1.1. The Junior Berwick Prize.

The Junior Berwick Prize, named after Professor W E H Berwick who was a Vice-President of the London Mathematical Society, is awarded by that Society in odd-numbered years was first awarded in 1947. W E H Berwick left money to establish two prizes which was given to the Society by his widow Daisy May Berwick. After 1999, the Junior Berwick Prize was renamed the Berwick Prize. The Berwick Prize for year X can only be awarded to a mathematician who, on 1 January of year X, is a member of the Society, is not already a Fellow of the Royal Society and has fewer than 15 years (full time equivalent) of involvement in mathematics at post-doctoral level, allowing for breaks in continuity, or who in the opinion of the Prizes Committee is at an equivalent stage in their career. It is awarded in recognition of an outstanding piece of mathematical research actually published by the Society during the eight years ending on 31 December of year X-1; and it may not be awarded to any person who has previously received the De Morgan Medal, the Pólya Prize, the Senior Berwick Prize, the Senior Whitehead Prize, the Naylor Prize or a Whitehead Prize.

1.2. Michael Atiyah awarded the 1961 Junior Berwick Prize.

The London Mathematical Society met on Thursday 15 June 1961 with the President, Hans Heilbronn, in the chair. The President announced that the Junior Berwick Prize for 1961 had been awarded to Dr M F Atiyah for his paper "Vector bundles over an elliptic curve", published in the Proceedings of the London Mathematical Society 7 (1957), 414-452.

1.3. Introduction to "Vector bundles over an elliptic curve".

The primary purpose of this paper is the study of algebraic vector bundles over an elliptic curve (defined over an algebraically closed field $k$). The interest of the elliptic curve lies in the fact that it provides the first non-trivial case, Grothendieck having shown that for a rational curve every vector bundle is a direct sum of line-bundles.

In order to provide the necessary background a certain amount of general material, not found in the literature, has been included. This consists of a brief discussion of 'Theorems A and B' and their relation with Universal bundles, a little on protective bundles, and some results on reduction of structure group. The case of vector bundles over an algebraic curve is treated in greater detail, and more precise results are obtained. In particular a refinement of Theorems A and B is given (Theorem 1) which seems to be a necessary preliminary in any attempt at classification of vector bundles. This concludes Part I of the paper.

Part II is devoted to the classification of vector bundles over an elliptic curve. The problem is completely solved and the main result is stated in Theorem 7. The characteristic of the field does not enter into this part of the problem, and the results are valid in both characteristic 0 and $p$.

In Part III we examine the operation of the tensor product. This is most easily expressed in terms of the ring $\hat{\mathcal{E}}$ generated by the vector bundles (cf. Part I, § 1). We show (Theorem 12) that $\hat{\mathcal{E}}$ is the tensor product of certain subrings $\hat{\mathcal{E}}_0$ and $\hat{\mathcal{E}}_p$ (over all primes $p$), and the ring structure of $\hat{\mathcal{E}}_0$ and $\hat{\mathcal{E}}_p$ is given by Theorems 8, 13, and 14. These results are all for the characteristic zero case, and we make only a few isolated remarks for the case of characteristic $p$.

We conclude the paper with a few brief applications of the results of Parts II and II.

2.1. The International Congress of Mathematicians 1966.

Georges de Rham, Chairman of the Fields Committee, made the following statement at the Opening Ceremony of the International Congress of Mathematicians held in Moscow in 1966:

Professor J C Fields, President of the International Congress of Mathematicians held in Toronto in 1924, proposed that two gold medals be awarded at each International Congress, for outstanding achievements in Mathematics. He set up a fund for that purpose, from out of the balance left over at the end of the Toronto Congress. In 1932, after his death, the International Congress held at Zürich decided to accept his proposal with thanks. As is well known, two medals have been presented at every Congress since held: Oslo 1936, Harvard 1950, Amsterdam 1954, Edinburgh 1958 and Stockholm 1962.

Following a tradition which has become well established, the Executive Committee of the International Mathematical Union appoints, in advance of the Congress, a special Committee to select the candidates. This time the Committee consists of Professors H Davenport, M Deuring, W Feller, M A Lavrentiev, J P Serre, D C Spencer, R Thorn, and I have been asked to be the chairman. Every one of the members has taken an active part in the deliberations. We have also consulted other experts. I thank them all for their valuable contribution.

The Memorandum of Fields says: "Because of the multiplicity of the branches of Mathematics and taking into account the fact that the interval between such Congresses is four years, it is felt that at least two medals should be available." In view of the vast development of Mathematics during the last forty years, it appears that this number could judiciously be increased to four. The Executive Committee of the International Mathematical Union has therefore viewed with sympathy the generous offer made by an anonymous donor to give this year two more medals. The Organising Committee of this Congress having agreed to this and the Medals Committee having accepted the responsibility to select four names, four medals will be awarded today. The medals have been struck by the Royal Mint in Ottawa. The name of the recipient is engraved on each of them. The name of Fields does not figure on them. Fields himself proposed to call them: "International Medals for outstanding discoveries in Mathematics". Each of them carries with it a cash prize which, this year, amounts to 1,500 Canadian dollars.

The Memorandum of Fields also contains the following: "In coming to its decision, the hands of the International Committee should be left as free as possible. It would be understood, however, that in making the awards, while it was in recognition of work already done, it was at the same time intended to be an encouragement for further achievements on the part of the recipients and a stimulus to renewed efforts on the part of the others."

On the basis of this text, and following precedent, we confine our choice to candidates under forty. We prepare a first list of about 30 names. We then looked for those whose work appeared to us the most important and the most striking, irrespective of any other consideration, setting aside any question of nationality. To our regret, we have had to give up several names which would have also deserved this distinction. Several young mathematicians of extra­ ordinary brilliance were among them. But because they are so young, there will be many Congresses before they reach forty and if they continue in their course, they will have every chance of receiving a medal. The choice was thus not easy. Nevertheless, after serious consideration and reflexion, we arrive at a conclusion without undue difficulty. The following four names, in alphabetical order, constitute our choice:

Michael Francis Atiyah,
Paul J Cohen,
Alexander Grothendieck,
Stephen Smale.

Unfortunately, A Grothendieck, was unable to come. May I call Messrs Atiyah, Cohen and Smale to come forward and receive these medals from the hands of Academician Keldysh. A brief account of their achievements will be given by noted mathematicians.

2.2. Henri Cartan on Michael Atiyah's achievements.

I will talk very briefly about Atiyah's work in three areas, which are closely linked to each other: K-theory, the index formula, and the "Lefschetz formula". I will leave aside other contributions to algebraic geometry or the theory of cobordism, although they are very interesting indeed, and I will also pass over in silence the very recent results, still unpublished, which the author himself will speak about in his lecture during this Congress.
...
In conclusion, we owe to Michael Atiyah several major contributions which closely link Topology and Analysis. Each of them was carried out collaboratively; without diminishing in any way the part that falls to such eminent collaborators as Hirzebruch, Singer or Bott, there is no doubt that in each case Atiyah's personal intervention was decisive. He gives us the example of a mathematician in whom the clarity of conceptions and the overall vision of phenomena are harmoniously combined with creative imagination, and also with the perseverance which leads to great achievements.

2.3. Michael Atiyah's lecture at the ICM 1966.

The subject matter of this talk lies in the area between Analysis and Algebraic Topology. More specifically, I want to discuss the relations between the analysis of linear partial differential operators of elliptic type and the algebraic topology of linear groups of finite-dimensional vector spaces. I will try to show that these two topics are intimately related, and that the study of each is of great importance for the development of the other.

The theory of elliptic differential equations has of course a long, and rich history, with its origins in the study of the Laplace equation and of the closely-associated Cauchy-Riemann equations. Its connection with topology, via the theory of holomorphic functions and Riemann surfaces, is equally classical. Its development in the last fifty years or so has however followed two rather separate courses.

On the one hand there has been the purely analytical development, the qualitative study of general elliptic operators. Here the emphasis has been on extending the basic theory of the Laplace operator to general operators of the same type - what we now call elliptic operators. The questions studied include regularity of solutions, boundary conditions and more recently the extension to suitable classes of integro-differential operators - now called pseudo-differential operators. On the whole this sort of work was carried out for domains in Euclidean space, though the extension to more general manifolds presents. nothing essentially new.

The second development has been the more detailed or quantitative study of the classical operators and their associated structures. This essentially includes the whole of algebraic geometry treated by topological and transcendental methods. The pioneering work in this field was of course done by Hodge some thirty years ago.

Roughly speaking we might say that the analysts were dealing with complicated operators and simple spaces (or were only asking simple questions), while the algebraic geometers and topologists were only dealing with simple operators but were studying rather general manifolds and asking more refined questions.

In recent times, the last five years or so, some serious attempts have been made to integrate these two different developments. Each seems now to have reached such a stage of maturity that it can confidently offer its services to the other half. For example, some of the ideas and techniques developed in the general theory of partial dif­ferential equations have been very successfully applied to the study of complex manifolds. My own interests, however, have been in the reverse direction and I would like to spend the rest of my time discussing the Riemann-Roch or Index Problem.

3.1. Royal Medal awarded to Michael Atiyah.

Michael Atiyah was awarded the Royal Medal by the Royal Society of London:-

... in recognition of his distinguished contributions to algebraic geometry and to the study of differential equations by the methods of algebraic topology.

3.2. Address of P S M Blackett, President of the Royal Society.

P S M Blackett, President of the Royal Society, addressed the Anniversary Meeting on 30 November 1968:

A Royal Medal is awarded to Professor M F Atiyah, F.R.S.

M F Atiyah is recognised as one of the world's outstanding mathematicians. Working with Hirzebruch he showed that important results in algebraic geometry had unexpected analogues in algebraic topology and differential geometry. In this work they developed 'K-theory', a theory that in the hands of Atiyah and others has led to the solution of many long-outstanding problems in differential topology. Later he introduced the 'bordism groups' and initiated a new and important branch of topology. More recently, working with Singer, he has shown that the Riemann-Roch theorem in algebraic geometry has an analogue in the theory of partial differential equations of elliptic type. Similarly, working with Bott, he has transformed the Lefschetz fixed-point formula in algebraic topology to an analogue in the theory of partial differential equations. In particular, these results relate the properties of the solutions of differential equations in a region to the topological properties of the region and its boundary.

Thus his wide and deep mathematical knowledge and his extraordinary insight have enabled him to found productive new theories and to display totally unexpected and very powerful connexions between algebraic geometry, algebraic and differential topology and the theory of partial differential equations.

4.1. London Mathematical Society meeting 20 June 1980.

The London Mathematical Society met at the Geological Society, Burlington House, London on Friday 20 June 1980. The President, C T C Wall, was in the Chair. About 50 members were present. The President presented the 1980 De Morgan Medal to M F Atiyah.

4.2. Citation for Michael Francis Atiyah.

Atiyah's extensive mathematical knowledge and keen insight have enabled him to found productive new theories and to reveal deep-seated connexions and analogies between diverse branches of mathematics. He has made important contributions to algebraic geometry, to differential geometry and topology, to the theory of group representations and to that of partial differential equations. His work in algebraic geometry continued the tradition of Baker, Hodge and Todd, and was a crucial contribution to the transformation of algebraic geometry by the introduction of topological methods. Together with Hirzebruch he developed K-theory, which in his hands and those of others led to the solutions of outstanding problems in homotopy theory and which in its various aspects has had repercussions in other areas. He established an entirely unexpected relation (by means of a spectral sequence)between the characters of a finite group and its homological properties.

He found, in joint work with Singer, an important result concerning elliptic partial differential equations, the Index Theorem, generalising the Riemann-Roch theorem. The formulation uses his earlier K-theory and led to numerous further extensions and generalisations: a fixed point theorem (jointly with Bott, and generalising a theorem of Lefschetz in algebraic topology); a version for actions of compact groups, giving new understanding of representation theory; versions for manifolds with boundary, leading to wholly new and intriguing invariants and to are-examination of the notion of ellipticity; and a version for noncompact "transversely elliptic" group actions, drawing once again on new ideas from commutative algebra and from the theory of distributions.

Recently he has investigated certain geometric objects, "instantons", introduced by physicists. A beautiful use of the "Penrose correspondence" led Atiyah (with Ward, Hitchin and Singer) to reinterpret certain manifolds with connexion in terms of holomorphic vector bundles over projective 3-space, and hence to obtain classifications: a piece of work which has excited great interest.

5.1. The Antonio Feltrinelli Prize.

Antonio Feltrinelli was born in Milan on 1 June 1887 to Giovanni Feltrinelli, nephew of Giacomo, founder of the Feltrinelli Collective Society from which the Fratelli Feltrinelli joint-stock company would later arise, with the various associated companies. A figure of notable importance in the Italian economic and financial field, Antonio Feltrinelli took over the management of the family group's affairs in 1935.

When, after the death of his brothers, he was left alone, he decided to use his personal fortune to found a large Italian cultural institution "similar to the Nobel Foundation". Therefore, in his will dated 15 March 1936 (published in 1942) he provided that an inalienable and perpetual fund be established to: "reward work, study, intelligence, in short, those men who stand out most in high works, in the arts, in the sciences, for they are the true benefactors of their country and of humanity."

The "Antonio Feltrinelli Fund" is managed by the Accademia Nazionale dei Lincei in order to award national and international prizes to people who have become illustrious in the sciences and arts.

5.2. Michael Atiyah writes about receiving the Prize.

I had commented with approval on the absence of Nobel Prizes in mathematics. Ironically a short while later I heard that I had been awarded the Feltrinelli Prize by the Italian Academy (the 'Lincei'). While lacking the fame or notoriety of the Nobel Prizes it was comparable financially and my mathematical predecessors were Hadamard, Lefschetz, and Leray: I was in excellent company. The presentations were to be made in Rome by the President of the Republic, and I was asked to prepare a speech of about 25 minutes on my work. This was clearly a serious task and I gave the matter much thought. Presumably I was supposed to survey my mathematical work in a way that would not be entirely incomprehensible to the distinguished audience at the Italian Academy. Eventually, with some misgiving, I produced "Speech on conferment of Antonio Feltrinelli Prize", and set off with my wife for the ceremony in Rome. Fortunately, Rome is not Stockholm and the Italians have a different way of doing things. There was no lack of ceremonial with imposing guardsmen in uniform and a beautifully furnished salon. Moreover, President Pertini duly turned up. However, as the proceedings progressed, time was running short, the President had other obligations to attend to and I was asked to compress my carefully prepared speech into 10 minutes! I hastily decided which passages to select. I cannot now remember what I omitted but no doubt I left out all the more mathematical bits. In a way it was a great relief. So, although my speech was essentially not given, the written version now serves a different purpose by providing a brief overview of my papers.

5.3. Speech on conferment of Antonio Feltrinelli Prize.

Atiyah began the paper as follows:

I feel deeply honoured by the award of the Antonio Feltrinelli Prize and for this opportunity of talking briefly about my work in mathematics. It is of course necessary for me to put things into historical perspective and to explain the role of many of my predecessors and collaborators. I shall, as far as possible, attempt to describe the development of ideas without entering into precise technicalities. Of course mathematics, like other sciences, rests ultimately on intricate and detailed arguments, but on an occasion such as this we shall have to overlook the fine design and concentrate on the broad sweep.

Large parts of mathematics centre ultimately around the solution of equations. At an elementary level we have algebraic equations, and, at a more sophisticated level, differential equations. Mathematicians study such equations attempting to obtain as much information as possible about their solutions, not just in a crude numerical sense but also in a qualitative and structural manner. One can distinguish broadly two levels of information concerning such solutions, the local and the global. Local information is for example concerned with studying all solutions near a given solution and the techniques for this rest on classical analysis developed and refined from Newton up to the nineteenth century. Global information by contrast is concerned with solutions which are "far apart" and here new ideas had to be developed which eventually led to the birth of the subject of topology, the most fundamental and primitive branch of geometry. Riemann, in the mid 19th century, in connection with algebraic equations, and later Poincaré, in connection with differential equations, were the pioneers and their work was notably extended in the first half of the 20th century by Lefschetz, Morse and Hodge.

6.1. The 1987 King Faisal International Prizes.

This month two Britons and a Nigerian will receive the prestigious King Faisal Foundation Prizes for science, medicine and service to Islam, according to an announcement by Prince Khaled Al-Faisal, governor of the Asir Region and director general of the King Faisal Foundation. The prizes, which include a SR350.000 ($93,333) cash grant, a gold medallion and a framed certificate for each winner, will be awarded to Sir Michael Atiyah, Dr Barrie Russell Jones and Sheikh Abubakar Mahmud Gumi.

6.2. Briton awarded the King Faisal Prize for Science.

Sir Michael Atiyah, winner of the King Faisal Prize for Science, is a geometry professor at Oxford University. Born in Great Britain and educated in Cairo, Egypt, and Manchester, England, Atiyah will receive the King Faisal Prize for his latest work in algebraic geometry and theoretical physics, in which he demonstrated that algebraic geometry can be used to obtain solutions to partial differential equations. Among physicists and mathematicians, this finding has led to a greater understanding of quantum field theory and general relativity.

7.1. The Copley Medal.

The Copley Medal is the Royal Society of London's oldest and most prestigious award. The medal is awarded for sustained, outstanding achievements in any field of science.

First awarded in 1731 following donations from Godfrey Copley FRS, it was initially awarded for the most important scientific discovery or for the greatest contribution made by experiment. The Copley Medal is thought to be the world's oldest scientific prize and it was awarded 170 years before the first Nobel Prize. Notable winners include Benjamin Franklin, Dorothy Hodgkin, Albert Einstein and Charles Darwin. The medal is of silver gilt, is awarded annually, alternating between the physical and biological sciences (odd and even years respectively), and is accompanied by a gift of £25,000.

7.2. The award to Michael Atiyah.

The 1988 Copley Medal is awarded to Michael Atiyah:-

... in recognition of his fundamental contributions to a wide range of topics in geometry, topology, analysis and theoretical physics.

7.3. The Copley Medal Award ceremony.

The President of the Royal Society of London, Sir George Porter,
awarded the Copley Medal to Michael at the Anniversary Meeting, 30 November 1988:

The Copley Medal is awarded to Sir Michael Atiyah, F.R.S., in recognition of his fundamental contributions to a wide range of topics in geometry, topology, analysis and theoretical physics.

For a quarter of a century, Sir Michael Atiyah has been the unrivalled leader of mathematical research in Britain. His work spans a range unequalled among his contemporaries, including algebraic topology, algebraic and differential geometry, analysis and theoretical physics. Many of his pioneering contributions have been the seeds for extensive later developments. His most celebrated work centres around the Atiyah-Singer Index Theorem, which shows how the topological behaviour of a manifold influences the behaviour of elliptic partial differential equations over it. This is one of the great insights of twentieth-century mathematics. In a series of papers, he and his collaborators have explored its many variants and ramifications, and have opened up new directions in many areas of mathematics, breaking through the old divisions between topology, differential equations and functional analysis. In recent years he has been a leader in developing links between geometry and particle physics, and many of the topics he has helped develop are in the forefront of research by mathematicians and physicists in this new and active area.

8.1. The Benjamin Franklin Medal.

In 1906, the United States Congress authorised a commemorative medal to mark the 200th anniversary of the birth of Benjamin Franklin. The first medal was presented "under the direction of the President of the United States" to the Republic of France. In recognition of its founder, subsequent medals were given to the American Philosophical Society for its use.

The Franklin Medal's meaning has changed over the course of its history. From 1937 to 1983 the medal was given for especially noteworthy contributions to the American Philosophical Society. From 1985 to 1991 the medal was the Society's highest award for the humanities and sciences. And in 1987, the Benjamin Franklin Medal for Distinguished Public Service was established to honour exceptional contributions to the general welfare. In 1993, when the Thomas Jefferson Medal was authorised by Congress, the Benjamin Franklin Medal was designated for recognition of distinguished achievement in the sciences. The medal is the Society's highest award for distinguished public service and the sciences.

8.2. The 1988 Benjamin Franklin Medal for Distinguished Achievement in the Sciences.

The Benjamin Franklin Medal for Distinguished Achievement in the Sciences was awarded to Sir Michael Atiyah by the American Philosophical Society in 1988:-

...in recognition of significant contributions to a remarkable range of mathematical topics, which established links between differential geometry, topology, and analysis; and creating useful mathematical tools for physicists.

8.3. The London Mathematical Society Report.

Sir Michael Atiyah, PRS, has been awarded the Benjamin Franklin Medal of the American Philosophical Society (APS). He was honoured for establishing links among differential geometry, topology and analysis, thereby creating useful mathematical tools for physicists. The Benjamin Franklin Medal is awarded in recognition of distinguished achievement in the sciences. Authorised by the United States Congress in 1906, the medal commemorates the 200th anniversary of the birth of Franklin, who founded the APS in 1743. Sir Michael received the Franklin Medal at the APS 250th Anniversary Celebration held in April 1993.

8.4. The American Mathematical Society Report.

Sir Michael Atiyah has been awarded the Benjamin Franklin Medal of the American Philosophical Society (APS). Atiyah is Master of Trinity College of Cambridge University, Director of the Isaac Newton Institute for the Mathematical Sciences at Cambridge, and President of the Royal Society. He was honoured for establishing links among differential geometry, topology, and analysis, thereby creating useful mathematical tools for physicists.

The Benjamin Franklin Medal is awarded in recognition of distinguished achievement in the sciences. Authorised by Congress in 1906, the medal commemorates the 200th anniversary of the birth of Franklin, who founded the Society in 1743. The Society has 694 elected members, most of them from the U.S. Atiyah was one of four receiving the Franklin Medal at the Society's 250th Anniversary Celebration, held 28 April through 1 May 1993.

9.1. The Jawaharlal Nehru Birth Centenary Medal.

The Jawaharlal Nehru Birth Centenary Medal is awarded by the Indian National Science Academy to recognise contributions in the area of International Cooperation in Science and Technology and Public Understanding of Science. The 'Jawaharlal Nehru Birth Centenary Medal' was instituted in 1990.

9.2. Michael Atiyah awarded the Jawaharlal Nehru Birth Centenary Medal.

Michael Atiyah was the second person to be awarded the Jawaharlal Nehru Birth Centenary Medal when he received the Medal in 1993.

10.1. Founding of the Charles University Prague.

In medieval times, Prague was one of the well respected world centres where traditions of Czech, German and Jewish cultures, art, philosophy and education melded together. All this led to an official foundation of a university in Prague in the fourteenth century. The Charles University (originally referred as "studium generale" of Prague) is the oldest university in Central Europe north of the Alps. It was founded on 7 April 1348 by the Bohemian King and Emperor of the Holy Roman Empire, Charles the Fourth.

At that time Charles University provided generally accepted education and degrees, which guaranteed the possibility of free exchange and migration of students and professors. The educational content and forms of education were almost the same among European Universities, and there was just one teaching language in the whole of Europe - Latin.

10.2. The 650th Anniversary.

Charles University, in Prague, began celebrating the 650th anniversary of its founding in December 1997. The Czech institution organised more than 150 events to mark the anniversary, including concerts, conferences, and symposia. The rector, Karel Maly, said the celebrations were intended as "a feast of European science and education." They culminated on 7 April 1998, the anniversary date.

10.3. Medal and Honorary Doctorate awarded to Michael Atiyah.

Michael Atiyah was proposed by the Faculty of Mathematics and Physics for a Medal and Honorary Doctorate for his:-

... exceptional contribution to the development of mathematics.

The award was made on 8 April 1998.

11.1 Founding of the Universidad Nacional Autónoma de México.

Archbishop Fray Juan de Zumárraga proposed having a university in New Spain in 1536. Viceroy Antonio de Mendoza joined this initiative and the Crown gave a positive response in 1547. It was not until 21 September 1551, however, before the Certificate of Creation of the Royal and Pontifical University of Mexico was issued. Its opening took place on 25 January 1553. It was organised in the image and likeness of the European universities of scholastic tradition, particularly that of Salamanca.

11.2. Medal and Honorary Doctorate awarded to Michael Atiyah.

Because of his connections with Mexico, Michael Atiyah was proposed for an Anniversary Medal and Honorary Doctorate. These were awarded during the 450th Anniversary celebrations in 2001.

12.1. Royal Society of Edinburgh Royal Medals.

Royal Society of Edinburgh Royal Medals are awarded for distinction and international repute in any of the following categories: life sciences; physical, engineering and informatic sciences; arts, humanities and social sciences; business, public service and public engagement..

These Medals were instituted by Her Majesty The Queen in the year 2000. They are awarded annually, to individuals who have achieved distinction and are of international repute in any of the following categories: life sciences; physical, engineering and informatic sciences; arts, humanities and social sciences; business, public service and public engagement. Candidates for the Royal Medals need not be RSE Fellows and should, preferably, have a Scottish connection, irrespective of place of domicile.

12.2. The Awards Ceremony.

The achievements of three people whose work has brought about public benefits on a global scale received royal recognition on Monday 29 October 2003. Sir Michael Atiyah, Lord Mackay of Clashfern and Sir Paul Nurse were presented with Royal Medals by the Duke of Edinburgh at a ceremony in The Royal Society of Edinburgh. The medallists were selected by the Royal Society of Edinburgh, Scotland's National Academy, in recognition of intellectual endeavour which has had a profound influence on people's lives, worldwide. Designed and produced in Scotland, three gold medals are awarded through the RSE each year.

12.3. Michael Atiyah awarded the Royal Society of Edinburgh Royal Medal.

Sir Michael Atiyah was awarded the Royal Society of Edinburgh Royal Medal in 2003:-

... for his profound and beneficial effect on the development of mathematics and science in the UK and Europe.

13.1. Abel Prize 2004 Citation

The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2004 to

Sir Michael Francis Atiyah
University of Edinburgh

and

Isadore M. Singer
Massachusetts Institute of Technology

... for their discovery and proof of the index theorem, bringing together topology, geometry and analysis, and their outstanding role in building new bridges between mathematics and theoretical physics.

The second Abel Prize has been awarded jointly to Michael Francis Atiyah and Isadore M Singer. The Atiyah-Singer index theorem is one of the great landmarks of twentieth-century mathematics, influencing profoundly many of the most important later developments in topology, differential geometry and quantum field theory. Its authors, both jointly and individually, have been instrumental in repairing a rift between the worlds of pure mathematics and theoretical particle physics, initiating a cross-fertilization which has been one of the most exciting developments of the last decades.

We describe the world by measuring quantities and forces that vary over time and space. The rules of nature are often expressed by formulas involving their rates of change, so-called differential equations. Such formulas may have an "index", the number of solutions of the formulas minus the number of restrictions which they impose on the values of the quantities being computed. The index theorem calculates this number in terms of the geometry of the surrounding space.

A simple case is illustrated by a famous paradoxical etching of M C Escher, "Ascending and Descending", where the people, going uphill all the time, still manage to circle the castle courtyard. The index theorem would have told them this was impossible!

The Atiyah-Singer index theorem was the culmination and crowning achievement of a more than one-hundred-year-old evolution of ideas, from Stokes's theorem, which students learn in calculus classes, to sophisticated modern theories like Hodge's theory of harmonic integrals and Hirzebruch's signature theorem.

The problem solved by the Atiyah-Singer theorem is truly ubiquitous. In the forty years since its discovery, the theorem has had innumerable applications, first in mathematics and then, beginning in the late 1970s, in theoretical physics: gauge theory, instantons, monopoles, string theory, the theory of anomalies etc.

At first, the applications in physics came as a complete surprise to both the mathematics and physics communities. Now the index theorem has become an integral part of their cultures. Atiyah and Singer, together and individually, have been tireless in their attempts to explain the insights of physicists to mathematicians. At the same time, they brought modern differential geometry and analysis as it applies to quantum field theory to the attention of physicists and suggested new directions in physics itself. This cross-fertilization continues to fruitful for both sciences.

Michael Francis Atiyah and Isadore M Singer are among the most influential mathematicians of the last century and are still working. With the index theorem they changed the landscape of mathematics. Over a period of twenty years they worked together on the index theorem and its ramifications.

Atiyah and Singer came originally from different fields of mathematics: Atiyah from algebraic geometry and topology, Singer from analysis. Their main contributions in their respective areas are also highly recognised. Atiyah's early work on meromorphic forms on algebraic varieties and his important 1961 paper on Thom complexes are such examples. Atiyah's pioneering work together with Friedrich Hirzebruch on the development of the topological analogue of Grothendieck's K-theory had numerous applications to classical problems of topology and turned out later to be deeply connected with the index theorem.

Singer established the subject of triangular operator algebras (jointly with Richard V Kadison). Singer's name is also associated with the Ambrose-Singer holonomy theorem and the Ray-Singer torsion invariant. Singer, together with Henry P McKean, pointed out the deep geometrical information hidden
in heat kernels, a discovery that had great impact.

The prize amount is NOK 6,000,000 (USD 875,000, GBP 475,000, EUR 710,000) and was awarded for the first time in 2003 to Jean-Pierre Serre.

13.2. Sir Michael Francis Atiyah biography.

Michael Francis Atiyah was born in 1929 in London. Atiyah got his B.A. and his doctorate from Trinity College, Cambridge. Atiyah has spent the greatest part of his academic career at Cambridge and Oxford. He has held many prominent positions; among them the highly prestigious Savilian Chair of Geometry at Oxford and that of Master of Trinity College, Cambridge. Atiyah has also been professor of mathematics at the Institute for Advanced Study in Princeton.

Atiyah rejuvenated British mathematics during his years at Oxford and Cambridge. He was also the driving force behind the creation of the Isaac Newton Institute for Mathematical Sciences in Cambridge and became its first director. Atiyah is now retired and an honorary professor at the University of Edinburgh.

Michael Francis Atiyah has received many honours during his career, including the Fields Medal (1966). He was elected a Fellow of the Royal Society in 1962 at the age of 32. He was awarded the Royal Medal of the Society in 1968 and its Copley Medal in 1988. Atiyah was president of the Royal Society from 1990 to 1995. Atiyah has served as president of London Maths Society (1974-1976). He has also played an important role in the shaping of today's European Mathematical Society (EMS).

Atiyah was responsible for the founding of the Inter-Academy Panel that brought together many of the world's academies of science. The Inter-Academy Panel has now been permanently established and will play a major role in the integration of scientific policy throughout the world. Atiyah also instigated the formation of the Association of European Academies (ALLEA). Atiyah has been president of Pugwash Conferences on Science and World Affairs.

13.3. Report of the prize ceremony.

After an audience with Their Majesties King Harald and Queen Sonja earlier in the day, it was time for the prize ceremony in the Great Hall of Oslo University. The main street of Oslo, Karl Johans gate, was decorated for the occasion with colourful Abel Prize banners. This year's prize winners arrived at the packed hall to the sound of Klaus Sanvik's recently-composed Abel Fanfare, performed by Sidsel Walstad on electric harp, followed by the arrival of the King and Queen.

Lars Walløe, the President of the Norwegian Academy of Science and Letters, welcomed those present to the ceremony. Before the official presentation, the audience were given a surprise in the form of a new arrangement of Michael Jackson's Billy Jean, performed by Sidsel Walstad (electric harp), Mocci Ryen (vocals) and Børre Flyen (percussion). The lively and youthful performance was appreciated, at least by Isadore Singer, who tapped his foot enthusiastically in time with the music.

The leader of the Abel Committee, Erling Størmer, briefly explained the reasons for the selection of Atiyah and Singer as this year's prize winners:-

The Atiyah-Singer index theorem is one of the most important mathematical results of the twentieth century. It has had an enormous impact on the further development of topology, differential geometry and theoretical physics. The theorem also provides us with a glimpse of the beauty of mathematical theory in that it explicitly demonstrates a deep connection between mathematical disciplines that appear to be completely separate.

After the address, His Majesty King Harald presented the Abel Prize to the two winners.

Sir Michael Atiyah commenced his acceptance speech by thanking colleagues who had made important contributions to the work, mentioning in particular Fritz Hirzebruch, Raoul Bott, Graeme Segal and Nigel Hitchin. He went on to explain that right from the time of Newton to that of Einstein there has been a close relationship between mathematics and the exploration of the natural world:-

One of the unexpected joys of my partnership with Is Singer has been that these links with physics have been reinforced during our time.

In conclusion, Atiyah said that in his opinion:-

Abel was really the first modern mathematician. His whole approach, with its generality, its insight and its elegance, set the tone for the next two centuries. If Abel had lived longer, he would have been the natural successor to the great Gauss: a statement with which I fully concur except for the qualification that Abel was a much nicer man, modest, friendly and likeable. I am proud to have a prize that bears his name.

The ceremony was concluded with Edvard Grieg's Halling before the King and Queen and the Abel Prize winners left the hall.

13.4. BBC report: Mathematicians share Abel Prize.

Sir Michael Atiyah and Isadore Singer have been awarded the Abel Prize for their outstanding work in mathematics.

The UK and US researchers developed what is now referred to as the Atiyah-Singer theorem about 40 years ago.

It concerns the use of differential equations and has allowed physicists to develop new theories about the cosmos.

The honour, from the Norwegian Academy of Science and Letters, was set up to mark a field of scientific endeavour that is overlooked by the Nobel Prizes.

In its citation for the £480,000 prize, the academy said their theorem "is one of the great landmarks of 20th Century mathematics, influencing profoundly many of the most important later developments in topology, differential geometry and quantum field theory".

In the spotlight

Sir Michael, 75, works out of Edinburgh University, and Singer, 79, is attached to the Massachusetts Institute of Technology.

The rules of nature can be expressed by differential equations, which are mathematical formulae based on rates of change.

These formulae can have and indices, which in the Atiyah-Singer theorem can be calculated in terms of the geometry of the surrounding space.

The mathematicians' contribution is said to have given modern theoretical physics a new tool to describe the forces and particles at work in the Universe. So-called superstring theory, which seeks to unify scientific understanding of all forces and all matter, is said to have been a great beneficiary of the work.

"I am delighted to win this prize with Sir Michael," Professor Singer said. "The work we did broke barriers between different branches of mathematics and that's probably its most important aspect.

"It has also had serious applications in theoretical physics. But most of all I appreciate the attention mathematics will be getting. It's well-deserved because mathematics is so basic to science and engineering."

Brilliant mind

Sir Michael was president of the Royal Society, the UK's academy of science, in the early 1990s. The present president, Lord May, said: "Sir Michael's outstanding achievements as a Mathematician have been recognised through an enviable array of awards and medals.

"The Abel Prize, effectively the Nobel for Mathematics, places him, quite rightly, at the top of his field on the international stage."

The prize is named after the brilliant Norwegian mathematician Niels Henrik Abel, who died in 1829, and was created in 2002.

The first prize, awarded last year, went to French mathematician Jean-Pierre Serre for his role in shaping algebraic geometry and number theory.

This year's prize is to be presented by Norway's King Harald at a May 25 ceremony in Oslo.

13.5. Science Report: Two Mathematicians Share Abel Prize.

Award honours theorem that brought together two branches of mathematics.

British and American mathematicians have won the second Abel Prize in mathematics. Michael Atiyah and Isadore Singer are to share the $875,000 award for their proof of a theorem that links two very different areas of mathematics-topology and differential equations.

Topology is the study of abstract objects such as 12-dimensional hyperspheres; differential equations is a discipline related to how mathematical functions vary in space and time and are extremely useful to physicists and engineers. The two disciplines each have their own theorems, their own discoveries, and their own intractable problems.

In the 1960s, Atiyah, currently at the University of Edinburgh, and Singer, now at the Massachusetts Institute of Technology, together forged a link between the two seemingly different fields. The so-called Atiyah-Singer index theorem is a tool "for using topological methods for proving powerful theorems in differential equations," says John Milnor, a mathematician at the University of Stony Brook in New York. "It was one of the major developments in 20th century mathematics."

Atiyah won the Fields Medal in 1966 and was knighted for this and other mathematical discoveries, whereas Singer received less high-profile adulation, although he is a member of the National Academy of Sciences. Nevertheless, says Milnor, both are worthy of the prize: "I think these are good people who deserve it."

13.6. Nature Report: Maths 'Nobel' awarded.

The Abel Prize, often described as a Nobel Prize for maths, has been awarded to two mathematicians for unifying swathes of mathematical theories that were once thought to be unrelated.

Sir Michael Atiyah and Isadore Singer worked together to create something called index theory, which helps to bring together branches of maths from topology to geometry. Their work can be described as a tool that helps scientists work out how many solutions there are to problems they are trying to unpick - such as how heat flows, or how an object moves.

"It is basically a formula that counts the number of solutions to another equation," says Atiyah.

"This theory is now a cornerstone of maths; it is one of the most fundamental results of the last 50 years," says Elmer Rees, a colleague of Atiyah's at Edinburgh University.

"It was as if an archaeologist had discovered exactly the same patterns on tombs in completely different parts of the world, proving that some underlying civilization had carved them all," says Marcus du Sautoy, a mathematician at Oxford University.

Atiyah and Singer devised index theory in the early 1960s, while Atiyah was based at Oxford University and Singer was at the Massachusetts Institute of Technology, Cambridge, where he still works.

Their theory also underpins the latest work on string theory, which tries to explain the fundamental nature of the universe by suggesting that matter is made of tiny 'strings' vibrating in many different dimensions.

Abel reward

The Norwegian Academy of Science and Letters established the Abel Prize in 2002, to commemorate the nineteenth century mathematician Niels Henrik Abel. There is no Nobel Prize for mathematics; awards such as the Abel and the Fields Medal help to fill the gap. King Harald of Norway will present the £475,000 prize at a ceremony in Oslo on 25 May.

Atiyah is a former president of the Royal Society, and more recently helped to establish the Millennium Prize, also known as the Clay Prize, which offers a US$1 million reward for solutions to any of the seven most vexing problems in mathematics. He spends much of his time publicising maths through radio broadcasts and public lectures.

Singer is a former member of the White House Science Council, and was vice-president of the American Mathematical Society in the 1970s.

13.7. Michael Atiyah's Interview after winning the Abel Prize.

Martin Raussen and Christian Skau interviewed Atiyah and Singer in Oslo on 24 May 2004.

MR and CS. First, we congratulate you for having been awarded the Abel Prize 2004. This prize has been given to you for "the discovery and the proof of the Index Theorem connecting geometry and analysis in a surprising way and your outstanding role in building new bridges between mathematics and theoretical physics". You have an impressive list of fine achievements in mathematics. Is the Index Theorem your most important result and the result you are most pleased with in your entire careers?

ATIYAH. First, I would like to say that I prefer to call it a theory, not a theorem. Actually, we have worked on it for 25 years and if I include all the related topics, I have probably spent 30 years of my life working on the area. So it is rather obvious that it is the best thing I have done.

MR and CS. We would like you to give us some comments on the history of the discovery of the Index Theorem. Were there precursors, conjectures in this direction already before you started? Were there only mathematical motivations or also physical ones?

ATIYAH. Mathematics is always a continuum, linked to its history, the past - nothing comes out of zero. And certainly the Index Theorem is simply a continuation of work that, I would like to say, began with Abel. So of course there are precursors. A theorem is never arrived at in the way that logical thought would lead you to believe or that posterity thinks. It is usually much more accidental, some chance discovery in answer to some kind of question. Eventually you can rationalise it and say that this is how it fits. Discoveries never happen as neatly as that. You can rewrite history and make it look much more logical, but actually it happens quite differently.

MR and CS. You worked out at least three different proofs with different strategies for the Index Theorem. Why did you keep on after the first proof? What different insights did the proofs give?

ATIYAH. I think it is said that Gauss had ten different proofs for the law of quadratic reciprocity. Any good theorem should have several proofs, the more the better. For two reasons: usually, different proofs have different strengths and weaknesses, and they generalise in different directions - they are not just repetitions of each other. And that is certainly the case with the proofs that we came up with. There are different reasons for the proofs, they have different histories and backgrounds. Some of them are good for this application, some are good for that application. They all shed light on the area. If you cannot look at a problem from different directions, it is probably not very interesting; the more perspectives, the better!

13.8. 'I'm a bit of a jack of all trades'.

There's no Nobel prize for maths, but Sir Michael Atiyah has won the next best thing. James Meek finds out what goes on inside the mind of a brilliant mathematician.

It doesn't matter how brilliant you are as a mathematician: you will never win the Nobel prize for maths, because there isn't one. There is, however, an N Abel prize, and Sir Michael Atiyah, who is a brilliant British mathematician, has won it. Sitting on a sofa in his big apartment, on a high floor of a modern block in the professorial quarter of south Edinburgh, Atiyah underplays the Nobel-sized bounty that comes with the prize: £480,000, which he will share with his fellow winner, Isadore Singer of the US. "If they'd given me this prize when I was younger, it would have been very useful," he says. "At my stage of life I don't know what I'm going to do with it. I'll probably use it for good causes. I'll probably give a little party."

Atiyah, who will be 75 tomorrow, won the most prestigious prize in maths, the Fields Medal, in 1966. "But it doesn't carry much money, and these days if a prize doesn't carry much money it doesn't get noticed," he says. Now he has the prestige prize and the money prize. Strictly speaking, it is called the Abel prize, after the 19th-century Norwegian mathematician Niels Abel. Since his first initial was "N", though, it seems reasonable to call it the N Abel prize.

The prize was awarded for the Atiyah-Singer Index Theorem, which the two men arrived at 40 years ago, and have worked on ever since. The citation said that the theorem is "one of the great landmarks of 20th-century mathematics, influencing profoundly many of the most important later developments in topology, differential geometry and quantum field theory." But what is it?

A bridge, says Atiyah, bringing together all the separate fields of maths - algebra, geometry, calculus, trigonometry, together with their myriad applications in economics, engineering and physics. "I'm a bit of a jack of all trades, I suppose. I don't specialise in any one. I pick up a bit here, a bit there, and if I find a connection between them, I get excited."

In 1990, Atiyah published a book called The Geometry and Physics of Knots, just as physicists were, with the help of his theorem, unravelling the mysteries of string theory, which promises to explain how the universe is made. Atiyah's knots were the kind you might tie with everyday string. String theory deals with invisible, conceptual, cross-dimensional items which physicists now believe underlie all matter. But the charm of maths, and Atiyah's bridge, is that his findings about ordinary knots could be used to illuminate the physicists' findings about cosmic "strings".

Atiyah's mother was Scottish. His father's family were Arabs, Anglophile Lebanese Christians who left the Ottoman empire for the British empire in the 19th century. His grandfather came to Khartoum in Sudan with General Kitchener. The young Atiyah grew up in imperial Khartoum in the 1930s and 1940s, where his father, Edward, worked as a liaison official between the Sudanese and the colonial authorities.

"My father's main dream was to go to Oxford. He wanted to convert himself into an Englishman," says Atiyah. "It didn't quite work out. When he came back to Sudan, he found he wasn't part of the English class structure, he was regarded as one of the lower classes, although he was Oxford-educated and regarded himself as culturally English. That turned him over a bit. He became an Arab nationalist to some extent. All his life was divided between wanting passionately to be English and yet sympathising with the Arab political position within the British empire."

Atiyah felt the same sense of divided loyalty. After a few years at the tiny Diocesan school in Khartoum he went to boarding school in Cairo. To the dismay of fellow pupils he was always trying to, "identify myself with the English ... I was wanting to pull myself away from the Arab background." Even though he spoke Arabic fluently, written Arabic was the only school subject he ever failed in.

At the end of the war, the family moved to England. After two years at Manchester Grammar School and two years' national service in the army, Atiyah went to Cambridge. Although his father used to joke that as a child Atiyah had an uncanny facility for making a profit when converting his pocket money from one currency to another, Atiyah does not remember any mathematical epiphany in his teens. He has a poor memory, he says, which is one of the reasons why he picked maths.

"Medicine, you've got to learn all this anatomy; law, you've got to learn all these legal cases; history, you have to read vast numbers of books. Mathematics - very few facts. That's why people with mathematical talent can do something very young, very early, they can soar off, they don't need to be burdened with vast amounts of facts. A few key things and off you go."

Atiyah uses images from the language of travellers to explain to lay people what it is like inside the mind of a brilliant mathematician. He talks about the journey to his greatest work in terms of a mountaineer. By exploring the whole country of maths, he says, "you get to the top of Mount Everest and look round. It's a long route, and when you get to the top, it's a big scene you can see."

Most of his work, he says, is pure thought. "I'm not the sort of person who does my mathematics writing on paper. I do that at the last stage of the game. I do my mathematics in my head. I sit down for a hard day's work and I write nothing all day. I just think. And I walk up and down because that helps keep me awake, it keeps the blood circulating, and I think and think.

"The main thrust of your thinking can only take place in big chunks of time, not only for hours but for days, weeks, you carry these ideas with you. You go out for a walk and you take your ideas with you. You go on a bus, you take a train, even when you go to sleep you wake up in the morning and you've got this enormously complicated set of ideas with you for long periods, maybe even for a year or two."

Vision is the key. Atiyah points out that, in the apparent banality of looking round a living room and identifying things, the brain is doing something extraordinary. "I look at this room, I see bookshelves, a piano, carpets. It's the same with mathematics. I look at a lot of mathematical things, I see how they are all related. You don't just see them - you've got to know what they mean, you've got to give significance to them. The brain has a really fantastic ability to do that.

"Vision is an enormously complicated process involving at least a dozen parts of the brain, each of which recognises something; one recognises a horizontal line, one recognises perspective, one recognises colour, one recognises motion. Somehow they're all integrated together ... You can't really visualise four dimensions or five dimensions but you get a rough idea of what it's like by comparison with three."

In 1955, Atiyah joined Princeton's Institute for Advanced Study, the gathering point for the most brilliant mathematical minds in the US. It was Albert Einstein's final academic home; he died a few months before Atiyah arrived. The institute's head was Robert Oppenheimer, the father of the atom bomb, who had been victimised by the McCarthy anti-communist witch hunt the previous year. Atiyah says these cold-war shadows, and scientists' responsibility for creating nuclear weapons, did not touch him then. "Ours was the first generation to have the mushroom cloud over our lives, so it was a very unsettled period internationally. But Princeton was a very ivory-tower place, a small intellectual retreat, very Europeanised. It wasn't like New York or Washington, it wasn't in the midst of the maelstrom ... we didn't discuss politics, although we were all concerned."

Decades later, after a stellar career of work, teaching, awards and professorships at Oxford, Princeton and Edinburgh, Atiyah returned to the issue of nuclear weapons. As president of the Royal Society in 1995, he made an unusually harsh attack on Britain's ownership of a nuclear arsenal. As a result, he was asked by Joseph Rotblat to become president of the Pugwash disarmament conferences. His views on Britain's nuclear weapons remain strong. "It was a crazy, misguided, post-imperial attempt to maintain British status. It wasted an awful lot of resources in terms of scientific manpower. Instead of doing what the Germans and Japanese did and turn out motor cars and become wealthy, we poured it down the drain."

His convictions culminated last year in Atiyah taking part in a political demonstration for the first time in his life. He marched in Edinburgh with thousands of other protesters against the invasion of Iraq.

The "war on terror", as presently run, is a self-perpetuating engine, he argues, turning out as many or more terrorists than it destroys or arrests. "I think Tony Blair really is in a terrible, terrible dilemma. So, of course, is George Bush, but George Bush you can make allowances for: he's just stupid. And Tony Blair is smart, and it's much more difficult to understand why he's driven himself into this position.

"The real fundamental cause of these things arises out of the Israeli-Palestinian problem. Much more generally, it's the influence of the west on its former colonies. The impact of the west is very complicated, some plus, some minus, but what's being implemented now is the negative part, imposing western power for economic, political, strategic reasons, which I think more and more leads to hostility, and the Israeli-Palestinian thing is at the core of that. As long as that's not stopped in a satisfactory way, the problem will continue. It is the terrible irony of the world that the Jews suffered terribly during the war in the Holocaust, and now are in some senses the cause of the next Holocaust."

13.9. Café Scientifique.

On Sunday 23 May, Sir Michael Atiyah participated in the first event in connection with the award of this year's Abel Prize. In collaboration with the Norwegian Association of Young Scientists, the British Council arranged a Café Scientifique at the Kafé Rust in Oslo, in which Atiyah gave an informal lecture on his chosen subject: Man versus machine - the brain and the computer, with the subtitle "Will a computer ever be awarded the Abel Prize?" Quentin Cooper, one of the BBC's most popular radio presenters, chaired the meeting, in which Sir Michael spoke for an hour to an audience of about 50 people. He pointed out that while computers are extremely adept at following pre-determined rules and that he himself is not surprised that, for example, very good chess programs have been developed, what is surprising is that people are still able to play chess on an equal footing with machines. In other words, computers are good at following rules, but what they are not able to do is to break the rules in a creative manner. As an example, he cited Niels Henrik Abel's proof of the impossibility of solving the general quintic equation. A computer would have continued to search for the solution, and would never have been able to break the rules, as Abel did, and look at the inverse of the problem. As a mathematician you have to know the rules, but to create something new you have to break those rules creatively, just like an artist or a musical composer.

After a brief interval, Quentin Cooper invited questions from the audience and a number of points were brought up that Atiyah addressed thoroughly and professionally.

After a highly successful meeting lasting almost two hours, Atiyah answered his own question, "Will a computer ever be awarded the Abel Prize?": - Only if the Abel Prize Committee is replaced by computers.

14.1. President's Medal of the Institute of Physics

Made on the recommendation of the President of the Institute of Physics, this medal is for both physicists and non-physicists who have contributed to physics in general and the of the Institute of Physics in particular.

14.2. President's Medal 2008.

The President's Medal 2008 is awarded to Professor Sir Michael Atiyah, University of Edinburgh:-

... in recognition of his outstanding contributions to a broad range of topics in mathematics, many of which have provided highly significant foundations to the development of theoretical physics; and of his eminent leadership within the scientific community.

15.1. Sir Michael Atiyah awarded the Grand Cross of Brazil.

Sir Michael Atiyah was awarded the Grand Cross of the National Order of Scientific Merit of Brazil for outstanding scientific achievement. He was presented with the Grand Cross in Brazil later in the year 2010.

16.1. The Grande Médaille of the Académie des Sciences.

Created in 1997, the Grande Médaille singles out every year a scientist in the international landscape for making a decisive contribution to the development of science, as testified both by the originality of his/her personal research projects and by its international standing and stimulating influence, so seminal indeed as to gain proper following from researchers.

16.2. Michael Atiyah awarded the Grande Médaille.

The Grande Médaille de l'Académie des sciences was awarded to Sir Michael Francis Atiyah on Tuesday 22 June 2010.

The Grande Médaille de l'Académie des sciences was awarded on Tuesday 22 June 2010 to Sir Michael Francis Atiyah for all of his mathematical work, for his fundamental contribution to the bringing together of mathematics and physics, for having formed and enlightened a generation of scientists by the quality of his writings and his presentations, and by the essential part he played in the organisation of the scientific community. Born in London in 1929, Michael Atiyah became Royal Society Professor at Oxford in 1972, and president of the Royal Society from 1990 to 1995. The Academy of Sciences, which elected him Foreign Associate in 1978, once again salutes him as a exceptional scientist for the breadth of his work and the breadth of his views.

Michael Atiyah received the Fields Medal in 1966 for the development of topological K-theory, for the establishment with Isadore Singer of the Atiyah-Singer index theorem, and for the proof with Raoul Bott of a fixed points formula. In 2004 with Isadore Singer, he received the Abel Prize for the Atiyah-Singer index theorem. This theorem, where the boundary-crossing influences of analysis, differential geometry and algebraic geometry, topology and number theory are expressed, was one of the essential stages from which the rapprochement between mathematics and physics has changed in nature. With Friedrich Hirzebruch, he led the Mathematische Arbeitstagung in Bonn for three decades, where hundreds of mathematicians and physicists met each year.

By becoming director of the Isaac Newton Institute in Cambridge in 1990 and president of the Royal Society, of which he was elected a member at the age of 32, Michael Atiyah put his authority at the service of the scientific community. He contributed to the creation of the Inter Academy Panel on International Issues (IAP), of All European Academies (ALLEA) and the European Mathematics Society.

16.3. Top award for a city academic who really counts.

One of the world's greatest living mathematicians has been awarded a prestigious honour by a 300-year-old French academic society.

Resident of Edinburgh, Sir Michael Atiyah will jet off to Paris in October 2010 to collect the Grande Medaille of the Institut de France Academie des Sciences - the first mathematician to receive the accolade. The society of leading French scholars was founded in 1666.

The award, which was created 13 years ago, is annually bestowed on foreign scholars who have significantly contributed to the development of science in an influential way. It is the highest honour the society can award foreign members.

Previous recipients include American atmospheric chemist Susan Solomon, Nasa astronaut Ronald Evans, and cancer researcher Robert Weinberg.

In a long and distinguished career, the 81-year-old previously won the Fields Medal in 1966, and was jointly awarded the Abel Prize in 2004 with his collaborator Professor Isadore Singer. Their Atiyah-Singer index theorem is one of the landmark discoveries in mathematics.

Innovative Mr Atiyah has lived in the Grange area of Edinburgh for the last 13 years and was made an honorary professor at Edinburgh University soon after he moved up from Cambridge. He also held the coveted post of president of the Royal Society of Edinburgh from 2005 to 2008.

Mr Atiyah, who has been a member of the Institut de France Academie des Sciences for 30 years, said: "I think I'm the first mathematician to win the award so it's a huge honour and I'm very pleased.

"I have been involved with the society for a long time and I have lots of good friends and colleagues there.

"I am going to Paris in October for a formal dinner in a grand hall, where I will have to make a speech in French. My French is a little rusty but I'll polish it up beforehand."

Mr Atiyah, who lives with his wife Lily and has two children and three grand-children, learned last month that he would be a prize-winner.

"It was a surprise because I had no idea I was even being considered," he said. "I get awards every now and again but this one is rather special because the French can be quite choosy." As his wife is an Edinburgh native, Mr Atiyah moved to the Scottish capital after a stint as master of Trinity College in Cambridge. He was also chancellor of the University of Leicester from 1995 to 2005.

"My wife Lily was brought up in Edinburgh and it was only fair to bring her back," he said. "I like hills and in Edinburgh I can see hills wherever I look. I used to live in Cambridge, which is quite flat. But Edinburgh is a beautiful city, one of the finest in the UK.

"If you retire somewhere where you have worked for a long time, you are a has been. I like to move about."

Mr Atiyah, who was born in London of a Lebanese father and a Scottish mother, came to Edinburgh after a career that took him to Oxford, Princeton and Cambridge. He was schooled in Egypt before moving to England with his family at the age of 16.

1968 Bonn University.

1969 University of Warwick.

1979 University of Durham.

1981 University of St Andrews.

1983 University of Chicago.

1983 Trinity College, Dublin.

1984 University of Cambridge.

1984 University of Edinburgh.

1985 University of London.

1985 University of Essex.

1987 University of Ghent.

1990 University of Reading.

1990 University of Helsinki.

1991 University of Leicester.

1992 Rutgers University.

1992 University of Salamanca.

1993 University of Montreal.

1993 University of Wales.

1993 University of Waterloo.

1994 Lebanese University.

1994 Queen's University, Kingston, Canada.

1994 University of Keele.

1994 University of Birmingham.

1995 Open University.

1996 UMIST.

1996 Chinese University of Hong Kong.

1997 Brown University.

1998 University of Oxford.

1998 Charles University, Prague.

1999 Heriot-Watt University.

2001 National University of Mexico.

2004 American University of Beirut

2005 University of York.

2006 Harvard University.

2007 Scuola Normale, Pisa.

2008 Polytechnic University of Catalonia