name	Sir Michael Atiyah
birth date	April 22, 1929
birth place	Hampstead, London, England
nationality	United Kingdom
field	Mathematics
work institutions	University of CambridgeUniversity of OxfordInstitute for Advanced StudyUniversity of LeicesterUniversity of Edinburgh
alma mater	Victoria College, AlexandriaManchester Grammar SchoolTrinity College, Cambridge
doctoral advisor	W. V. D. Hodge
doctoral students	Simon DonaldsonK. David ElworthyNigel HitchinFrances KirwanPeter KronheimerRuth LawrenceGeorge LusztigJack MoravaIan R. PorteousBrian SandersonGraeme SegalDavid O. Tall
awards	Fields Medal (1966)Copley Medal (1988)Abel Prize (2004)
footnotes	}}

Sir Michael Francis Atiyah, OM, FRS, FRSE (born 22 April 1929) is a British mathematician working in geometry.

Atiyah grew up in Sudan and Egypt but spent most of his academic life in the United Kingdom at Oxford and Cambridge, and in the United States at the Institute for Advanced Study. He has been president of the Royal Society (1990–1995), master of Trinity College, Cambridge (1990–1997), chancellor of the University of Leicester (1995–2005), and president of the Royal Society of Edinburgh (2005–2008). He is currently retired, and is an honorary professor at the University of Edinburgh.

Atiyah's mathematical collaborators include Raoul Bott, Friedrich Hirzebruch and Isadore Singer, and his students include Graeme Segal, Nigel Hitchin and Simon Donaldson. Together with Hirzebruch, he laid the foundations for topological K-theory, an important tool in algebraic topology, which, informally speaking, describes ways in which spaces can be twisted. His best known result, the Atiyah–Singer index theorem, was proved with Singer in 1963 and is widely used in counting the number of independent solutions to differential equations. Some of his more recent work was inspired by theoretical physics, in particular instantons and monopoles, which are responsible for some subtle corrections in quantum field theory. He was awarded the Fields Medal in 1966, the Copley Medal in 1988, and the Abel Prize in 2004.

Biography

Atiyah was born in Hampstead, London, to Lebanese writer Edward Atiyah and Scot Jean Atiyah (née Levens). Patrick Atiyah, professor of law, is his brother; he has one other brother, Joe, and a sister, Selma. He went to primary school at the Diocesan school in Khartoum, Sudan (1934–1941) and to secondary school at Victoria College in Cairo and Alexandria (1941–1945); the school was also attended by European nobility displaced by the Second World War and some future leaders of Arab nations. He returned to England and Manchester Grammar School for his HSC studies (1945–1947) and did his national service with the Royal Electrical and Mechanical Engineers (1947–1949). His undergraduate and postgraduate studies took place at Trinity College, Cambridge (1949–1955). He was a doctoral student of William V. D. Hodge and was awarded a doctorate in 1955 for a thesis entitled ''Some Applications of Topological Methods in Algebraic Geometry''.

Atiyah married Lily Brown on 30 July 1955, with whom he has three sons. He spent the academic year 1955–1956 at the Institute for Advanced Study, Princeton, then returned to Cambridge University, where he was a research fellow and assistant lecturer (1957–1958), then a university lecturer and tutorial fellow at Pembroke College (1958–1961). In 1961, he moved to the University of Oxford, where he was a reader and professorial fellow at St Catherine's College (1961–1963). He became Savilian Professor of Geometry and a professorial fellow of New College, Oxford from 1963 to 1969. He then took up a three year professorship at the Institute for Advanced Study in Princeton after which he returned to Oxford as a Royal Society Research Professor and professorial fellow of St Catherine's College. He was president of the London Mathematical Society from 1974 to 1976.

Atiyah has been active on the international scene, for instance as president of the Pugwash Conferences on Science and World Affairs from 1997 to 2002. He also contributed to the foundation of the InterAcademy Panel on International Issues, the Association of European Academies (ALLEA), and the European Mathematical Society (EMS).

Within the United Kingdom, he was involved in the creation of the Isaac Newton Institute for Mathematical Sciences in Cambridge and was its first director (1990–1996). He was President of the Royal Society (1990–1995), Master of Trinity College, Cambridge (1990–1997), Chancellor of the University of Leicester (1995–2005), and president of the Royal Society of Edinburgh (2005–2008). He is now retired and is an honorary professor at the University of Edinburgh.

Collaborations

Atiyah has collaborated with many other mathematicians. His three main collaborations were with Raoul Bott on the Atiyah–Bott fixed-point theorem and many other topics, with Isadore M. Singer on the Atiyah–Singer index theorem, and with Friedrich Hirzebruch on topological K-theory, all of whom he met at the Institute for Advanced Study in Princeton in 1955. His other collaborators include J. Frank Adams (Hopf invariant problem), Jürgen Berndt (projective planes), Roger Bielawski (Berry–Robbins problem), Howard Donnelly (L-functions), Vladimir G. Drinfeld (instantons), Johan L. Dupont (singularities of vector fields), Lars Gårding (hyperbolic differential equations), Nigel J. Hitchin (monopoles), William V. D. Hodge (Integrals of the second kind), Michael Hopkins (K-theory), Lisa Jeffrey (topological Lagrangians), John D. S. Jones (Yang–Mills theory), Juan Maldacena (M-theory), Yuri I. Manin (instantons), Nick S. Manton (Skyrmions), Vijay K. Patodi (Spectral asymmetry), A. N. Pressley (convexity), Elmer Rees (vector bundles), Wilfried Schmid (discrete series representations), Graeme Segal (equivariant K-theory), Alexander Shapiro (Clifford algebras), L. Smith (homotopy groups of spheres), Paul Sutcliffe (polyhedra), David O. Tall (lambda rings), John A. Todd (Stiefel manifolds), Cumrun Vafa (M-theory), Richard S. Ward (instantons) and Edward Witten (M-theory, topological quantum field theories).

His later research on gauge field theories, particularly Yang–Mills theory, stimulated important interactions between geometry and physics, most notably in the work of Edward Witten.

Atiyah's many students include Peter Braam 1987, Simon Donaldson 1983, K. David Elworthy 1967, Howard Fegan 1977, Eric Grunwald 1977, Nigel Hitchin 1972, Lisa Jeffrey 1991, Frances Kirwan 1984, Peter Kronheimer 1986, Ruth Lawrence 1989, George Lusztig 1971, Jack Morava 1968, Michael Murray 1983, Peter Newstead 1966, Ian R. Porteous 1961, John Roe 1985, Brian Sanderson 1963, Rolph Schwarzenberger 1960, Graeme Segal 1967, David Tall 1966, and Graham White 1982.

Other contemporary mathematicians who influenced Atiyah include Roger Penrose, Lars Hörmander, Alain Connes and Jean-Michel Bismut. Atiyah said that the mathematician he most admired was Hermann Weyl, and that his favorite mathematicians from before the 20th century were Bernhard Riemann and William Rowan Hamilton.

Mathematical work

The six volumes of Atiyah's collected papers include most of his work, except for his commutative algebra textbook and a few works written since 2004.

Algebraic geometry (1952–1958)

Atiyah's early papers on algebraic geometry (and some general papers) are reprinted in the first volume of his collected works.

As an undergraduate Atiyah was interested in classical projective geometry, and wrote his first paper: a short note on twisted cubics. He started research under W. V. D. Hodge and won the Smith's prize for 1954 for a sheaf-theoretic approach to ruled surfaces, which encouraged Atiyah to continue in mathematics, rather than switch to his other interests—architecture and archaeology. His PhD thesis with Hodge was on a sheaf-theoretic approach to Solomon Lefschetz's theory of integrals of the second kind on algebraic varieties, and resulted in an invitation to visit the Institute for Advanced Study in Princeton for a year. While in Princeton he classified vector bundles on an elliptic curve (extending Grothendieck's classification of vector bundles on a genus 0 curve), by showing that any vector bundle is a sum of (essentially unique) indecomposable vector bundles, and then showing that the space of indecomposable vector bundles of given degree and positive dimension can be identified with the elliptic curve. He also studied double points on surfaces, giving the first example of a flop, a special birational transformation of 3-folds that was later heavily used in Mori's work on minimal models for 3-folds. Atiyah's flop can also be used to show that the universal marked family of K3 surfaces is non-Hausdorff.

K theory (1959–1974)

Atiyah's works on K-theory, including his book on K-theory are reprinted in volume 2 of his collected works.

The simplest example of a vector bundle is the Möbius band (pictured on the right): a strip of paper with a twist in it, which represents a rank 1 vector bundle over a circle (the circle in question being the centerline of the Möbius band). K-theory is a tool for working with higher dimensional analogues of this example, or in other words for describing higher dimensional twistings: elements of the K-group of a space are represented by vector bundles over it, so the Möbius band represents an element of the K-group of a circle.

Topological K-theory was discovered by Atiyah and Friedrich Hirzebruch who were inspired by Grothendieck's proof of the Grothendieck–Riemann–Roch theorem and Bott's work on the periodicity theorem. This paper only discussed the zeroth K-group; they shortly after extended it to K-groups of all degrees, giving the first (nontrivial) example of a generalized cohomology theory.

Several results showed that the newly introduced K-theory was in some ways more powerful than ordinary cohomology theory. Atiyah and Todd used K-theory to improve the lower bounds found using ordinary cohomology by Borel and Serre for the James number, describing when a map from a complex Stiefel manifold to a sphere has a cross section. (Adams and Grant-Walker later showed that the bound found by Atiyah and Todd was best possible.) Atiyah and Hirzebruch used K-theory to explain some relations between Steenrod operations and Todd classes that Hirzebruch had noticed a few years before. The original solution of the Hopf invariant one problem operations by J. F. Adams was very long and complicated, using secondary cohomology operations. Atiyah showed how primary operations in K-theory could be used to give a short solution taking only a few lines, and in joint work with Adams also proved analogues of the result at odd primes.

The Atiyah–Hirzebruch spectral sequence relates the ordinary cohomology of a space to its generalized cohomology theory. (Atiyah and Hirzebruch used the case of K-theory, but their method works for all cohomology theories).

Atiyah showed that for a finite group ''G'', the K-theory of its classifying space, ''BG'', is isomorphic to the completion of its character ring: :$K(BG)\; \backslash cong\; R(G)^\{\backslash wedge\}.$ The same year they proved the result for ''G'' any compact connected Lie group. Although soon the result could be extended to ''all'' compact Lie groups by incorporating results from Graeme Segal's thesis, that extension was complicated. However a simpler and more general proof was produced by introducing equivariant K-theory, ''i.e.'' equivalence classes of ''G''-vector bundles over a compact ''G''-space ''X''. It was shown that under suitable conditions the completion of the equivariant K-theory of ''X'' is isomorphic to the ordinary K-theory of a space, $X\_G$, which fibred over ''BG'' with fibre ''X'': :$K\_G(X)^\{\backslash wedge\}\; \backslash cong\; K(X\_G).$

The original result then followed as a corollary by taking ''X'' to be a point: the left hand side reduced to the completion of ''R(G)'' and the right to ''K(BG)''. See Atiyah–Segal completion theorem for more details.

He defined new generalized homology and cohomology theories called bordism and cobordism, and pointed out that many of the deep results on cobordism of manifolds found by R. Thom, C. T. C. Wall, and others could be naturally reinterpreted as statements about these cohomology theories. Some of these cohomology theories, in particular complex cobordism, turned out to be some of the most powerful cohomology theories known.

He introduced the J-group ''J''(''X'') of a finite complex ''X'', defined as the group of stable fiber homotopy equivalence classes of sphere bundles; this was later studied in detail by J. F. Adams in a series of papers, leading to the Adams conjecture.

With Hirzebruch he extended the Grothendieck–Riemann–Roch theorem to complex analytic embeddings, and in a related paper they showed that the Hodge conjecture for integral cohomology is false. The Hodge conjecture for rational cohomology is, as of 2008, a major unsolved problem.

The Bott periodicity theorem was a central theme in Atiyah's work on K-theory, and he repeatedly returned to it, reworking the proof several times to understand it better. With Bott he worked out an elementary proof, and gave another version of it in his book. With Bott and Shapiro he analysed the relation of Bott periodicity to the periodicity of Clifford algebras; although this paper did not have a proof of the periodicity theorem, a proof along similar lines was shortly afterwards found by R. Wood. In he found a proof of several generalizations using elliptic operators; this new proof used an idea that he used to give a particularly short and easy proof of Bott's original periodicity theorem.

Index theory (1963–1984)

Atiyah's work on index theory is reprinted in volumes 3 and 4 of his collected works.

The index of a differential operator is closely related to the number of independent solutions (more precisely, it is the differences of the numbers of independent solutions of the differential operator and its adjoint). There are many hard and fundamental problems in mathematics that can easily be reduced to the problem of finding the number of independent solutions of some differential operator, so if one has some means of finding the index of a differential operator these problems can often be solved. This is what the Atiyah–Singer index theorem does: it gives a formula for the index of certain differential operators, in terms of topological invariants that look quite complicated but are in practice usually straightforward to calculate.

Several deep theorems, such as the Hirzebruch–Riemann–Roch theorem, are special cases of the Atiyah–Singer index theorem. In fact the index theorem gave a more powerful result, because its proof applied to all compact complex manifolds, while Hirzebruch's proof only worked for projective manifolds. There were also many new applications: a typical one is calculating the dimensions of the moduli spaces of instantons. The index theorem can also be run "in reverse": the index is obviously an integer, so the formula for it must also give an integer, which sometimes gives subtle integrality conditions on invariants of manifolds. A typical example of this is Rochlin's theorem, which follows from the index theorem.

The index problem for elliptic differential operators was posed in 1959 by Gel'fand. He noticed the homotopy invariance of the index, and asked for a formula for it by means of topological invariants. Some of the motivating examples included the Riemann–Roch theorem and its generalization the Hirzebruch–Riemann–Roch theorem, and the Hirzebruch signature theorem. Hirzebruch and Borel had proved the integrality of the  genus of a spin manifold, and Atiyah suggested that this integrality could be explained if it were the index of the Dirac operator (which was rediscovered by Atiyah and Singer in 1961).

The first announcement of the Atiyah–Singer theorem was their 1963 paper. The proof sketched in this announcement was inspired by Hirzebruch's proof of the Hirzebruch–Riemann–Roch theorem and was never published by them, though it is described in the book by Palais. Their first published proof was more similar to Grothendieck's proof of the Grothendieck–Riemann–Roch theorem, replacing the cobordism theory of the first proof with K-theory, and they used this approach to give proofs of various generalizations in a sequence of papers from 1968 to 1971.

Instead of just one elliptic operator, one can consider a family of elliptic operators parameterized by some space ''Y''. In this case the index is an element of the K-theory of ''Y'', rather than an integer. If the operators in the family are real, then the index lies in the real K-theory of ''Y''. This gives a little extra information, as the map from the real K theory of ''Y'' to the complex K theory is not always injective.

With Bott, Atiyah found an analogue of the Lefschetz fixed-point formula for elliptic operators, giving the Lefschetz number of an endomorphism of an elliptic complex in terms of a sum over the fixed points of the endomorphism. As special cases their formula included the Weyl character formula, and several new results about elliptic curves with complex multiplication, some of which were initially disbelieved by experts. Atiyah and Segal combined this fixed point theorem with the index theorem as follows. If there is a compact group action of a group ''G'' on the compact manifold ''X'', commuting with the elliptic operator, then one can replace ordinary K theory in the index theorem with equivariant K-theory. For trivial groups ''G'' this gives the index theorem, and for a finite group ''G'' acting with isolated fixed points it gives the Atiyah–Bott fixed point theorem. In general it gives the index as a sum over fixed point submanifolds of the group ''G''.

Atiyah solved a problem asked independently by Hörmander and Gel'fand, about whether complex powers of analytic functions define distributions. Atiyah used Hironaka's resolution of singularities to answer this affirmatively. An ingenious and elementary solution was found at about the same time by J. Bernstein, and discussed by Atiyah.

As an application of the equivariant index theorem, Atiyah and Hirzeburch showed that manifolds with effective circle actions have vanishing Â-genus. (Lichnerowicz showed that if a manifold has a metric of positive scalar curvature then the Â-genus vanishes.)

With Elmer Rees, Atiyah studied the problem of the relation between topological and holomorphic vector bundles on projective space. They solved the simplest unknown case, by showing that all rank 2 vector bundles over projective 3-space have a holomorphic structure. Horrocks had previously found some non-trivial examples of such vector bundles, which were later used by Atiyah in his study of instantons on the 4-sphere.

Atiyah, Bott and Vijay K. Patodi gave a new proof of the index theorem using the heat equation.

If the manifold is allowed to have boundary, then some restrictions must be put on the domain of the elliptic operator in order to ensure a finite index. These conditions can be local (like demanding that the sections in the domain vanish at the boundary) or more complicated global conditions (like requiring that the sections in the domain solve some differential equation). The local case was worked out by Atiyah and Bott, but they showed that many interesting operators (e.g., the signature operator) do not admit local boundary conditions. To handle these operators, Atiyah, Patodi and Singer introduced global boundary conditions equivalent to attaching a cylinder to the manifold along the boundary and then restricting the domain to those sections that are square integrable along the cylinder. This resulted in a series of papers on spectral asymmetry, which were later unexpectedly used in theoretical physics, in particular in Witten's work on anomalies.

The fundamental solutions of linear hyperbolic partial differential equations often have Petrovsky lacunas: regions where they vanish identically. These were studied in 1945 by I. G. Petrovsky, who found topological conditions describing which regions were lacunas. In collaboration with Bott and Lars Gårding, Atiyah wrote three papers updating and generalizing Petrovsky's work.

Atiyah showed how to extend the index theorem to some non-compact manifolds, acted on by a discrete group with compact quotient. The kernel of the elliptic operator is in general infinite dimensional in this case, but it is possible to get a finite index using the dimension of a module over a von Neumann algebra; this index is in general real rather than integer valued. This version is called the ''L² index theorem,'' and was used by Atiyah and Schmid to give a geometric construction, using square integrable harmonic spinors, of Harish-Chandra's discrete series representations of semisimple Lie groups. In the course of this work they found a more elementary proof of Harish-Chandra's fundamental theorem on the local integrability of characters of Lie groups.

With H. Donnelly and I. Singer, he extended Hirzebruch's formula (relating the signature defect at cusps of Hilbert modular surfaces to values of L-functions) from real quadratic fields to all totally real fields.

Gauge theory (1977–1985)

Many of his papers on gauge theory and related topics are reprinted in volume 5 of his collected works. A common theme of these papers is the study of moduli spaces of solutions to certain non-linear partial differential equations, in particular the equations for instantons and monopoles. This often involves finding a subtle correspondence between solutions of two seemingly quite different equations. An early example of this which Atiyah used repeatedly is the Penrose transform, which can sometimes convert solutions of a non-linear equation over some real manifold into solutions of some linear holomorphic equations over a different complex manifold.

In a series of papers with several authors, Atiyah classified all instantons on 4 dimensional Euclidean space. It is more convenient to classify instantons on a sphere as this is compact, and this is essentially equivalent to classifing instantons on Euclidean space as this is conformally equivalent to a sphere and the equations for instantons are conformally invariant. With Hitchin and Singer he calculated the dimension of the moduli space of irreducible self-dual connections (instantons) for any principle bundle over a compact 4-dimensional Riemannian manifold. For example, the dimension of the space of SU₂ instantons of rank ''k''>0 is 8''k''−3. To do this they used the Atiyah–Singer index theorem to calculate the dimension of the tangent space of the moduli space at a point; the tangent space is essentially the space of solutions of an elliptic differential operator, given by the linearization of the non-linear Yang–Mills equations. These moduli spaces were later used by Donaldson to construct his invariants of 4-manifolds. Atiyah and Ward used the Penrose correspondence to reduce the classification of all instantons on the 4-sphere to a problem in algebraic geometry. With Hitchin he used ideas of Horrocks to solve this problem, giving the ADHM construction of all instantons on a sphere; Manin and Drinfeld found the same construction at the same time, leading to a joint paper by all four authors. Atiyah reformulated this construction using quaternions and wrote up a leisurely account of this classification of instantons on Euclidean space as a book.

Atiyah's work on instanton moduli spaces was used in Donaldson's work on Donaldson theory. Donaldson showed that the moduli space of (degree 1) instantons over a compact simply connected 4-manifold with positive definite intersection form can be compactified to give a cobordism between the manifold and a sum of copies of complex projective space. He deduced from this that the intersection form must be a sum of one dimensional ones, which led to several spectacular applications to smooth 4-manifolds, such as the existence of non-equivalent smooth structures on 4 dimensional Euclidean space. Donaldson went on to use the other moduli spaces studied by Atiyah to define Donaldson invariants, which revolutionized the study of smooth 4-manifolds, and showed that they were more subtle than smooth manifolds in any other dimension, and also quite different from topological 4-manifolds. Atiyah described some of these results in a survey talk.

Green's functions for linear partial differential equations can often be found by using the Fourier transform to convert this into an algebraic problem. Atiyah used a non-linear version of this idea. He used the Penrose transform to convert the Green's function for the conformally invariant Laplacian into a complex analytic object, which turned out to be essentially the diagonal embedding of the Penrose twistor space into its square. This allowed him to find an explicit formula for the conformally invariant Green's function on a 4-manifold.

In his paper with Jones, he studied the topology of the moduli space of SU(2) instantons over a 4-sphere. They showed that the natural map from this moduli space to the space of all connections induces epimorphisms of homology groups in a certain range of dimensions, and suggested that it might induce isomorphisms of homology groups in the same range of dimensions. This became known as the Atiyah–Jones conjecture, and was later proved by several mathematicians.

Harder and M. S. Narasimhan described the cohomology of the moduli spaces of stable vector bundles over Riemann surfaces by counting the number of points of the moduli spaces over finite fields, and then using the Weil conjectures to recover the cohomology over the complex numbers. Atiyah and R. Bott used Morse theory and the Yang–Mills equations over a Riemann surface to reproduce and extending the results of Harder and Narasimhan.

An old result due to Schur and Horn states that the set of possible diagonal vectors of an Hermitian matrix with given eigenvalues is the convex hull of all the permutations of the eigenvalues. Atiyah proved a generalization of this that applies to all compact symplectic manifolds acted on by a torus, showing that the image of the manifold under the moment map is a convex polyhedron, and with Pressley gave a related generalization to infinite dimensional loop groups.

Duistermaat and Heckman found a striking formula, saying that the push-forward of the Liouville measure of a moment map for a torus action is given exactly by the stationary phase approximation (which is in general just an asymptotic expansion rather than exact). Atiyah and Bott showed that this could be deduced from a more general formula in equivariant cohomology, which was a consequence of well-known localization theorems. Atiyah showed that the moment map was closely related to geometric invariant theory, and this idea was later developed much further by his student F. Kirwan. Witten shortly after applied the Duistermaat–Heckman formula to loop spaces and showed that this formally gave the Atiyah–Singer index theorem for the Dirac operator; this idea was lectured on by Atiyah.

With Hitchin he worked on magnetic monopoles, and studied their scattering using an idea of Nick Manton. His book with Hitchin gives a detailed description of their work on magnetic monopoles. The main theme of the book is a study of a moduli space of magnetic monopoles; this has a natural Riemannian metric, and a key point is that this metric is complete and hyperkahler. The metric is then used to study the scattering of two monopoles, using a suggestion of N. Manton that the geodesic flow on the moduli space is the low energy approximation to the scattering. For example, they show that a head-on collision between two monopoles results in 90-degree scattering, with the direction of scattering depending on the relative phases of the two monopoles. He also studied monopoles on hyperbolic space.

Atiyah showed that instantons in 4 dimensions can be identified with instantons in 2 dimensions, which are much easier to handle. There is of course a catch: in going from 4 to 2 dimensions the structure group of the gauge theory changes from a finite dimensional group to an infinite dimensional loop group. This gives another example where the moduli spaces of solutions of two apparently unrelated nonlinear partial differential equations turn out to be essentially the same.

Atiyah and Singer found that anomalies in quantum field theory could be interpreted in terms of index theory of the Dirac operator; this idea later became widely used by physicists.

Later work (1986 onwards)

Many of the papers in the 6th volume of his collected works are surveys, obituaries, and general talks. Since its publication, Atiyah has continued to publish, including several surveys, a popular book, and another paper with Segal on twisted K-theory.

One paper is a detailed study of the Dedekind eta function from the point of view of topology and the index theorem.

Several of his papers from around this time study the connections between quantum field theory, knots, and Donaldson theory. He introduced the concept of a topological quantum field theory, inspired by Witten's work and Segal's definition of a conformal field theory. His book describes the new knot invariants found by Vaughan Jones and Edward Witten in terms of topological quantum field theories, and his paper with L. Jeffrey explains Witten's Lagrangian giving the Donaldson invariants.

He studied skyrmions with Nick Manton, finding a relation with magnetic monopoles and instantons, and giving a conjecture for the structure of the moduli space of two skyrmions as a certain subquotient of complex projective 3-space.

Several papers were inspired by a question of M. Berry, who asked if there is a map from the configuration space of ''n'' points in 3-space to the flag manifold of the unitary group. Atiyah gave an affirmative answer to this question, but felt his solution was too computational and studied a conjecture that would give a more natural solution. He also related the question to Nahm's equation.

With Juan Maldacena and Cumrun Vafa, and E. Witten he described the dynamics of M-theory on manifolds with G₂ holonomy. These papers seem to be the first time that Atiyah has worked on exceptional Lie groups.

In his papers with M. Hopkins and G. Segal he returned to his earlier interest of K-theory, describing some twisted forms of K-theory with applications in theoretical physics.

Controversies (Atiyah-Raju Case)

C._K._Raju suggested using functional differential equations in 1994. He argued that functional differential equations meant a paradigm shift in physics, and could explain quantum mechanics. Atiyah echoed exactly the same thing in his Einstein lecture of October 2005, except that he did not acknowledge Raju's already published work. The Atiyah-Raju case is one of the cases of misconduct investigated by the Society for Scientific Values that acknowledged that Raju has a prima facie case against Atiyah. .

Awards and honours

In 1966, when he was thirty-seven years old, he was awarded the Fields Medal, for his work in developing K-theory, a generalized Lefschetz fixed-point theorem and the Atiyah–Singer theorem, for which he also won the Abel Prize jointly with Isadore Singer in 2004. Among other prizes he has received are the Royal Medal of the Royal Society in 1968, the De Morgan Medal of the London Mathematical Society in 1980, the Antonio Feltrinelli Prize from the Accademia Nazionale dei Lincei in 1981, the King Faisal International Prize for Science in 1987, the Copley Medal of the Royal Society in 1988, the Benjamin Franklin Medal of the American Philosophical Society in 1993, the Jawaharlal Nehru Birth Centenary Medal of the Indian National Science Academy in 1993, the President's Medal from the Institute of Physics in 2008 and the Grande Médaille of the French Academy of Sciences in 2010.

He was elected a foreign member of the National Academy of Sciences, the American Academy of Arts and Sciences (1969), the Academie des Sciences, the Akademie Leopoldina, the Royal Swedish Academy, the Royal Irish Academy, the Royal Society of Edinburgh, the American Philosophical Society, the Indian National Science Academy, the Chinese Academy of Science, the Australian Academy of Science, the Russian Academy of Science, the Ukrainian Academy of Science, the Georgian Academy of Science, the Venezuela Academy of Science, the Norwegian Academy of Science and Letters, the Royal Spanish Academy of Science, the Accademia dei Lincei and the Moscow Mathematical Society.

Atiyah has been awarded honorary degrees by the universities of Bonn, Warwick, Durham, St. Andrews, Dublin, Chicago, Cambridge, Edinburgh, Essex, London, Sussex, Ghent, Reading, Helsinki, Salamanca, Montreal, Wales, Lebanon, Queen's (Canada), Keele, Birmingham, UMIST, Brown, Heriot–Watt, Mexico, Oxford, Hong Kong (Chinese University), The Open University, American University of Beirut, the Technical University of Catalonia and Leicester.

Atiyah was made a Knight Bachelor in 1983 and made a member of the Order of Merit in 1992.

The Michael Atiyah building at the University of Leicester and the Michael Atiyah Chair in Mathematical Sciences at the American University of Beirut were named after him.

Notes

References

Books by Atiyah

This subsection lists all books written by Atiyah; it omits a few books that he edited. . A classic textbook covering standard commutative algebra. . Reprinted as . . Reprinted as . . Reprinted as . . Reprinted as . . . . . . . First edition (1967) reprinted as . . Reprinted as . . . .

Selected papers by Atiyah

. Reprinted in . . Reprinted in . . Reprinted in . . Reprinted in . Formulation of the Atiyah "Conjecture" on the rationality of the L²-Betti numbers. . An announcement of the index theorem. Reprinted in . . This gives a proof using K theory instead of cohomology. Reprinted in . . This reformulates the result as a sort of Lefschetz fixed point theorem, using equivariant K theory. Reprinted in . . This paper shows how to convert from the K-theory version to a version using cohomology. Reprinted in . This paper studies families of elliptic operators, where the index is now an element of the K-theory of the space parametrizing the family. Reprinted in . . This studies families of real (rather than complex) elliptic operators, when one can sometimes squeeze out a little extra information. Reprinted in . . This states a theorem calculating the Lefschetz number of an endomorphism of an elliptic complex. Reprinted in . (reprinted in )and . Reprinted in . These give the proofs and some applications of the results announced in the previous paper. ; Reprinted in . ; . Reprinted in .

Other references

. Reprinted in volume 1 of his collected works, p. 65–75, ISBN 0-387-13619-3. On page 120 Gel'fand suggests that the index of an elliptic operator should be expressible in terms of topological data. . This describes the original proof of the index theorem. (Atiyah and Singer never published their original proof themselves, but only improved versions of it.) . . .

External links

Michael Atiyah tells his life story at Web of Stories

The celebrations of Michael Atiyah's 80th birthday in Edinburgh, 20-24 April 2009

Mathematical descendants of Michael Atiyah

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Name	William Thurston
Birth date	October 30, 1946
Nationality	American
Field	Mathematics
Work institutions	Cornell UniversityUniversity of California, DavisUniversity of California, BerkeleyPrinceton University
Alma mater	New College of FloridaUniversity of California, Berkeley
Doctoral advisor	Morris Hirsch
Doctoral students	Richard CanaryBenson FarbDavid GabaiWilliam GoldmanSteven KerckhoffYair Minsky
Prizes	Fields Medal (1982)Oswald Veblen Prize in Geometry (1976)National Academy of Sciences (1983) }}

William Paul Thurston (born October 30, 1946) is an American mathematician. He is a pioneer in the field of low-dimensional topology. In 1982, he was awarded the Fields Medal for his contributions to the study of 3-manifolds. He is currently a professor of mathematics and computer science at Cornell University (since 2003).

Mathematical contributions

Foliations

His early work, in the early 1970s, was mainly in foliation theory, where he had a dramatic impact. His more significant results include:

The proof that every Haefliger structure on a manifold can be integrated to a foliation (this implies, in particular that every manifold with zero Euler characteristic admits a foliation of codimension one).

The construction of a continuous family of smooth, codimension one foliations on the three-sphere whose Godbillon–Vey invariant (after Claude Godbillon and Jacques Vey) takes every real value.

With John Mather, he gave a proof that the cohomology of the group of homeomorphisms of a manifold is the same whether the group is considered with its discrete topology or its compact-open topology.

In fact, Thurston resolved so many outstanding problems in foliation theory in such a short period of time that, according to Thurston, it led to a kind of exodus from the field, where advisors counselled students against going into foliation theory because Thurston was "cleaning out the subject" (see "On Proof and Progress in Mathematics", especially section 6 ).

The geometrization conjecture

His later work, starting around the late 1970s, revealed that hyperbolic geometry played a far more important role in the general theory of 3-manifolds than was previously realised. Prior to Thurston, there were only a handful of known examples of hyperbolic 3-manifolds of finite volume, such as the Seifert–Weber space. The independent and distinct approaches of Robert Riley and Troels Jørgensen in the mid-to-late 1970s showed that such examples were less atypical than previously believed; in particular their work showed that the figure eight knot complement was hyperbolic. This was the first example of a hyperbolic knot.

Inspired by their work, Thurston took a different, more explicit means of exhibiting the hyperbolic structure of the figure eight knot complement. He showed that the figure eight knot complement could be decomposed as the union of two regular ideal hyperbolic tetrahedra whose hyperbolic structures matched up correctly and gave the hyperbolic structure on the figure eight knot complement. By utilizing Haken's normal surface techniques, he classified the incompressible surfaces in the knot complement. Together with his analysis of deformations of hyperbolic structures, he concluded that all but 10 Dehn surgeries on the figure eight knot resulted in irreducible, non-Haken non-Seifert-fibered 3-manifolds. These were the first such examples; previously it had been believed that except for certain Seifert fiber spaces, all irreducible 3-manifolds were Haken. These examples were actually hyperbolic and motivated his next revolutionary theorem.

Thurston proved that in fact most Dehn fillings on a cusped hyperbolic 3-manifold resulted in hyperbolic 3-manifolds. This is his celebrated hyperbolic Dehn surgery theorem.

To complete the picture, Thurston proved a hyperbolization theorem for Haken manifolds. A particularly important corollary is that many knots and links are in fact hyperbolic. Together with his hyperbolic Dehn surgery theorem, this showed that closed hyperbolic 3-manifolds existed in great abundance.

The geometrization theorem has been called ''Thurston's Monster Theorem,'' due to the length and difficulty of the proof. Complete proofs were not written up until almost 20 years later. The proof involves a number of deep and original insights which have linked many apparently disparate fields to 3-manifolds.

Thurston was next led to formulate his geometrization conjecture. This gave a conjectural picture of 3-manifolds which indicated that all 3-manifolds admitted a certain kind of geometric decomposition involving eight geometries, now called Thurston model geometries. Hyperbolic geometry is the most prevalent geometry in this picture and also the most complicated. A proof to that conjecture follows from the recent work of Grigori Perelman (2002–2003).

Orbifold theorem

In his work on hyperbolic Dehn surgery, Thurston realized that orbifold structures naturally arose. Such structures had been studied prior to Thurston, but his work, particularly the next theorem, would bring them to prominence. In 1981, he announced the orbifold theorem, an extension of his geometrization theorem to the setting of 3-orbifolds. Two teams of mathematicians around 2000 finally finished their efforts to write down a complete proof, based mostly on Thurston's lectures given in the early 1980s in Princeton. His original proof relied partly on Hamilton's work on the Ricci flow.

Education and career

Thurston was born in Washington, D.C. to a homemaker and an aeronautical engineer. He received his bachelors degree from New College (now New College of Florida) in 1967. For his undergraduate thesis he developed an intuitionist foundation for topology. Following this, he earned a doctorate in mathematics from the University of California, Berkeley, in 1972. His Ph.D. advisor was Morris W. Hirsch and his dissertation was on ''Foliations of Three-Manifolds which are Circle Bundles''.

After completing his Ph.D., he spent a year at the Institute for Advanced Study, then another year at MIT as Assistant Professor. In 1974, he was appointed Professor of Mathematics at Princeton University. In 1991, he returned to UC-Berkeley as Professor of Mathematics and in 1993 became Director of the Mathematical Sciences Research Institute. In 1996, he moved to University of California, Davis. In 2003, he moved again to become Professor of Mathematics at Cornell University.

His Ph.D. students include Richard Canary, Renaud Dreyer, David Gabai, William Goldman, Benson Farb, Detlef Hardorp, Craig Hodgson, Richard Kenyon, Steven Kerckhoff, Robert Meyerhoff, Yair Minsky, Lee Mosher, Igor Rivin, Oded Schramm, Richard Schwartz, Martin Bridgeman, William Floyd and Jeffrey Weeks. His son, Dylan Thurston, is an assistant professor of mathematics at Barnard College, Columbia University.

Thurston has turned his attention in recent years to mathematical education and bringing mathematics to the general public. He has served as mathematics editor for Quantum Magazine, a youth science magazine, and as head of The Geometry Center. As director of Mathematical Sciences Research Institute from 1992 to 1997, he initiated a number of programs designed to increase awareness of mathematics among the public.

In 2005 Thurston won the first AMS Book Prize, for ''Three-dimensional Geometry and Topology''. The prize ''recognizes an outstanding research book that makes a seminal contribution to the research literature''.

Thurston has an Erdős number of 2.

Selected works

William Thurston, ''The geometry and topology of 3-manifolds'', Princeton lecture notes (1978–1981).

William Thurston. ''Three-dimensional geometry and topology. Vol. 1''. Edited by Silvio Levy. Princeton Mathematical Series, 35. Princeton University Press, Princeton, NJ, 1997. x+311 pp. ISBN 0-691-08304-5

William Thurston, ''Hyperbolic structures on 3-manifolds''. I. Deformation of acylindrical manifolds. Ann. of Math. (2) 124 (1986), no. 2, 203–246.

William Thurston, ''Three-dimensional manifolds, Kleinian groups and hyperbolic geometry'', Bull. Amer. Math. Soc. (N.S.) 6 (1982), 357–381.

William Thurston. ''On the geometry and dynamics of diffeomorphisms of surfaces''. Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, 417–431

Epstein, David B. A.; Cannon, James W.; Holt, Derek F.; Levy, Silvio V. F.; Paterson, Michael S.; Thurston, William P. ''Word processing in groups''. Jones and Bartlett Publishers, Boston, MA, 1992. xii+330 pp. ISBN 0-86720-244-0

Eliashberg, Yakov M.; Thurston, William P. ''Confoliations''. University Lecture Series, 13. American Mathematical Society, Providence, RI, 1998. x+66 pp. ISBN 0-8218-0776-5

See also

References

External links

Thurston's page at Cornell

Category:Topologists Category:Members of the United States National Academy of Sciences Category:American mathematicians Category:Fields Medalists Category:University of California, Berkeley faculty Category:Cornell University faculty Category:University of California, Davis Category:New College of Florida alumni Category:1946 births Category:Living people Category:Sloan Research Fellowships Category:University of California, Berkeley alumni

zh-min-nan:William Thurston de:William Thurston es:William Thurston fr:William Thurston ko:윌리엄 서스턴 it:William Thurston ht:William Thurston nl:William Thurston ja:ウィリアム・サーストン no:William Thurston pnb:ولیم تھرسٹن pl:William Thurston pt:William Thurston ru:Тёрстон, Уильям Пол sk:William Thurston sl:William Thurston fi:William Thurston uk:Вільям Терстон zh:威廉·瑟斯顿

Name	Stephen Hawking
Birth name	Stephen William Hawking
Birth date	January 08, 1942
Birth place	Oxford, England, United Kingdom
Residence	United Kingdom
Nationality	British
Fields	Applied mathematicsTheoretical physicsCosmology
Workplaces	Cambridge UniversityCalifornia Institute of TechnologyPerimeter Institute for Theoretical Physics
Alma mater	Oxford UniversityCambridge University
Doctoral advisor	Dennis Sciama
Academic advisors	Robert Berman
Doctoral students	Bruce AllenRaphael BoussoFay DowkerMalcolm PerryBernard CarrGary GibbonsHarvey ReallDon PageTim PrestidgeRaymond LaflammeJulian Luttrell
Known for	Black holesTheoretical cosmologyQuantum gravityHawking radiation
Influences	Dikran TahtaAlbert Einstein
Awards
Spouse	Jane Hawking(m. 1965–1991, divorced)Elaine Mason(m. 1995–2006, divorced)
Signature	Hawkingsig.svg }}

Stephen William Hawking, CH, CBE, FRS, FRSA (born 8 January 1942) is a British theoretical physicist and cosmologist, whose scientific books and public appearances have made him an academic celebrity. He is an Honorary Fellow of the Royal Society of Arts, a lifetime member of the Pontifical Academy of Sciences, and in 2009 was awarded the Presidential Medal of Freedom, the highest civilian award in the United States.

Hawking was the Lucasian Professor of Mathematics at the University of Cambridge for 30 years, taking up the post in 1979 and retiring on 1 October 2009. He is now Director of Research at the Centre for Theoretical Cosmology in the Department of Applied Mathematics and Theoretical Physics at the University of Cambridge. He is also a Fellow of Gonville and Caius College, Cambridge and a Distinguished Research Chair at the Perimeter Institute for Theoretical Physics in Waterloo, Ontario. He is known for his contributions to the fields of cosmology and quantum gravity, especially in the context of black holes. He has also achieved success with works of popular science in which he discusses his own theories and cosmology in general; these include the runaway best seller ''A Brief History of Time'', which stayed on the British ''Sunday Times'' best-sellers list for a record-breaking 237 weeks.

Hawking's key scientific works to date have included providing, with Roger Penrose, theorems regarding gravitational singularities in the framework of general relativity, and the theoretical prediction that black holes should emit radiation, which is today known as Hawking radiation (or sometimes as Bekenstein–Hawking radiation).

Hawking has a motor neurone disease that is related to amyotrophic lateral sclerosis, a condition that has progressed over the years and has left him almost completely paralysed.

Early life

Stephen Hawking was born on 8 January 1942 to Dr. Frank Hawking, a research biologist, and Isobel Hawking. He had two younger sisters, Philippa and Mary, and an adopted brother, Edward. Though Hawking's parents were living in North London, they moved to Oxford while his mother was pregnant with Stephen, desiring a safer location for the birth of their first child. (London was under attack at the time by the Luftwaffe.) According to Hawking, a German V-2 missile struck only a few streets away.

After Hawking was born, the family moved back to London, where his father headed the division of parasitology at the National Institute for Medical Research. In 1950, Hawking and his family moved to St Albans, Hertfordshire, where he attended St Albans High School for Girls from 1950 to 1953. (At that time, boys could attend the Girls' school until the age of ten.) From the age of eleven, he attended St Albans School, where he was a good, but not exceptional, student. When asked later to name a teacher who had inspired him, Hawking named his mathematics teacher Dikran Tahta. He maintains his connection with the school, giving his name to one of the four houses and to an extracurricular science lecture series. He has visited it to deliver one of the lectures and has also granted a lengthy interview to pupils working on the school magazine, ''The Albanian''.

Hawking was always interested in science. Inspired by his mathematics teacher, he originally wanted to study the subject at university. However, Hawking's father wanted him to apply to University College, Oxford, where his father had attended. As University College did not have a mathematics fellow at that time, it would not accept applications from students who wished to read that discipline. Hawking therefore applied to read natural sciences, in which he gained a scholarship. Once at University College, Hawking specialised in physics. His interests during this time were in thermodynamics, relativity, and quantum mechanics. His physics tutor, Robert Berman, later said in ''The New York Times Magazine'':

Hawking was passing, but his unimpressive study habits resulted in a final examination score on the borderline between first and second class honours, making an "oral examination" necessary. Berman said of the oral examination:

After receiving his B.A. degree at Oxford in 1962, he stayed to study astronomy. He decided to leave when he found that studying sunspots, which was all the observatory was equipped for, did not appeal to him and that he was more interested in theory than in observation. He left Oxford for Trinity Hall, Cambridge, where he engaged in the study of theoretical astronomy and cosmology.

Career

Almost as soon as he arrived at Cambridge, he started developing symptoms of amyotrophic lateral sclerosis (ALS, known colloquially in the United States as Lou Gehrig's disease), a type of motor neurone disease which would cost him almost all neuromuscular control. During his first two years at Cambridge, he did not distinguish himself, but, after the disease had stabilised and with the help of his doctoral tutor, Dennis William Sciama, he returned to working on his PhD.

Hawking was elected as one of the youngest Fellows of the Royal Society in 1974, was created a Commander of the Order of the British Empire in 1982, and became a Companion of Honour in 1989. Hawking is a member of the Board of Sponsors of the ''Bulletin of the Atomic Scientists''.

In 1974, he accepted the Sherman Fairchild Distinguished Scholar visiting professorship at the California Institute of Technology (Caltech) to work with his friend, Kip Thorne, who was a faculty member there. He continues to have ties with Caltech, spending a month each year there since 1992.

Hawking's achievements were made despite the increasing paralysis caused by the ALS. By 1974, he was unable to feed himself or get out of bed. His speech became slurred so that he could be understood only by people who knew him well. In 1985, he caught pneumonia and had to have a tracheotomy, which made him unable to speak at all. A Cambridge scientist built a device that enables Hawking to write onto a computer with small movements of his body, and then have a voice synthesiser speak what he has typed.

Research fields

Hawking's principal fields of research are theoretical cosmology and quantum gravity.

In the late 1960s, he and his Cambridge friend and colleague, Roger Penrose, applied a new, complex mathematical model they had created from Albert Einstein's theory of general relativity. This led, in 1970, to Hawking proving the first of many singularity theorems; such theorems provide a set of sufficient conditions for the existence of a gravitational singularity in space-time. This work showed that, far from being mathematical curiosities which appear only in special cases, singularities are a fairly generic feature of general relativity.

He supplied a mathematical proof, along with Brandon Carter, Werner Israel and D. Robinson, of John Wheeler's no-hair theorem – namely, that any black hole is fully described by the three properties of mass, angular momentum, and electric charge.

Hawking also suggested upon analysis of gamma ray emissions that after the Big Bang, primordial mini black holes were formed. With Bardeen and Carter, he proposed the four laws of black hole mechanics, drawing an analogy with thermodynamics. In 1974, he calculated that black holes should thermally create and emit subatomic particles, known today as Bekenstein-Hawking radiation, until they exhaust their energy and evaporate.

In collaboration with Jim Hartle, Hawking developed a model in which the universe had no boundary in space-time, replacing the initial singularity of the classical Big Bang models with a region akin to the North Pole: one cannot travel north of the North Pole, as there is no boundary. While originally the no-boundary proposal predicted a closed universe, discussions with Neil Turok led to the realisation that the no-boundary proposal is also consistent with a universe which is not closed.

Along with Thomas Hertog at CERN, in 2006 Hawking proposed a theory of "top-down cosmology," which says that the universe had no unique initial state, and therefore it is inappropriate for physicists to attempt to formulate a theory that predicts the universe's current configuration from one particular initial state. Top-down cosmology posits that in some sense, the present "selects" the past from a superposition of many possible histories. In doing so, the theory suggests a possible resolution of the fine-tuning question: It is inevitable that we find our universe's present physical constants, as the current universe "selects" only those past histories that led to the present conditions. In this way, top-down cosmology provides an anthropic explanation for why we find ourselves in a universe that allows matter and life, without invoking an ensemble of multiple universes.

Hawking's many other scientific investigations have included the study of quantum cosmology, cosmic inflation, helium production in anisotropic Big Bang universes, large N cosmology, the density matrix of the universe, topology and structure of the universe, baby universes, Yang-Mills instantons and the S matrix, anti de Sitter space, quantum entanglement and entropy, the nature of space and time, including the arrow of time, spacetime foam, string theory, supergravity, Euclidean quantum gravity, the gravitational Hamiltonian, Brans-Dicke and Hoyle-Narlikar theories of gravitation, gravitational radiation, and wormholes.

At a George Washington University lecture in honour of NASA's fiftieth anniversary, Hawking theorised on the existence of extraterrestrial life, believing that "primitive life is very common and intelligent life is fairly rare."

Losing an old bet

Hawking was in the news in July 2004 for presenting a new theory about black holes which goes against his own long-held belief about their behaviour, thus losing a bet he made with Kip Thorne and John Preskill of Caltech. Classically, it can be shown that information crossing the event horizon of a black hole is lost to our universe, and that thus all black holes are identical beyond their mass, electrical charge and angular velocity (the "no hair theorem"). The problem with this theorem is that it implies the black hole will emit the same radiation regardless of what goes into it, and as a consequence that if a pure quantum state is thrown into a black hole, an "ordinary" mixed state will be returned. This runs counter to the rules of quantum mechanics and is known as the black hole information paradox.

Human spaceflight

At the fiftieth anniversary of NASA in 2008, Hawking gave a keynote speech on the final frontier exhorting and inspiring the space technology community on why we (the human race) explore space.

At the celebration of his sixty-fifth birthday on 8 January 2007, Hawking announced his plan to take a zero-gravity flight in 2007 to prepare for a sub-orbital spaceflight in 2009 on Virgin Galactic's space service. Billionaire Richard Branson pledged to pay all expenses for the latter, costing an estimated £100,000. Stephen Hawking's zero-gravity flight in a "''Vomit Comet''" of Zero Gravity Corporation, during which he experienced weightlessness eight times, took place on 26 April 2007. He became the first quadriplegic to float in zero-gravity. This was the first time in forty years that he moved freely, without his wheelchair. The fee is normally US$3,750 for 10–15 plunges, but Hawking was not required to pay the fee. A bit of a futurist, Hawking was quoted before the flight saying: }} In an interview with ''The Daily Telegraph'', he suggested that space was the Earth's long term hope. He continued this theme at a 2008 Charlie Rose interview.

Existence and nature of extraterrestrial life

Hawking has indicated that he is almost certain that alien life exists in other parts of the universe and uses a mathematical basis for his assumptions. "To my mathematical brain, the numbers alone make thinking about aliens perfectly rational. The real challenge is to work out what aliens might actually be like." He believes alien life not only certainly exists on planets but perhaps even in other places, like within stars or even floating in outer space. He also warns that a few of these species might be intelligent and threaten Earth. Contact with such species might be devastating for humanity. "If aliens visit us, the outcome would be much as when Columbus landed in America, which didn't turn out well for the Native Americans," he said. He advocated that, rather than try to establish contact, humans should try to avoid contact with alien life forms.

Illness

Stephen Hawking is severely disabled by a motor neurone disease known as Amyotrophic lateral sclerosis (ALS), sometimes known as Lou Gehrig's disease. Hawking's illness is markedly different from typical ALS because if confirmed, Hawking's case would make for the most protracted case ever documented. A survival for more than ten years after diagnosis is uncommon for ALS; the longest documented durations, other than Hawking's, are 32 and 39 years and these cases were termed benign because of the lack of the typical progressive course.

When he was young, he enjoyed riding horses. At Oxford, he coxed a rowing team, which, he stated, helped relieve his immense boredom at the university. Symptoms of the disorder first appeared while he was enrolled at University of Cambridge; he lost his balance and fell down a flight of stairs, hitting his head. Worried that he would lose his genius, he took the Mensa test to verify that his intellectual abilities were intact. The diagnosis of motor neurone disease came when Hawking was 21, shortly before his first marriage, and doctors said he would not survive more than two or three years. Hawking gradually lost the use of his arms, legs, and voice, and as of 2009 has been almost completely paralysed.

During a visit to the research centre CERN in Geneva in 1985, Hawking contracted pneumonia, which in his condition was life-threatening as it further restricted his already limited respiratory capacity. He had an emergency tracheotomy, and as a result lost what remained of his ability to speak. He has since used an electronic voice synthesiser to communicate.

The DECtalk DTC01 voice synthesiser he uses, which has an American English accent, is no longer being produced. Asked why he has still kept it after so many years, Hawking mentioned that he has not heard a voice he likes better and that he identifies with it. Hawking is said to be looking for a replacement since, aside from being obsolete, the synthesiser is both large and fragile by current standards. As of mid 2009, he was said to be using NeoSpeech's VoiceText speech synthesiser.

In Hawking's many media appearances, he appears to speak fluently through his synthesiser, but in reality, it is a tedious drawn-out process. Hawking's setup uses a predictive text entry system, which requires only the first few characters in order to auto-complete the word, but as he is only able to use his cheek for data entry, constructing complete sentences takes time. His speeches are prepared in advance, but having a live conversation with him provides insight as to the complexity and work involved. During a TED Conference talk, it took him seven minutes to answer a question.

He describes himself as lucky, despite his disease. Its slow progression has allowed him time to make influential discoveries and has not hindered him from having, in his own words, "a very attractive family." When his wife, Jane, was asked why she decided to marry a man with a three-year life expectancy, she responded, "Those were the days of atomic gloom and doom, so we all had a rather short life expectancy." On 20 April 2009, Cambridge University released a statement saying that Hawking was "very ill" with a chest infection, and was admitted to Addenbrooke's Hospital. The following day, it was reported that his new condition was "comfortable" and he would make a full recovery from the infection.

As popular science advocate

Hawking has played himself on numerous television shows and has been portrayed in many more. He has played himself on a ''Red Dwarf'' anniversary special, played a hologram of himself on the episode "Descent" of ''Star Trek: The Next Generation'', appeared in a skit on ''Late Night with Conan O'Brien'', and appeared on the Discovery Channel special ''Alien Planet''. He has also played himself in several episodes of ''The Simpsons'' and ''Futurama'', and has had an action figure made of his ''Simpsons'' likeness. In 2008, Hawking was the subject of and featured in the documentary series ''Stephen Hawking, Master of the Universe'' for Channel 4. In September 2008, Hawking presided over the unveiling of the 'Chronophage' (time-eating) Corpus Clock at Corpus Christi College Cambridge. His actual synthesiser voice was used on parts of the Pink Floyd song "Keep Talking" from the 1994 album ''The Division Bell'', as well as on Turbonegro's "Intro: The Party Zone" on their 2005 album ''Party Animals'', Wolfsheim's "Kein Zurück (Oliver Pinelli Mix)".

Recognition

Acclaim

On 19 December 2007, a statue of Hawking by renowned late artist Ian Walters was unveiled at the Centre for Theoretical Cosmology, University of Cambridge. In May 2008, the statue of Hawking was unveiled at the African Institute for Mathematical Sciences in Cape Town. The Stephen W. Hawking Science Museum in San Salvador, El Salvador is named in honour of Stephen Hawking, citing his scientific distinction and perseverance in dealing with adversity. Stephen Hawking Building in Cambridge opened on 17 April 2007. The building belongs to Gonville and Caius College and is used as an undergraduate accommodation and conference facility.

Distinctions

Hawking's belief that the lay person should have access to his work led him to write a series of popular science books in addition to his academic work. The first of these, ''A Brief History of Time'', was published on 1 April 1988 by Hawking, his family and friends, and some leading physicists. It surprisingly became a best-seller and was followed by ''The Universe in a Nutshell'' (2001). Both books have remained highly popular all over the world. A collection of essays titled ''Black Holes and Baby Universes'' (1993) was also popular. His book, ''A Briefer History of Time'' (2005), co-written by Leonard Mlodinow, aims to update his earlier works and make them accessible to an even wider audience. In 2007 Hawking and his daughter, Lucy Hawking, published ''George's Secret Key to the Universe'', a children's book focusing on science that has been described as "like ''Harry Potter'', but without the magic." The book includes information on Hawking radiation.

Hawking supports the children's charity SOS Children's Villages UK.

Awards and honours

1975 Eddington Medal

1976 Hughes Medal of the Royal Society

1979 Albert Einstein Medal

1981 Franklin Medal

1982 Order of the British Empire (Commander)

1985 Gold Medal of the Royal Astronomical Society

1986 Member of the Pontifical Academy of Sciences

1988 Wolf Prize in Physics

1989 Prince of Asturias Awards in Concord

1989 Companion of Honour

1999 Julius Edgar Lilienfeld Prize of the American Physical Society

2003 Michelson Morley Award of Case Western Reserve University

2006 Copley Medal of the Royal Society 2008 Fonseca Prize of the University of Santiago de Compostela 2009 Presidential Medal of Freedom, the highest civilian honour in the United States

Personal life

Hawking revealed that he did not see much point in obtaining a doctorate if he were to die soon. Hawking later said that the real turning point was his 1965 marriage to Jane Wilde, a language student. After gaining his PhD at Trinity Hall, he became first a Research Fellow, and later on a Professorial Fellow at Gonville and Caius College. Jane Hawking (née Wilde), Hawking's first wife, cared for him until 1991 when the couple separated, reportedly because of the pressures of fame and his increasing disability. They had three children: Robert, Lucy, and Timothy. Hawking then married his nurse, Elaine Mason (who was previously married to David Mason, the designer of the first version of Hawking's talking computer), in 1995. In October 2006, Hawking filed for divorce from his second wife amid claims by former nurses that she had abused him.

In 1999, Jane Hawking published a memoir, ''Music to Move the Stars'', detailing the marriage and his breakdown; in 2010 she published a revised version, ''Travelling to Infinity, My Life with Stephen''. Hawking's daughter, Lucy, is a novelist. Their oldest son, Robert, emigrated to the United States, married, and has a son. After a period of estrangement, Hawking and his first family were reconciled in 2007.

His view on how to live life is to "seek the greatest value of our action".

Hawking was asked about his IQ in a 2004 newspaper interview, and replied, "I have no idea. People who boast about their I.Q. are losers." Yet when asked "Are you saying you are not a genius?", Hawking replied "I hope I'm near the upper end of the range."

Hawking strongly opposed the US-led Iraq War, calling it "a war crime" and "based on lies". In 2004, he personally attended a demonstration against the war in Trafalgar Square, and participated in a public reading of the names of Iraqi war victims.

Religious views

In his early work, Hawking spoke of "God" in a metaphorical sense, such as in ''A Brief History of Time'': "If we discover a complete theory, it would be the ultimate triumph of human reason—for then we should know the mind of God." In the same book he suggested the existence of God was unnecessary to explain the origin of the universe. His 2010 book ''The Grand Design'' and interviews with the ''Telegraph'' and the Channel 4 documentary ''Genius of Britain'', clarify that he does "not believe in a personal God". Hawking writes, "The question is: is the way the universe began chosen by God for reasons we can't understand, or was it determined by a law of science? I believe the second." He adds, "Because there is a law such as gravity, the Universe can and will create itself from nothing."

His ex-wife, Jane, said during their divorce proceedings that he was an atheist. Hawking has stated that he is "not religious in the normal sense" and he believes that "the universe is governed by the laws of science. The laws may have been decreed by God, but God does not intervene to break the laws." In an interview published in ''The Guardian'' newspaper, Hawking regarded the concept of Heaven as a myth, stating that there is "no heaven or afterlife" and that such a notion was a "fairy story for people afraid of the dark."

Hawking contrasted religion and science in 2010, saying: "There is a fundamental difference between religion, which is based on authority, [and] science, which is based on observation and reason. Science will win because it works."

Selected publications

Technical

''Singularities in Collapsing Stars and Expanding Universes'' with Dennis William Sciama, 1969 Comments on Astrophysics and Space Physics Vol 1 No.1

''The Large Scale Structure of Space-Time'' with George Ellis, 1973, ISBN 0-521-09906-4

''The Nature of Space and Time'' with Roger Penrose, foreword by Michael Atiyah, New Jersey: Princeton University Press, 1996, ISBN 0-691-05084-8

''The Large, the Small, and the Human Mind'', (with Abner Shimony, Nancy Cartwright, and Roger Penrose), Cambridge University Press, 1997, ISBN 0-521-56330-5 (hardback), ISBN 0-521-65538-2 (paperback), Canto edition: ISBN 0-521-78572-3

''Information Loss in Black Holes'', Cambridge University Press, 2005

''God Created the Integers: The Mathematical Breakthroughs That Changed History'', Running Press, 2005 ISBN 0762419229

Popular

''A Brief History of Time'', (Bantam Press 1988) ISBN 055305340X

''Black Holes and Baby Universes and Other Essays'', (Bantam Books 1993) ISBN 0553374117

''The Universe in a Nutshell'', (Bantam Press 2001) ISBN 055380202X

''On The Shoulders of Giants. The Great Works of Physics and Astronomy'', (Running Press 2002) ISBN 076241698X

''A Briefer History of Time'', coauthored with Leonard Mlodinow, (Bantam Books 2005) ISBN 0553804367

''The Grand Design'', coauthored with Leonard Mlodinow, (Bantam Press 2010) ISBN 0553805371

Children's fiction

These are co-written with his daughter Lucy.

''George's Secret Key to the Universe'', (Random House, 2007) ISBN 9780385612708

''George's Cosmic Treasure Hunt'', (Simon & Schuster, 2009) ISBN 9781416986713

Films and series

''A Brief History of Time'' (1991)

''Stephen Hawking's Universe'' (1997)

''Horizon: The Hawking Paradox'' (2005)

''Masters of Science Fiction'' (2007)

''Stephen Hawking: Master of the Universe'' (2008) ''Into The Universe with Stephen Hawking'' (2010) A list of Hawking's publications through the year 2002 is available on his website.

See also

Basic concepts of quantum mechanics

Many-worlds interpretation, or flexiverse

Susskind–Hawking battle – see Black hole information paradox

References

Further reading

A layman's guide to Stephen Hawking

Ferguson, Kitty (1991). ''Stephen Hawking: Quest For A Theory of Everything''. Franklin Watts. ISBN 0-553-29895-X

Highly influential in the field. A much cited centennial survey.

Pickover, Clifford, ''Archimedes to Hawking: Laws of Science and the Great Minds Behind Them'', Oxford University Press, 2008, ISBN 978-0195336115

External links

Stephen Hawking's web site

Stephen Hawking's page on Academia.edu

The Life of Stephen Hawking – slideshow by ''The First Post''

Video: Stephen Hawking – discussion of two views of the universe

Videos: Stephen Hawking's concept of God, The role of God within the no boundary cosmology and Imaginary time

;Dated

Stephen Hawking says universe not created by God

Public Lectures, including debate with Roger Penrose 1996–2006

''Hawking celebrates own brief history'', 7 January 2002, BBC

"Leaping the Abyss", interview in ''Reason'' by Gregory Benford 2002-04-01

2002-03-24

''Black holes turned "inside out"'', 22 July 2004, BBC

''Return of the time lord'', Interview about "A Brief History of Time", 27 September 2005, The Guardian.

Stephen Hawking touches on God and science – Physicist says Pope John Paul told scientists not to study universe's origins msnbc June 2006

Press Release from the Catholic League on misquote of Pope by Hawking 2006-06-16

BBC interview 2008-12-05

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af:Stephen Hawking ar:ستيفن هوكينج az:Stiven Hokinq bn:স্টিফেন হকিং zh-min-nan:Stephen Hawking be:Стывен Уільям Хокінг bg:Стивън Хокинг bs:Stephen Hawking ca:Stephen Hawking cs:Stephen Hawking cy:Stephen Hawking da:Stephen Hawking de:Stephen Hawking et:Stephen Hawking el:Στήβεν Χώκινγκ es:Stephen Hawking eo:Stephen Hawking ext:Stephen Hawking eu:Stephen Hawking fa:استیون هاوکینگ fr:Stephen Hawking ga:Stephen Hawking gl:Stephen Hawking ko:스티븐 호킹 hi:स्टीफन हॉकिंग hr:Stephen Hawking io:Stephen Hawking id:Stephen Hawking is:Stephen Hawking it:Stephen Hawking he:סטיבן הוקינג jv:Stephen Hawking kn:ಸ್ಟೀಫನ್‌ ಹಾಕಿಂಗ್ ka:სტივენ ჰოუკინგი ht:Stephen Hawking la:Stephanus Hawking lv:Stīvens Hokings lb:Stephen Hawking lt:Stephen Hawking hu:Stephen Hawking mk:Стивен Хокинг ml:സ്റ്റീഫൻ ഹോക്കിങ് mr:स्टीफन हॉकिंग ms:Stephen Hawking mn:Стивен Хокинг my:စတီဖင်ဟော့ကင်း nah:Stephen Hawking nl:Stephen Hawking ja:スティーヴン・ホーキング no:Stephen Hawking nn:Stephen Hawking pnb:سٹیفن ہاکنگ pl:Stephen Hawking pt:Stephen Hawking ro:Stephen Hawking ru:Хокинг, Стивен Уильям sq:Stephen Hawking simple:Stephen Hawking sk:Stephen Hawking sl:Stephen Hawking sr:Стивен Хокинг sh:Stephen Hawking fi:Stephen Hawking sv:Stephen Hawking tl:Stephen Hawking ta:ஸ்டீபன் ஹோக்கிங் te:స్టీఫెన్ హాకింగ్ th:สตีเฟน ฮอว์คิง tg:Стивен Ҳокинг tr:Stephen Hawking uk:Стівен Гокінг ur:سٹیفن ہاکنگ vi:Stephen Hawking war:Stephen Hawking yi:סטיווען האקינג zh-yue:霍金 bat-smg:Stephen Hawking zh:史蒂芬·霍金

name	David Gabai
nationality
fields	Mathematics
workplaces	Princeton UniversityCaltech
alma mater	Princeton University MIT
doctoral advisor	William Thurston
Known for	low-dimensional topology
awards	Oswald Veblen Prize in Geometry (2004) }}

David Gabai, a mathematician, is currently a professor at Princeton University. Focused on low-dimensional topology and hyperbolic geometry, he is a leading researcher in those subjects.

David Gabai received his S.B. degree from the MIT in 1976 and his Ph.D. from Princeton in 1980 under the direction of William Thurston. During his Ph.D., he obtained foundational results on the foliations of 3-manifolds; these results are the basis of many research areas in geometric topology now.

After positions at Harvard and U Penn, he spent most of the years between 1986–2001 at Caltech, and has been at Princeton University since 2001.

In 2004, David Gabai was awarded the Oswald Veblen Prize in Geometry, which is one of the most prestigious prizes in geometric topology, given every 3 years by the American Mathematical Society. In 2011, he was elected to the United States National Academy of Sciences.

David Gabai has played a key role in the field of topology of 3-manifolds in the last 3 decades. Some of the foundational results he and his collaborators have proved are as follows: Existence of taut foliation in 3-manifolds, Property R Conjecture, foundation of essential laminations, Seifert fiber space conjecture, rigidity of homotopy hyperbolic 3-manifolds, weak hyperbolization for 3-manifolds with genuine lamination, Smale Conjecture for hyperbolic 3-manifolds, Marden's Tameness Conjecture, Weeks's manifold being the minimum volume closed hyperbolic 3-manifold.

References

External links

Category:Year of birth missing (living people) Category:Living people Category:Topologists Category:Princeton University faculty Category:Members of the United States National Academy of Sciences

de:David Gabai

Tian Gang (; born November 1958) is a Chinese mathematician and an academician of the Chinese Academy of Sciences. He is known for his contributions to geometric analysis and quantum cohomology, among other fields. He was born in Nanjing, China, was a professor of mathematics at MIT from 1995–2006 (holding the chair of Simons Professor of Mathematics from 1996), but now divides his time between Princeton University and Peking University. His employment at Princeton started from 2003, and now he is entitled Higgins Professor of Mathematics; starting 2005, he has been the director of Beijing International Center for Mathematical Research (BICMR).

Biography

Tian graduated from Nanjing University in 1982, and received a master's degree from Peking University in 1984. In 1988, he received a Ph.D. in mathematics from Harvard University, after having studied under Shing-Tung Yau. This work was so exceptional he was invited to present it at the Geometry Festival that year. In 1998, he was appointed as a Cheung Kong Scholar professor at the School of Mathematical Sciences at Peking University, under the "Cheung Kong Scholars Programme" (长江计划) of the Ministry of Education. Later his appointment was changed to Cheung Kong Scholar chair professorship. He was awarded the Alan T. Waterman Award in 1994, and the Veblen Prize in 1996. In 2004 Tian was inducted into the American Academy of Arts and Sciences.

Mathematical contributions

Much of Tian's earlier work was about the existence of Kähler–Einstein metrics on complex manifolds under the direct of Yau. In particular he solved the existence question for Kähler–Einstein metrics on compact complex surfaces with positive first Chern class, and showed that hypersurfaces with a Kähler–Einstein metric are stable in the sense of geometric invariant theory. He proved that a Kähler manifold with trivial canonical bundle has trivial obstruction space, known as the Bogomolov–Tian–Todorov theorem.

He (jointly with Jun Li) constructed the moduli spaces of maps from curves in both algebraic geometry and symplectic geometry and studied the obstruction theory on these moduli spaces. He also (jointly with Y. Ruan) showed that the quantum cohomology ring of a symplectic manifold is associative.

In 2006, together with John Morgan of Columbia University, amongst others, Tian helped verify the proof of the Poincaré conjecture given by Grigori Perelman.

Publications

(Selected)

Tian, Gang. Smoothness of the universal deformation space of compact Calabi-Yau manifolds and its Petersson-Weil metric. Mathematical aspects of string theory (San Diego, Calif., 1986), 629--646, Adv. Ser. Math. Phys., 1, World Sci. Publishing, Singapore, 1987.

Tian, Gang. On Kähler-Einstein metrics on certain Kähler manifolds with $C\sb 1(M)>0$. Invent. Math. 89 (1987), no. 2, 225--246.

Tian, G.; Yau, Shing-Tung. Complete Kähler manifolds with zero Ricci curvature. I. J. Amer. Math. Soc. 3 (1990), no. 3, 579--609.

Tian, G. On Calabi's conjecture for complex surfaces with positive first Chern class. Invent. Math. 101 (1990), no. 1, 101--172.

Tian, Gang. On a set of polarized Kähler metrics on algebraic manifolds. J. Differential Geom. 32 (1990), no. 1, 99--130.

Ruan, Yongbin; Tian, Gang. A mathematical theory of quantum cohomology. J. Differential Geom. 42 (1995), no. 2, 259--367.

Tian, Gang. Kähler-Einstein metrics with positive scalar curvature. Invent. Math. 130 (1997), no. 1, 1--37.

Ruan, Yongbin; Tian, Gang. Higher genus symplectic invariants and sigma models coupled with gravity. Invent. Math. 130 (1997), no. 3, 455--516.

Li, Jun; Tian, Gang. Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties. J. Amer. Math. Soc. 11 (1998), no. 1, 119--174.

Liu, Gang; Tian, Gang. Floer homology and Arnold conjecture. J. Differential Geom. 49 (1998), no. 1, 1--74.

Liu, Xiaobo; Tian, Gang. Virasoro constraints for quantum cohomology. J. Differential Geom. 50 (1998), no. 3, 537--590.

Tian, Gang. Gauge theory and calibrated geometry. I. Ann. of Math. (2) 151 (2000), no. 1, 193--268.

Tian, Gang; Zhu, Xiaohua. Uniqueness of Kähler-Ricci solitons. Acta Math. 184 (2000), no. 2, 271--305.

Cheeger, J.; Colding, T. H.; Tian, G. On the singularities of spaces with bounded Ricci curvature. Geom. Funct. Anal. 12 (2002), no. 5, 873--914.

Tao, Terence; Tian, Gang. A singularity removal theorem for Yang-Mills fields in higher dimensions. J. Amer. Math. Soc. 17 (2004), no. 3, 557--593.

Tian, Gang; Viaclovsky, Jeff. Bach-flat asymptotically locally Euclidean metrics. Invent. Math. 160 (2005), no. 2, 357--415.

Cheeger, Jeff; Tian, Gang. Curvature and injectivity radius estimates for Einstein 4-manifolds. J. Amer. Math. Soc. Vol. 19, No. 2 (2006), 487--525.

Morgan, John; Tian, Gang. Ricci flow and the Poincaré conjecture. Clay Mathematics Monographs, 3. American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2007, 525pp.

Song, Jian; Tian, Gang. The Kähler-Ricci flow on surfaces of positive Kodaira dimension. Invent. Math. 170 (2007), no. 3, 609--653.

Chen, X. X.; Tian, G. Geometry of Kähler metrics and foliations by holomorphic discs. Publ. Math. Inst. Hautes Études Sci. No. 107 (2008), 1--107.

Kołodziej, Sławomir; Tian, Gang A uniform $L^\infty$L∞ estimate for complex Monge-Ampère equations. Math. Ann. 342 (2008), no. 4, 773–787.

Mundet i Riera, I.; Tian, G. A compactification of the moduli space of twisted holomorphic maps. Adv. Math. 222 (2009), no. 4, 1117–1196.

Rivière, Tristan; Tian, Gang The singular set of 1-1 integral currents. Ann. of Math. (2) 169 (2009), no. 3, 741–794.

Tian, Gang Finite-time singularity of Kähler-Ricci flow. Discrete Contin. Dyn. Syst. 28 (2010), no. 3, 1137–1150.

Students

Vladimir Bozin, MIT, 2004;

Xiaodong Cao, MIT, 2002;

Sandra Francisco, MIT, 2005;

Zuoliang Hou, MIT, 2004;

Ljudmila Kamenova, MIT, 2006;

Peng Lu State, University of New York at Stony Brook, 1996;

Zhiqin Lu, New York University, 1997;

Dragos Oprea, MIT, 2005;

Yanir Rubinstein, MIT, 2008;

Sema Salur, Michigan State University, 2000;

Bianca Santoro, MIT, 2006;

Natasa Sesum, MIT, 2004;

Jake Solomon, MIT, 2006;

Michael Usher, MIT, 2004;

Lijing Wang, MIT, 2003;

Hao Wu, MIT, 2004;

Zhiyu Wu, Columbia University, 1998;

Baozhong Yang, MIT, 2000;

Zhou Zhang, MIT, 2006;

Hans-Joachim Hein, Princeton, 2010;

Richard Bamler, Princeton, 2011;

Chi Li, Princeton, 2011;

Mohammad Farajzadeh Tehrani, Princeton 2011;

Giulia Saccà, Princeton, 2012;

Guangbo Xu, Princeton 2012;

References

External links

Veblen prize citation

M.I.T. home page for Gang Tian (probably out of date).

Category:1958 births Category:20th-century mathematicians Category:21st-century mathematicians Category:Chinese mathematicians Category:Harvard University alumni Category:Members of the Chinese Academy of Sciences Category:Peking University alumni Category:Peking University faculty Category:Fellows of the American Academy of Arts and Sciences Category:Princeton University faculty Category:Living people

de:Gang Tian ja:田剛 pt:Gang Tian zh:田刚