geometric quantisation of moduli spaces. In the setting developed by Konstant, Kirillov and Souriau, the first step to quantise the set of smooth real functions on a manifold is to associate to each such function {f} a first order differential operator {F}. Next, one needs to choose a polarisation (this means that one needs to split the phase-space in position coordinates and momenta and choose a complexification) and finally quantise – there is a very serious difficulty though: proving that the resulting quantisation is independent of the choice of polarisation. Hitchin vanquished this problem by introducing yet again a new mathematical concept: the famous Hitchin connection on the moduli space [3].

A more recent example is the notion of generalised complex structure [4], which is an analogue of complex structure not on the tangent bundle $T$ of a manifold, but on the direct sum of the tangent and cotangent bundles $T \oplus T^*$. This structure includes many other mathematical structures (such as symplectic structure and Calabi-Yau structure) as special cases and lies at the basis of most recent results in topological string theory.

Hitchin keeps the mathematicians busy too, indeed his celebrated integrable systems [2] provided the basis for the formulation of the geometric Langlands correspondence that in turn has given food for thought to a huge number of pure mathematicians who have dominated all the International Congress of Mathematicians since the nineties.

In the best tradition of longevity established by the founder of the Shaw prize himself, who died at 107 years of age, Nigel Hitchin defies the stereotype that mathematicians need to be in their early thirties to produce great results. When already in his sixties, Hitchin introduced the notion of co-Higgs bundle [5], an adaptation of Simpson's Higgs bundles in which the tangent bundle is replaced by the cotangent bundle. This inspiring work will surely focus a great deal of research in the future.

On a personal note, Nigel is a lovely man who has served the mathematical community in the UK and overseas by taking on really heavy responsibilities such as Chairman of the Research Assessment Exercise Pure Mathematics Panel (twice!), President of the London Mathematical Society and many others. He has a wonderful sense of humour and cooks the best pheasant in red wine sauce I ever tasted.