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Simon Donaldson extras
We give three autobiographies of Simon Donaldson. The first we have constructed from the Heidelberg Laureate Forum interview he gave in 2020. The other two are autobiographies, one from the book Mathematicians, the other written at the time he won the Shaw Prize in 2009. This is followed by an essay on the Shaw Prize 2009. We then give the citation for Donaldson winning the 2015 Breakthrough Prize in Mathematics, and finally the relevant part of the citation for the 2020 Wolf Prize.
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1. The HLF Portraits: Simon Donaldson.
I was born into a scientific household. My mother did a degree in natural sciences in Cambridge. She didn't pursue it - her generation didn't after they got married. My father was an engineer. He started his career in the navy but he left the navy about the time he got married. Then he worked in the physiology laboratory in Cambridge. He was building the apparatus for experiments. He was working with well-known people such as Andrew Huxley trying to understand the nervous system. I was one of four children. There were plentiful supplies of books but my parents allowed all the children to develop their own interests. What tended to be even more conspicuous than books, there were various projects. My father always had projects he was doing in the house - he built model airplanes, he would mend things in the house - there was a lot of activity and an interest in doing things.
I'm the third of four, three boys and a girl. My sister is older then me. There were underlying expectations but these were very general. I went to a prep school in Cambridge, a feeder school for an independent secondary school. I think most of the things I know I learnt there. I was most interested in history at that time - the history school masters were keen on me. I didn't do extraordinarily well nor was I particularly hard working.
When I was twelve the family moved because my father got a different job. Actually it was working with some of the same people but rather than working for Cambridge University they set up a research unit funded by the Medical Research Council. At that time my father was working on artificial vision and they had some success in doing that but maybe it was a bit before its time. I have the impression that now everything has gone much further. Maybe some of the technology wasn't quite there at that time. They were also working on some less ambitious things, basically when you stimulate the nervous system artificially. He shared the struggles of his work with his children.
We moved to Kent, near to a town called Sevenoaks which is a commuter town twenty miles south of London. I went to Sevenoaks School which is one of the slightly anomalous schools. It was a private school but since there was no local grammar school so half of the pupils were actually supported by the state and half were private. It had a broader social mix than one might think for an independent school. It had a reputation of being quite innovative - practised lots of trendy theories of education - this was about 1970. I didn't completely like it to begin with. It took me some years to settle in to this school. I had grown up thinking I had a path to the schools of Cambridge that I was going to follow and then this was different. But actually the disruption was probably quite good for me. For a time I was not especially happy there.
Part of my problem was my passion for boats all through this time. At that point in my life I wanted to be a designer of yachts. Going back to the school it was fortunate that sailing was a big thing at the school so through the sailing I became happy in the school. I would design yachts, not constructing them of course, but I was quite serious about it. I had books about the calculations you must make and all the things you must think of. There was more mathematics involved with these things than I had learnt at school. My father showed me some mathematical techniques that I was using describing curves by the superimposition of sine waves.
So I was intrigued by all that - it was about this time that I became interested in mathematics - partly from the yacht design and partly because I found it very interesting and beautiful. Another thing was that it fitted in with what I was capable of doing in the sense that most of my family would be doing more projects where you actually built something - they would make a boat or make an airplane or something. But I hadn't got on very well with physically making things. I could make things, but not physical things, I could make things that existed as ideas. They were like projects but they were not practical projects. Projects of the mind - somehow I found that very congenial. I wanted to learn calculus when I heard about it; some years before it would be covered in school, I would study it.
I should have spoken about my grandfather - my mother's father - he wasn't a mathematician but he was a retired schoolmaster. He was extraordinarily keen on my development; in fact he took a great interest so he would give me many many books of all kinds but, in particular, at the stage we are talking about now he would go down to the book shops in Cambridge and buy me some suitable maths books. Some of them are still on these shelves. I might have had some interest that stayed dormant if there hadn't been a book to develop it. He took a special interest in all his grandchildren, but probably particularly in me. There were some excellent mathematics teachers in the school - they did take a special interest in me - but I wouldn't say they were the seed - they were nurturing.
From the time I was about fourteen, I gave up the idea of being a yacht designer - I decided I wanted to be a mathematician. That was presented to me like someone might want to be a concert pianist - many may want to do it but few will succeed. But in reality its not quite like that. But that was how I thought about it at the time. This was some very desirable thing to try to do but I had a plan B - to go and be an accountant or something like that. Most of my family went to Cambridge and, particularly in mathematics, it had the long tradition of being the main centre. Quite a big element at that time is the competitive tradition - there were scholarships to Oxford and Cambridge and financially they weren't worth very much but they but in the world of people at schools applying to Oxford and Cambridge they were a big thing. In the end I got a scholarship - the point I'm trying to make is that there is an element of competition in mathematics of being the smartest quickest person which, at this period, was a significant thing. Also in Cambridge there is a competition to be the top. I never was the top - it was a competition I would enter and I would do reasonably but I was never outstanding.
I became intrigued by the branch of mathematics called geometry which is a bit distant from what most people will think of as geometry. It does involve working with ideas in a visual way even if they are very abstract ideas. In the Cambridge course at that time there was almost nothing about geometry but there was one small course and there were applied mathematics courses like relativity and such things. The fact that it was not really taught added to the intrigue - a secret subject. As I soon found out, just down the road in Oxford, there were lots of people who did exactly that. In Cambridge there were people more around Stephen Hawking who were in a sense doing the same kind of thing but from an applied direction. Part of this is that at Cambridge mathematics is divided into pure and applied but I had decided I was going to be pure but a lot of activity was in applied. At Oxford they didn't make the same distinction - it is more unified.
I applied to Oxford for a Ph.D. I did alright in my exams in Cambridge. There was someone in Cambridge who was very important to me. There were several but the person I'm thinking of is Frank Adams. He was a topologist and I somehow managed to impress him. He was one of my referees for Oxford. He was very kind. After all my exams at Cambridge, Adams wrote to me saying, "I was one of your examiners and I want to congratulate you on what you wrote in these exams," This was kind of him. I didn't answer the questions to get the maximum marks in the least time - it wasn't my way of doing things. I tend to know a little about many things rather than knowing a lot about one particular thing. I bring together different ideas.
At Oxford it was two people who were important to me, Atiyah and Penrose. I went to Oxford and it was the activity and culture that developed around them that was important. One thing is that they are both remarkable people and particularly Atiyah is a very charismatic person who would have been at the top of any area he wanted. But also there was a wider movement at that time in mathematics partly in less division into compartments also specifically in extending the connection between geometry and physics. That was a much more general thing that was going on - a general intellectual movement; in particular Oxford was at the centre of that. It was exactly the right thing for me. I had then, and I have now, very little knowledge of advanced physics. I had a good background in classical physics but I have no, or essentially very little, technical knowledge of the front line of physics. But certain ideas and questions arose in physics which could be understood in mathematical terms. The lack of technical knowledge of physics was not a major barrier. I remember someone telling me at that time, "Don't try to learn quantum field theory until you have a tenured job."
Hitchin was my formal director. He suggested the direction for my dissertation, particularly work he'd done a few years before with Atiyah. But I don't think he meant me to work on the conjecture in general. What he suggested was that I checked some constructions for a special case. He never really said, "This is your project." Nevertheless, I started to try to solve the conjecture. I should say that I didn't meet Atiyah in the first year I was in Oxford. I worked on this conjecture. The thing there was that all these geometers in Oxford didn't do analysis - weren't experts on partial differential equations. I'm exaggerating, of course, but there was a kind of a gap there intellectually, particularly with non-linear partial differential equations. But partly because I had a lot of analysis in Cambridge and partly because it is something I find congenial, I embarked on solving this problem using techniques from analysis, particularly partial differential equations.
I was struggling to understand the techniques from partial differential equations that I had to learn for myself. I'm trying to understand something, then the way is to do thought experiments - if this is true what would be the consequences. So doing that, I had this idea about how the solutions to this equation could behave. That gave me this picture which gave a space for parameterising all the solutions. I looked at the properties of this space. On the other direction, on the side of topology, if you have such a space there are basic consequences so I put these consequences together and it transpired that this would be some new information about the manifolds - the 3-dimensional manifolds. I was not starting off thinking it was special - I didn't even know this information was new. At one time I thought it would be an interesting new approach to something well-known.
I had a colleague whom I shared an office with, Mike Hopkins, who is now a very distinguished topologist. He filled me in on some crucial information. I realised that this information would be important - new information. It depended on both knowing about the partial differential equations and about the topology, so maybe the Cambridge background was rather good there because we had not only learnt the geometry in a formal way, but had learnt other things. So with those two things anyone would see you got information but there were not too many people who had that combination of expertise. Once I'd made this observation it was obvious that I should go full steam ahead to develop it into a much bigger theory of how these spaces could give information about 4-dimensional manifolds. It took me a month or so to find out what it was good for. At this stage it was all thought experiments. If this is true then this follows, then this, etc. There was a lot of hard work filling in all the proofs and details.
For my first year at Oxford I was working with Hitchin but then he suggested I move to Atiyah. It was about a month after I started working with Atiyah that I came along and said, "I've got this formula," He was excited. It would be reasonable to say, if it is very broadly interpreted, that I'm still working somewhere in between partial differential equation approaches and with topological ideas. The precise mix of these things have evolved over the years. Another way to look at it is to say that from the early 1980s we suddenly discovered a lot more about these 4-dimensional manifolds - at least for 10 years after that there was an explosion of knowledge but after that the pace of discovery slowed down. There are still fundamental questions we have no way of attacking. So I still really like to feel I'm attacking these questions but because they are not really accessible I work round them. But they are still the centre of my interest. If you asked what I really want to understand, then it is these questions about 4-dimensional manifolds and not just the answers to the questions but why are these connections coming from physics and things like that. Interaction between these fields had thrived. It is for younger people who have a deeper understanding of quantum field theory.
2. The Mathematicians autobiography of Simon Donaldson.
My father had a large influence on my development into a mathematician, at least in a general way. I have an early memory of him saying with relish : "... and then I shall be able to get back to research" (presumably, after completing some chores which he had described to me). I had no idea what "research" might be, but from that time the word was tinged with glamour and romance.
My father was an engineer (my two brothers followed him) and was often sceptical of overly theoretical work. "All they produce is paper" - I hear him fulminating - "I bet they haven't touched a screwdriver in years". (Although this was not meant completely seriously; he had a deep interest in all kinds of science). Our house was always full of projects of a creative, practical kind: building model aircraft and so on. My own attempts in this direction were usually less successful; my vision of what I wanted to achieve outran my patience and ability to bring it about. So that was partly how I moved towards mathematics, where vision was not trammelled by irksome practical difficulties. Another thing that was very important was that I was fascinated by sailing, sailing boats, and anything nautical. So, when I was about thirteen, I decided that my career was to be a yacht designer, and began to design some. (I had no intention of actually building these yachts - that could wait until I had wealthy clients - so the enterprise was not limited by practicalities.) I went into this deeply. To design a yacht you need to calculate volumes, areas, moments, and so on from your plans. So it was quite natural for me to learn more mathematics. Gradually, the mathematics became the centre of my interests and the yacht designing fell into the background.
I was also lucky to have excellent mathematics teachers. It was important for me to be able to do well in mathematics and physics at school. A large influence in that direction came from my grandfather. He was a teacher of modern languages and, at a younger age, encouraged my interest in history and academic things generally.
By the time I was about sixteen, I had a fairly definite idea that what I wanted to be was a mathematician and some notion of what that was. I would investigate various questions that occurred to me, almost never making any definite progress. So in a way I was precocious (although definitely not in the sense of being particularly good at tests of the Mathematical Olympiad kind or, later, the undergraduate exams in Cambridge). This made the transition to life as a proper research mathematician, as a doctoral student in Oxford with Nigel Hitchin and Michael Atiyah, comparatively easy for me.
I mostly work by drawing pictures (vestiges of my yacht plans?) so I was naturally attracted to geometry. When I was an undergraduate, it was not so easy to learn about differential geometry, since it did not really come into the standard course, but this added to the allure of the subject. Holding fast to my metaphorical paternal screwdriver, I prefer problems that are quite concrete and specific, where one feels one is actually producing and working with some definite mathematical object.
I was blessed with good fortune at the start of my research career. At that time (1980) the Yang-Mills equations, arising in particle physics, were making a big impact in pure mathematics, particularly in connection with geometry and Roger Penrose's twistor theory. The project Hitchin suggested to me involved a rather different kind of question, connecting differential and algebraic geometry but veering more towards analysis and partial differential equations. Happily - with rather different purposes in mind, Karen Uhlenbeck and Cliff Taubes had, in the few years before, gone a long way in developing the relevant analytical techniques. Of course this was before the Internet made it so easy to find papers, and I remember the exciting day when I received Uhlenbeck's preprints by mail from the USA. This was when I was a first-year doctoral student. A good approach in research, I find, is to imagine what should be true, i.e. a picture of what properties some mathematical objects have, and then explore the consequences. If the consequences lead to a contradiction, that shows that the picture needs to be modified; on the other hand, if the consequences fit in with what one knows otherwise and lead to some interesting further predictions, that is good evidence for the correctness of the picture. Following this strategy (although certainly not consciously) and exploring the properties of Yang-Mills instantons, I stumbled on, at the beginning of my second year, an entirely unexpected application of these to the topology of four-dimensional manifolds. The two main themes of my research in the twenty-seven years since then have been extending this and, in a different direction, developing the links between algebraic geometry, differential geometry, and partial differential equations.
3. The Shaw Prize autobiography of Simon K Donaldson.
I was born in 1957 in Cambridge, England, the third of four children. At that time, my father worked as an electrical engineer in the Physiology Department of the University. My mother had been brought up in Cambridge and had taken a Science degree there. When I was 12 we moved to a village in Kent, following my father's appointment to lead a team in London developing neurological implants.
The passion of my youth was sailing. Through this, I became interested in the design of boats, and in turn in mathematics. From the age of about 16, I spent much time studying books, puzzling over problems and trying to explore. I did well in mathematics and physics at school, but not outstandingly so.
In 1976 I returned to Cambridge for my first degree. The subject I liked best was geometry, although there was rather little of this in the Cambridge course at that time and my main training was in analysis, topology and traditional Mathematical Physics. The word "geometry" may convey a misleading impression. The modern subject is very far from Euclid's, and it is perhaps better to think of vector calculus and, for example, the geometrical notion of the "flux" of a vector field.
In 1980 I moved to Oxford to work for a doctorate, supervised by Nigel Hitchin. This was an exciting time in Oxford. Penrose's "twistor theory" was dominant, an early example of the now-pervasive interaction between geometry and fundamental physics. Sir Michael Atiyah, who supervised my work later, was a driving force in this and a few years before he, Hitchin, Drinfeld and Manin had done renowned work on Yang-Mills instantons, using twistor theory and geometry of complex variables. These instantons solve generalisations of Maxwell's equations.
In my thesis I studied two different, but related, topics which have developed into the two themes of most of my subsequent research. The first theme is the interaction between differential and algebraic geometry. The problem that Hitchin proposed to me was to relate the instantons over complex spaces to "bundles" studied by algebraic geometers. What was unusual, in terms of the Oxford environment, was that I tried to tackle this problem using analytical techniques. This kind of approach was by no means new in other problems and in other parts of the world, but was not a strong tradition in the UK. I learnt the trade by studying preprints of Cliff Taubes and Karen Uhlenbeck, which opened up the analytical approach to Yang-Mills theory.
My original focus was on the case of a complex space, but certain central questions made sense on any 4-dimensional manifold. Thinking about these, and combining with an existence result proved by Taubes, led to the other topic of my thesis: the application of Yang-Mills theory to 4-manifold topology. This was quite unexpected. While the basic argument "stared me in the face", thanks to my training in topology, hard work was required to carry it through in detail.
Now I turn to the decade 1983-93. I spent a year in Princeton, and met my wife, Nora, during a visit to the University of Maryland. Our three children, Andres, Jane and Nicholas were born, joining my step-daughter Adriana. (Andres and Jane have both now gained degrees from Cambridge and Nicholas is nearing the end of high school. Adriana is a mathematics teacher.) In research, my focus was on developing the topological applications into a general theory. This was as part of a large endeavour by many mathematicians around the world. I wrote one monograph, with Kronheimer, about these ideas and another dealing with Floer's theory, which extends them to 3-dimensional manifolds. I was made a Professor in Oxford in 1985 and was fortunate to have many research students.
From about 1994 I developed a different research strand, introducing techniques into symplectic topology. The ideas meshed in well with the results obtained by Taubes around that time. After two decades of dramatic development, 4-manifold theory has reached a much more steady state and, while a great deal is known, there are huge areas where we are entirely ignorant. This strand was an example of an attempt to find some new approach.
We spent the year 1997-8 in Stanford, and I moved back to my current position at Imperial College, London. My wife runs a Medical Statistics Unit at King's College. There was little geometry in Imperial then, but now, thanks largely to the drive of my colleague Richard Thomas, we have one of the main centres for research in this area. My work over the past decade has in a sense returned to my thesis problem, but extended into Riemannian geometry. This is an area with a longer history and the problems are much harder, but the theory is developing in an exciting way. Many excellent young mathematicians around the world are entering the field and contributing to these developments.
7 October 2009, Hong Kong
4. The Shaw Prize essay 2009.
Over the past 30 years, geometry in 3 and 4 dimensions has been totally revolutionised by new ideas emerging from theoretical physics. Old problems have been solved but, more importantly, new vistas have been opened up which will keep mathematicians busy for decades to come.
While the initial spark has come from physics (where it was extensively pursued by Edward Witten), the detailed mathematical development has required the full armoury of non-linear analysis, where deep technical arguments have to be carefully guided by geometric insight and topological considerations.
The two main pioneers who both initiated and developed key aspects of this new field are Simon K Donaldson and Clifford H Taubes. Together with their students, they have established an active school of research which is both wide-ranging, original and deep. Most of the results, including some very recent ones, are due to them.
To set the scene, it is helpful to look back over the previous two centuries. The 19th century was dominated by the geometry of 2-dimensional surfaces, starting with the work of Abel on algebraic functions, and developing into the theory of complex Riemann surfaces. By the beginning of the 20th century, Poincaré had introduced topological ideas which were to prove so fruitful, notably in the work of Hodge on higher dimensional algebraic geometry and also in the global analysis of dynamical systems.
In the latter half of the 20th century there was spectacular progress in understanding the topology of higher dimensional manifolds and fairly complete results were obtained in dimensions 5 or greater. The two "low dimensions" of 3 and 4, arguably the most important for the real physical world, presented serious difficulties but these were expected to be surmounted, along established lines, in the near future.
In the 1980's this complacent view was shattered by the impact of new ideas coming from physics. The first breakthrough was made by Simon K Donaldson in his PhD thesis where he used the Yang-Mills equations of -gauge theory to study 4-dimensional smooth (differentiable) manifolds. Specifically, Donaldson studied the moduli (or parameter) space of all -instantons, solutions of the self-dual Yang-Mills equations (which minimise the Yang-Mills functional), and used it as a tool to derive results about the 4-manifold. This instanton moduli space depends on a choice of Riemannian metric on the 4-manifold but Donaldson was able to produce results which were independent of the metric.
There are serious analytical difficulties in carrying out this programme and Donaldson had to rely on the earlier work of Karen Uhlenbeck and Clifford H Taubes. As these new ideas were developed and expanded by Donaldson, Taubes and others, spectacular results came tumbling out. Here is an abbreviated list, which shows the wide and unexpected gulf between topological 4-manifold (where the problems had just been solved by Michael Freedman) and smooth 4-manifold:
(1) Many compact topological 4-manifold which have no smooth structure.
(2) Many inequivalent smooth structures on compact 4-manifold.
(3) Uncountably many inequivalent smooth structures on Euclidean 4-space.
(4) New invariants of smooth structures.
The invariants in (4) were first introduced by Donaldson using his instanton moduli space. Subsequently, an alternative and somewhat simpler approach emerged, again from physics, in the form of Seiberg-Witten theory. Here, one just counted the finite number of solutions of the Seiberg-Witten equations (i.e. the moduli space was now zero dimensional).
(2) Many inequivalent smooth structures on compact 4-manifold.
(3) Uncountably many inequivalent smooth structures on Euclidean 4-space.
(4) New invariants of smooth structures.
One of Taubes' great achievements was to relate Seiberg-Witten invariants to those introduced earlier by Gromov for symplectic manifolds. Such manifolds occur both as phase spaces in classical mechanics and in complex algebraic geometry, through the Kahler metrics inherited from projective space and exploited by Hodge. Although symplectic manifolds need not carry a complex structure, they always carry an almost (i.e. non-integrable) complex structure. Gromov introduced the idea of "pseudo-holomorphic curves" on symplectic manifolds and obtained invariants by suitably counting such curves. Taubes, in a series of long and difficult papers, proved that, for a symplectic 4-manifold, the Seiberg-Witten invariants essentially coincide with the Gromov-Witten invariants (an extension of the Gromov invariants). The key step in the work of Taubes is the construction of a pseudo-holomorphic curve from a solution of the Seiberg-Witten equations. This is fundamental since it connects gauge theory (a theory of potentials and fields) to sub-varieties (curves). Roughly, it represents a kind of non-linear duality.
In fact, extending complex algebraic geometry to symplectic manifolds (of any even dimension) was again pioneered by Donaldson who proved various existence theorems such as the existence of symplectic submanifolds. In the apparently large gap between algebraic geometry and theoretical physics, symplectic manifolds form a natural bridge and the recent results of Donaldson, Taubes and others provide, so to speak, a handrail across the bridge.
All this work in 4 dimensions has an impact on 3 dimensions, especially through the work of Andreas Floer, and Taubes has made many contributions in this direction. His most outstanding result is his very recent proof, in 3 dimensions, of a long-standing conjecture of Alan Weinstein. This asserts the existence of a closed orbit for a Reeb vector field on a contact 3-manifold. Contact 3-manifolds arise naturally as level sets of Hamiltonian functions (energy) on a symplectic 4-manifold, and the Weinstein conjecture now asserts the existence of a closed orbit of the Hamiltonian vector field. This latest tour de force of Taubes exhibits his real power as a geometric analyst.
In recent years Donaldson has turned his attention to the hard problem of finding Hermitian metrics of constant scalar curvature on compact complex manifolds. The famous solution by Yau of the Calabi conjecture is an example of such problems. Donaldson has recast the constant scalar curvature problem in terms of moment maps, an idea derived from symplectic geometry which played a key role in gauge theory. This construction of metrics is a much deeper problem, being extremely non-linear but Donaldson has already made incisive progress on the analytical questions involved. This new work of Donaldson represents an exciting new advance which is currently attracting much attention.
This quick summary of the contributions of both Donaldson and Taubes shows how they have transformed our understanding of 3 and 4 dimensions. New ideas from physics, together with deep and delicate analysis in a topological framework, have been the hallmark of their work. They are fully deserving of the Shaw Prize in Mathematical Sciences for 2009.
Mathematical Sciences Selection Committee
The Shaw Prize
7 October 2009, Hong Kong
5. Simon Donaldson: 2015 Breakthrough Prize in Mathematics.
Simon Donaldson was awarded the 2015 Breakthrough Prize in Mathematics: "For the new revolutionary invariants of four-dimensional manifolds and for the study of the relation between stability in algebraic geometry and in global differential geometry, both for bundles and for Fano varieties."
The Science
We experience the world in three dimensions: up and down, left and right, forward and back. But according to Einstein, there are actually four: his theory of general relativity integrates time with the three spatial dimensions. And while we can't visualise four dimensions, we can analyse them with mathematics. Simon Donaldson has transformed our understanding of four-dimensional shapes, showing which ones can be "tamed" with the kind of equations that mathematicians can solve, and which can't. In the process, he both provided powerful new tools for physicists and incorporated new ideas from physics into mathematics.
Comments by Simon Donaldson
It has been my great good fortune that my career has spanned a period of exceptionally exciting developments in my field. Ideas and techniques from different areas - topology, physics, differential equations, geometry - have become interwoven in ways that no one would have predicted half a century ago. It is a privilege to have been able to witness this and take some part in it. Mathematics has a long time scale. One of the pleasantest things is, looking back in time, to contemplate how the developments we see in our lifetimes fit into the longer term. And, looking forward in time, we have confidence that the problems that seem to us intractable will yield to future advances, invisible to us now. I owe enormous thanks to my advisors, Michael Atiyah and Nigel Hitchin, and to all the mathematicians who made Oxford in the 1980s such a special place. What I learned then has underpinned my whole career. Looking forward, rather than back, it has been my great good fortune to have been able to watch the development of many extraordinary research students. I would like to thank my wife, Nora, my parents, and all my family for their support.
6. Sir Simon Kirwan Donaldson: Wolf Prize Laureate in Mathematics 2020.
Simon Donaldson and Yakov Eliashberg were jointly awarded the Wolf Prize: "for their contributions to differential geometry and topology."
Sir Simon Kirwan Donaldson (born 1957, Cambridge, U.K.) is an English mathematician known for his work on the topology of smooth (differentiable) four-dimensional manifolds and Donaldson-Thomas theory.
Donaldson's passion of youth was sailing. Through this, he became interested in the design of boats, and in turn in mathematics. Donaldson gained a BA degree in mathematics from Pembroke College, Cambridge in 1979, and in 1980 began postgraduate work at Worcester College, Oxford.
As a graduate student, Donaldson made a spectacular discovery on the nature or 4-dimensional geometry and topology which is considered one of the great events of 20th century mathematics. He showed there are phenomena in 4-dimensions which have no counterpart in any other dimension. This was totally unexpected, running against the perceived wisdom of the time.
Not only did Donaldson make this discovery but he also produced new tools with which to study it, involving deep new ideas in global nonlinear analysis, topology, and algebraic geometry.
After gaining his DPhil degree from Oxford University in 1983, Donaldson was appointed a Junior Research Fellow at All Souls College, Oxford, he spent the academic year 1983-84 at the Institute for Advanced Study in Princeton, and returned to Oxford as Wallis Professor of Mathematics in 1985. After spending one year visiting Stanford University, he moved to Imperial College London in 1998. Donaldson is currently a permanent member of the Simons Center for Geometry and Physics at Stony Brook University and a Professor in Pure Mathematics at Imperial College London.
Donaldson's work is remarkable in its reversal of the usual direction of ideas from mathematics being applied to solve problems in physics.
A trademark of Donaldson's work is to use geometric ideas in infinite dimensions, and deep non-linear analysis, to give new ways to solve partial differential equations (PDE). In this way he used the Yang-Mills equations, which has its origin in quantum field theory, to solve problems in pure mathematics (Kähler manifolds) and changed our understanding of symplectic manifolds. These are the phase spaces of classical mechanics, and he has shown that large parts of the powerful theory of algebraic geometry can be extended to them.
Applying physics to problems or pure mathematics was a stunning reversal of the usual interaction between the subjects and has helped develop a new unification of the subjects over the last 20 years, resulting in great progress in both. His use of moduli (or parameter) spaces of solutions of physical equations - and the interpretation of this technique as a form of quantum field theory - is now pervasive throughout many branches of modem mathematics and physics as a way to produce "Donaldson-type Invariants" of geometries of all types. In the last 5 years he has been making great progress with special geometries crucial to string theory in dimensions six ("Donaldson-Thomas theory"), seven and eight.
Professor Simon Donaldson is awarded the Wolf Prize for his leadership in geometry in the last 35 years. His work has been a unique combination of novel ideas in global non-linear analysis, topology, algebraic geometry, and theoretical physics, following his fundamental work on 4-manifolds and gauge theory. Especially remarkable is his recent work on symplectic and Kähler geometry.
Last Updated December 2023