当代数学大师迈克尔阿蒂亚证明黎曼猜想

当代数学大师迈克尔阿蒂亚证明黎曼猜想
    去9月24日，89岁高龄的当代数学大师迈克尔阿蒂亚先生亲自前往海德堡高级数学论坛公开证明黎曼猜想，此举吸引了全球数学界的目光。
   阿蒂亚指出：黎曼猜想的核心问题是素数分布定理，两者关系密切。
国人对此事件关注不够。
附件1: 阿蒂亚简历：
附件2 证明会场情况
袁萌   陈启清 2月18日1
附件1: 阿蒂亚简历：
Michael Atiyah
OM FRS FRSE FMedSci FAA FREng

Michael Atiyah in 2007
Born
Michael Francis Atiyah
22 April 1929
Hampstead, London, England
Died
11 January 2019 (aged 89)
Residence
United Kingdom
Nationality
British, Lebanese [1]
Known for
Atiyah–Singer index theorem
Atiyah–Segal completion theorem
Awards
Berwick Prize (1961)
Fields Medal (1966)
Royal Medal (1968)
De Morgan Medal (1980)
Copley Medal (1988)
Abel Prize (2004)

Education
Victoria College, Alexandria
Manchester Grammar School
Trinity College, Cambridge (BA, PhD)
Scientific career
Institutions
University of Oxford - New College, Oxford and St Catherine's College, Oxford
Institute for Advanced Study
University of Leicester
University of Edinburgh
Pembroke College, Cambridge
Thesis
Some Applications of Topological Methods in Algebraic Geometry (1955)
Doctoral advisor
W. V. D. Hodge[2][3]
Doctoral students
Simon Donaldson
K. David Elworthy
Nigel Hitchin[4]
Lisa Jeffrey
Frances Kirwan
Peter Kronheimer
Ruth Lawrence
George Lusztig
Ian R. Porteous
Graeme Segal
David O. Tall[3]
Other notable students
Edward Witten

Sir Michael Francis Atiyah OM FRS FRSE FMedSci FAA FREng[5] (/əˈtiːə/; 22 April 1929 – 11 January 2019) was a British-Lebanese mathematician specialising in geometry.[6]
Atiyah grew up in Sudan and Egypt but spent most of his academic life in the United Kingdom at University of Oxford and University of Cambridge, and in the United States at the Institute for Advanced Study.[7] He was the President of the Royal Society (1990–1995), founding director of the Isaac Newton Institute (1990–1996), master of Trinity College, Cambridge (1990–1997), chancellor of the University of Leicester (1995–2005), and the President of the Royal Society of Edinburgh (2005–2008). From 1997 until his death, he was an honorary professor at the University of Edinburgh.[8]
Atiyah's mathematical collaborators included Raoul Bott, Friedrich Hirzebruch[9] and Isadore Singer, and his students included 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 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 and the Abel Prize in 2004.

Contents
1 Education and early life
2 Career and research
2.1 Collaborations
2.2 Algebraic geometry (1952–1958)
2.3 K theory (1959–1974)
2.4 Index theory (1963–1984)
2.5 Gauge theory (1977–1985)
2.6 Later work (1986–2019)
3 Bibliography
3.1 Books
3.2 Selected papers
3.3 Awards and honours
4 Personal life
5 References
5.1 Sources
6 External links
Education and early life
 

Great Court of Trinity College, Cambridge, where Atiyah was a student and later Master
Atiyah was born on 22 April 1929 in Hampstead, London, England, the son of Jean (née Levens) and Edward Atiyah.[10] His mother was Scottish and his father was a Lebanese Orthodox Christian. He had two brothers, Patrick (deceased) and Joe, and a sister, Selma (deceased).[11] Atiyah 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.[12] 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).[13] He was a doctoral student of William V. D. Hodge[3] and was awarded a doctorate in 1955 for a thesis entitled Some Applications of Topological Methods in Algebraic Geometry.[2][3]
Career and research[edit]
 

The Institute for Advanced Study in Princeton, where Atiyah was professor from 1969 to 1972
Atiyah 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, Cambridge (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).[13] 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.[13]
I started out by changing local currency into foreign currency everywhere I travelled as a child and ended up making money. That's when my father realised that I would be a mathematician some day.
Michael Atiyah[14]
Atiyah was president of the Pugwash Conferences on Science and World Affairs from 1997 to 2002.[15] 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).[16]
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),[15] Chancellor of the University of Leicester (1995–2005),[15] and president of the Royal Society of Edinburgh (2005–2008).[17] From 1997 until his death in 2019 he was an honorary professor in the University of Edinburgh. He was a Trustee of the James Clerk Maxwell Foundation.[citation needed]
Collaborations[edit]
 

The old Mathematical Institute (now the Department of Statistics) in Oxford, where Atiyah supervised many of his students
Atiyah collaborated with many 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,[18] all of whom he met at the Institute for Advanced Study in Princeton in 1955.[19] His other collaborators included; 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[20] (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).[21]
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.[22]
If you attack a mathematical problem directly, very often you come to a dead end, nothing you do seems to work and you feel that if only you could peer round the corner there might be an easy solution. There is nothing like having somebody else beside you, because he can usually peer round the corner.
Michael Atiyah[23]
Atiyah's students included; 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.[3]
Other contemporary mathematicians who influenced Atiyah include Roger Penrose, Lars Hörmander, Alain Connes and Jean-Michel Bismut.[24] Atiyah said that the mathematician he most admired was Hermann Weyl,[25] and that his favourite mathematicians from before the 20th century were Bernhard Riemann and William Rowan Hamilton.[26]
The seven volumes of Atiyah's collected papers include most of his work, except for his commutative algebra textbook;[27] the first five volumes are divided thematically and the sixth and seventh arranged by date.
Algebraic geometry (1952–1958)[edit]
Main article: Algebraic geometry
 

A twisted cubic curve, the subject of Atiyah's first paper
Atiyah's early papers on algebraic geometry (and some general papers) are reprinted in the first volume of his collected works.[28]
As an undergraduate Atiyah was interested in classical projective geometry, and wrote his first paper: a short note on twisted cubics.[29] He started research under W. V. D. Hodge and won the Smith's prize for 1954 for a sheaf-theoretic approach to ruled surfaces,[30] which encouraged Atiyah to continue in mathematics, rather than switch to his other interests—architecture and archaeology.[31] 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.[32] While in Princeton he classified vector bundles on an elliptic curve (extending Alexander 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,[33] and then showing that the space of indecomposable vector bundles of given degree and positive dimension can be identified with the elliptic curve.[34] He also studied double points on surfaces,[35] giving the first example of a flop, a special birational transformation of 3-folds that was later heavily used in Shigefumi Mori's work on minimal models for 3-folds.[36] Atiyah's flop can also be used to show that the universal marked family of K3 surfaces is non-Hausdorff.[37]
K theory (1959–1974)[edit]
Main article: K-theory
 

A Möbius band is the simplest non-trivial example of a vector bundle.
Atiyah's works on K-theory, including his book on K-theory[38] are reprinted in volume 2 of his collected works.[39]
The simplest nontrivial 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.[40]
Topological K-theory was discovered by Atiyah and Friedrich Hirzebruch[41] 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,[42] 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[43] 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[44] 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[45] also proved analogues of the result at odd primes.
 

Michael Atiyah and Friedrich Hirzebruch (right), the creators of topological K-theory
The Atiyah–Hirzebruch spectral sequence relates the ordinary cohomology of a space to its generalized cohomology theory.[42] (Atiyah and Hirzebruch used the case of K-theory, but their method works for all cohomology theories).
Atiyah showed[46] that for a finite group G, the K-theory of its classifying space, BG, is isomorphic to the completion of its character ring:
K ( B G ) ≅ R ( G ) ∧ . {\displaystyle K(BG)\cong R(G)^{\wedge }.}

The same year[47] 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,[48] 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.[49] 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 {\displaystyle X_{G}}
, which fibred over BG with fibre X:
K G ( X ) ∧ ≅ K ( X G ) . {\displaystyle K_{G}(X)^{\wedge }\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 René Thom, C. T. C. Wall, and others could be naturally reinterpreted as statements about these cohomology theories.[50] Some of these cohomology theories, in particular complex cobordism, turned out to be some of the most powerful cohomology theories known.
Algebra is the offer made by the devil to the mathematician. The devil says: `I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvellous machine.'
Michael Atiyah[51]
He introduced[52] 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,[52] and in a related paper [53] 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.[54]
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,[55] and gave another version of it in his book.[56] With Bott and Shapiro he analysed the relation of Bott periodicity to the periodicity of Clifford algebras;[57] although this paper did not have a proof of the periodicity theorem, a proof along similar lines was shortly afterwards found by R. Wood. He found a proof of several generalizations using elliptic operators;[58] this new proof used an idea that he used to give a particularly short and easy proof of Bott's original periodicity theorem.[59]
Index theory (1963–1984)[edit]
 

Isadore Singer (in 1977), who worked with Atiyah on index theory
Main article: Atiyah–Singer index theorem
Atiyah's work on index theory is reprinted in volumes 3 and 4 of his collected works.[60][61]
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.[citation needed]  
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.[citation needed]
The most useful piece of advice I would give to a mathematics student is always to suspect an impressive sounding Theorem if it does not have a special case which is both simple and non-trivial.
Michael Atiyah[62]
The index problem for elliptic differential operators was posed in 1959 by Gel'fand.[63] 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.[64] 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.[65] Their first published proof [66] 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.[67] 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.[68]
 

Atiyah's former student Graeme Segal (in 1982), who worked with Atiyah on equivariant K-theory
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.[69] 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.[70] 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.[71]
Atiyah[72] 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.[73]
As an application of the equivariant index theorem, Atiyah and Hirzebruch showed that manifolds with effective circle actions have vanishing Â-genus.[74] (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.[75] 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.
 

Raoul Bott, who worked with Atiyah on fixed point formulas and several other topics
Atiyah, Bott and Vijay K. Patodi[76] 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, and also introduced the Atiyah–Patodi–Singer eta invariant. This resulted in a series of papers on spectral asymmetry,[77] which were later unexpectedly used in theoretical physics, in particular in Witten's work on anomalies.
 

The lacunas discussed by Petrovsky, Atiyah, Bott and Gårding are similar to the spaces between shockwaves of a supersonic object.
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.[78]
Atiyah[79] 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[80] 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.[81]
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.[82]
Gauge theory (1977–1985)[edit]
Main article: Gauge theory
 

On the left, two nearby monopoles of the same polarity repel each other, and on the right two nearby monopoles of opposite polarity form a dipole. These are abelian monopoles; the non-abelian ones studied by Atiyah are more complicated.
Many of his papers on gauge theory and related topics are reprinted in volume 5 of his collected works.[83] 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 classifying instantons on Euclidean space as this is conformally equivalent to a sphere and the equations for instantons are conformally invariant. With Hitchin and Singer[84] he calculated the dimension of the moduli space of irreducible self-dual connections (instantons) for any principal bundle over a compact 4-dimensional Riemannian manifold (the Atiyah–Hitchin–Singer theorem). For example, the dimension of the space of SU2 instantons of rank k>0 is 8k−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.[85] 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.[86] Atiyah reformulated this construction using quaternions and wrote up a leisurely account of this classification of instantons on Euclidean space as a book.[87]
The mathematical problems that have been solved or techniques that have arisen out of physics in the past have been the lifeblood of mathematics.
Michael Atiyah[88]
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.[89]
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.[90] 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,[91] 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.[92]
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.[93] 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.[94]
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,[95] and with Pressley gave a related generalization to infinite-dimensional loop groups.[96]
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 [97] 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[98] 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.[99]
With Hitchin he worked on magnetic monopoles, and studied their scattering using an idea of Nick Manton.[100] His book [101] 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 hyperkähler. 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.[102]
Atiyah showed[103] 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;[104] this idea later became widely used by physicists.
Later work (1986–2019)[edit]
 

Edward Witten, whose work on invariants of manifolds and topological quantum field theories was influenced by Atiyah
Many of the papers in the 6th volume[105] of his collected works are surveys, obituaries, and general talks. Atiyah continued to publish subsequently, including several surveys, a popular book,[106] and another paper with Segal on twisted K-theory.
One paper[107] 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.[108] His book[109] 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[110] explains Witten's Lagrangian giving the Donaldson invariants.
He studied skyrmions with Nick Manton,[111] 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[112] were inspired by a question of Jonathan Robbins (called the Berry–Robbins problem), 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, and introduced the Atiyah conjecture on configurations.
But for most practical purposes, you just use the classical groups. The exceptional Lie groups are just there to show you that the theory is a bit bigger; it is pretty rare that they ever turn up.
Michael Atiyah[113]
With Juan Maldacena and Cumrun Vafa,[114] and E. Witten[115] 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[116] and G. Segal[117] he returned to his earlier interest of K-theory, describing some twisted forms of K-theory with applications in theoretical physics.
In October 2016, he claimed[118] a short proof of the non-existence of complex structures on the 6-sphere. His proof, like many predecessors, is considered flawed by the mathematical community, even after the proof was rewritten in a revised form.[119][120]
In September 2018, at the 2018 Heidelberg Laureate Forum, he claimed a simple proof of the Riemann hypothesis, one of the most important and challenging problems in mathematics. His claims have been rejected by the mathematical community.[121][122]
Bibliography[edit]
Books[edit]
This subsection lists all books written by Atiyah; it omits a few books that he edited.
Atiyah, Michael F.; Macdonald, Ian G. (1969), Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., MR 0242802. A classic textbook covering standard commutative algebra.
Atiyah, Michael F. (1970), Vector fields on manifolds, Arbeitsgemeinschaft für Forschung des Landes Nordrhein-Westfalen, Heft 200, Cologne: Westdeutscher Verlag, MR 0263102. Reprinted as (Atiyah 1988b, item 50).
Atiyah, Michael F. (1974), Elliptic operators and compact groups, Lecture Notes in Mathematics, Vol. 401, Berlin, New York: Springer-Verlag, MR 0482866. Reprinted as (Atiyah 1988c, item 78).
Atiyah, Michael F. (1979), Geometry of Yang–Mills fields, Scuola Normale Superiore Pisa, Pisa, MR 0554924. Reprinted as (Atiyah 1988e, item 99).
Atiyah, Michael F.; Hitchin, Nigel (1988), The geometry and dynamics of magnetic monopoles, M. B. Porter Lectures, Princeton University Press, doi:10.1515/9781400859306, ISBN 978-0-691-08480-0, MR 0934202. Reprinted as (Atiyah 2004, item 126).
Atiyah, Michael F. (1988a), Collected works. Vol. 1 Early papers: general papers, Oxford Science Publications, The Clarendon Press Oxford University Press, ISBN 978-0-19-853275-0, MR 0951892.
Atiyah, Michael F. (1988b), Collected works. Vol. 2 K-theory, Oxford Science Publications, The Clarendon Press Oxford University Press, ISBN 978-0-19-853276-7, MR 0951892.
Atiyah, Michael F. (1988c), Collected works. Vol. 3 Index theory: 1, Oxford Science Publications, The Clarendon Press Oxford University Press, ISBN 978-0-19-853277-4, MR 0951892.
Atiyah, Michael F. (1988d), Collected works. Vol. 4 Index theory:2, Oxford Science Publications, The Clarendon Press Oxford University Press, ISBN 978-0-19-853278-1, MR 0951892.
Atiyah, Michael F. (1988e), Collected works. Vol. 5 Gauge theories, Oxford Science Publications, The Clarendon Press Oxford University Press, ISBN 978-0-19-853279-8, MR 0951892.
Atiyah, Michael F. (1989), K-theory, Advanced Book Classics (2nd ed.), Addison-Wesley, ISBN 978-0-201-09394-0, MR 1043170. First edition (1967) reprinted as (Atiyah 1988b, item 45).
Atiyah, Michael F. (1990), The geometry and physics of knots, Lezioni Lincee. [Lincei Lectures], Cambridge University Press, doi:10.1017/CBO9780511623868, ISBN 978-0-521-39521-2, MR 1078014. Reprinted as (Atiyah 2004, item 136).
Atiyah, Michael F. (2004), Collected works. Vol. 6, Oxford Science Publications, The Clarendon Press Oxford University Press, ISBN 978-0-19-853099-2, MR 2160826.
Atiyah, Michael F. (2007), Siamo tutti matematici (Italian: We are all mathematicians), Roma: Di Renzo Editore, p. 96, ISBN 978-88-8323-157-5
Atiyah, Michael F.; Iagolnitzer, Daniel; Chong, Chitat (2015), Fields Medallists' Lectures (3rd Edition), World Scientific, doi:10.1142/9652, ISBN 978-981-4696-18-0.
Selected papers[edit]
Atiyah, Michael F. (1961), "Characters and cohomology of finite groups", Inst. Hautes Études Sci. Publ. Math., 9: 23–64, doi:10.1007/BF02698718. Reprinted in (Atiyah 1988b, paper 29).
Atiyah, Michael F.; Hirzebruch, Friedrich (1961), "Vector bundles and homogeneous spaces", Proc. Sympos. Pure Math. AMS, 3: 7–38. Reprinted in (Atiyah 1988b, paper 28).
Atiyah, Michael F.; Segal, Graeme B. (1969), "Equivariant K-Theory and Completion", Journal of Differential Geometry, 3: 1–18. Reprinted in (Atiyah 1988b, paper 49).
Atiyah, Michael F. (1976), "Elliptic operators, discrete groups and von Neumann algebras", Colloque "Analyse et Topologie" en l'Honneur de Henri Cartan (Orsay, 1974), Asterisque, 32–33, Soc. Math. France, Paris, pp. 43–72, MR 0420729. Reprinted in (Atiyah 1988d, paper 89). Formulation of the Atiyah "Conjecture" on the rationality of the L2-Betti numbers.
Atiyah, Michael F.; Singer, Isadore M. (1963), "The Index of Elliptic Operators on Compact Manifolds", Bull. Amer. Math. Soc., 69 (3): 322–433, doi:10.1090/S0002-9904-1963-10957-X. An announcement of the index theorem. Reprinted in (Atiyah 1988c, paper 56).
Atiyah, Michael F.; Singer, Isadore M. (1968a), "The Index of Elliptic Operators I", Annals of Mathematics, 87 (3): 484–530, doi:10.2307/1970715, JSTOR 1970715. This gives a proof using K theory instead of cohomology. Reprinted in (Atiyah 1988c, paper 64).
Atiyah, Michael F.; Segal, Graeme B. (1968), "The Index of Elliptic Operators: II", Annals of Mathematics, Second Series, 87 (3): 531–545, doi:10.2307/1970716, JSTOR 1970716. This reformulates the result as a sort of Lefschetz fixed point theorem, using equivariant K theory. Reprinted in (Atiyah 1988c, paper 65).
Atiyah, Michael F.; Singer, Isadore M. (1968b), "The Index of Elliptic Operators III", Annals of Mathematics, Second Series, 87 (3): 546–604, doi:10.2307/1970717, JSTOR 1970717. This paper shows how to convert from the K-theory version to a version using cohomology. Reprinted in (Atiyah 1988c, paper 66).
Atiyah, Michael F.; Singer, Isadore M. (1971), "The Index of Elliptic Operators IV", Annals of Mathematics, Second Series, 93 (1): 119–138, doi:10.2307/1970756, JSTOR 1970756 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 (Atiyah 1988c, paper 67).
Atiyah, Michael F.; Singer, Isadore M. (1971), "The Index of Elliptic Operators V", Annals of Mathematics, Second Series, 93 (1): 139–149, doi:10.2307/1970757, JSTOR 1970757. This studies families of real (rather than complex) elliptic operators, when one can sometimes squeeze out a little extra information. Reprinted in (Atiyah 1988c, paper 68).
Atiyah, Michael F.; Bott, Raoul (1966), "A Lefschetz Fixed Point Formula for Elliptic Differential Operators", Bull. Am. Math. Soc., 72 (2): 245–50, doi:10.1090/S0002-9904-1966-11483-0. This states a theorem calculating the Lefschetz number of an endomorphism of an elliptic complex. Reprinted in (Atiyah 1988c, paper 61).
Atiyah, Michael F.; Bott, Raoul (1967), "A Lefschetz Fixed Point Formula for Elliptic Complexes: I", Annals of Mathematics, Second Series, 86 (2): 374–407, doi:10.2307/1970694, JSTOR 1970694 (reprinted in (Atiyah 1988c, paper 61))and Atiyah, Michael F.; Bott, Raoul (1968), "A Lefschetz Fixed Point Formula for Elliptic Complexes: II. Applications", Annals of Mathematics, Second Series, 88 (3): 451–491, doi:10.2307/1970721, JSTOR 1970721. Reprinted in (Atiyah 1988c, paper 62). These give the proofs and some applications of the results announced in the previous paper.
Atiyah, Michael F.; Bott, Raoul; Patodi, Vijay K. (1973), "On the heat equation and the index theorem" (PDF), Invent. Math., 19 (4): 279–330, Bibcode:1973InMat..19..279A, doi:10.1007/BF01425417, MR 0650828; Atiyah, Michael F.; Bott, R.; Patodi, V. K. (1975), "Errata", Invent. Math., 28 (3): 277–280, Bibcode:1975InMat..28..277A, doi:10.1007/BF01425562, MR 0650829 Reprinted in (Atiyah 1988d, paper 79, 79a).
Atiyah, Michael F.; Schmid, Wilfried (1977), "A geometric construction of the discrete series for semisimple Lie groups", Invent. Math., 42: 1–62, Bibcode:1977InMat..42....1A, doi:10.1007/BF01389783, MR 0463358; Atiyah, Michael F.; Schmid, Wilfried (1979), "Erratum", Invent. Math., 54 (2): 189–192, Bibcode:1979InMat..54..189A, doi:10.1007/BF01408936, MR 0550183. Reprinted in (Atiyah 1988d, paper 90).
Atiyah, Michael (2010), Edinburgh Lectures on Geometry, Analysis and Physics, arXiv:1009.4827v1
Awards and honours[edit]
 

The premises of the Royal Society, where Atiyah was president from 1990 to 1995
In 1966, when he was thirty-seven years old, he was awarded the Fields Medal,[123] 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.[124] Among other prizes he has received are the Royal Medal of the Royal Society in 1968,[125] 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,[126] the Copley Medal of the Royal Society in 1988,[127] the Benjamin Franklin Medal for Distinguished Achievement in the Sciences of the American Philosophical Society in 1993,[128] the Jawaharlal Nehru Birth Centenary Medal of the Indian National Science Academy in 1993,[129] the President's Medal from the Institute of Physics in 2008,[130] the Grande Médaille of the French Academy of Sciences in 2010[131] and the Grand Officier of the French Légion d'honneur in 2011.[132]
So I don't think it makes much difference to mathematics to know that there are different kinds of simple groups or not. It is a nice intellectual endpoint, but I don't think it has any fundamental importance.
Michael Atiyah, commenting on the classification of finite simple groups[113]
He was elected a foreign member of the National Academy of Sciences, the American Academy of Arts and Sciences (1969),[133] the Académie 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.[13][15] In 2012, he became a fellow of the American Mathematical Society.[134] He was also appointed as a Honorary Fellow[5] of the Royal Academy of Engineering[5] in 1993.
Atiyah was awarded honorary degrees by the universities of Birmingham, Bonn, Chicago, Cambridge, Dublin, Durham, Edinburgh, Essex, Ghent, Helsinki, Lebanon, Leicester, London, Mexico, Montreal, Oxford, Reading, Salamanca, St. Andrews, Sussex, Wales, Warwick, the American University of Beirut, Brown University, Charles University in Prague, Harvard University, Heriot–Watt University, Hong Kong (Chinese University), Keele University, Queen's University (Canada), The Open University, University of Waterloo, Wilfrid Laurier University, Technical University of Catalonia, and UMIST.[13][15][135][136]
I had to wear a sort of bulletproof vest after that!
Michael Atiyah, commenting on the reaction to the previous quote[137]
Atiyah was made a Knight Bachelor in 1983[13] and made a member of the Order of Merit in 1992.[15]
The Michael Atiyah building[138] at the University of Leicester and the Michael Atiyah Chair in Mathematical Sciences[139] at the American University of Beirut were named after him.
Personal life[edit]
Atiyah married Lily Brown on 30 July 1955, with whom he had three sons, John, David and Robin. Atiyah's eldest son John died on 24 June 2002 while on a walking holiday in the Pyrenees with his wife Maj-Lis. Lily Atiyah died on 13 March 2018 at the age of 90.[6][11][13]
Sir Michael Atiyah died on 11 January 2019, aged 89.[140][141]
References[edit]
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^ Hitchin, Nigel J. (1972). Differentiable manifolds : the space of harmonic spinors. bodleian.ox.ac.uk (DPhil thesis). University of Oxford. OCLC 500473357. EThOS uk.bl.ethos.459281.
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^ Institute for Advanced Study: A Community of Scholars Archived 6 January 2013 at the Wayback Machine
^ "Atiyah's CV" (PDF).
^ Atiyah, Michael (2014). "Friedrich Ernst Peter Hirzebruch 17 October 1927 – 27 May 2012". Biographical Memoirs of Fellows of the Royal Society. 60: 229–247. doi:10.1098/rsbm.2014.0010.
^ ATIYAH, Sir Michael (Francis). ukwhoswho.com. Who's Who. 2014 (online edition via Oxford University Press ed.). A & C Black, an imprint of Bloomsbury Publishing plc.  (subscription required)
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^ Atiyah, Michael (2016). "The Non-Existent Complex 6-Sphere". arXiv:1610.09366 [math.DG].
^ What is the current understanding regarding complex structures on the 6-sphere? (MathOverflow), archived from the original on 24 September 2018, retrieved 24 September 2018
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^ "Skepticism surrounds renowned mathematician's attempted proof of 160-year-old hypothesis". Science | AAAS. 24 September 2018. Archived from the original on 26 September 2018. Retrieved 26 September 2018.
^ "Riemann hypothesis likely remains unsolved despite claimed proof". Archived from the original on 24 September 2018. Retrieved 24 September 2018.
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Sources[edit]
Boyer, Charles P.; Hurtubise, J. C.; Mann, B. M.; Milgram, R. J. (1993), "The topology of instanton moduli spaces. I. The Atiyah–Jones conjecture", Annals of Mathematics, Second Series, 137 (3): 561–609, doi:10.2307/2946532, ISSN 0003-486X, JSTOR 2946532, MR 1217348
Barth, Wolf P.; Hulek, Klaus; Peters, Chris A.M.; Van de Ven, Antonius (2004), Compact Complex Surfaces, Berlin: Springer, p. 334, ISBN 978-3-540-00832-3
Gel'fand, Israel M. (1960), "On elliptic equations", Russ. Math. Surv., 15 (3): 113–123, Bibcode:1960RuMaS..15..113G, doi:10.1070/rm1960v015n03ABEH004094. 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.
Harder, G.; Narasimhan, M. S. (1975), "On the cohomology groups of moduli spaces of vector bundles on curves", Mathematische Annalen, 212 (3): 215–248, doi:10.1007/BF01357141, ISSN 0025-5831, MR 0364254, archived from the original on 5 March 2016, retrieved 30 September 2013
Matsuki, Kenji (2002), Introduction to the Mori program, Universitext, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4757-5602-9, ISBN 978-0-387-98465-0, MR 1875410
Palais, Richard S. (1965), Seminar on the Atiyah–Singer Index Theorem, Annals of Mathematics Studies, 57, S.l.: Princeton Univ Press, ISBN 978-0-691-08031-4. This describes the original proof of the index theorem. (Atiyah and Singer never published their original proof themselves, but only improved versions of it.)
Segal, Graeme B. (1968), "The representation ring of a compact Lie group", Inst. Hautes Études Sci. Publ. Math., 34: 113–128, doi:10.1007/BF02684592.
Yau, Shing-Tung; Chan, Raymond H., eds. (1999), "Sir Michael Atiyah: a great mathematician of the twentieth century", Asian J. Math., International Press, 3 (1): 1–332, ISBN 978-1-57146-080-6, MR 1701915, archived from the original on 8 August 2008.
Yau, Shing-Tung, ed. (2005), The Founders of Index Theory: Reminiscences of Atiyah, Bott, Hirzebruch, and Singer, International Press, p. 358, ISBN 978-1-57146-120-9, archived from the original on 7 February 2006.
External links[edit]
Michael Atiyah
at Wikipedia's sister projects
Quotations from Wikiquote
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
"Sir Michael Atiyah on math, physics and fun", superstringtheory.com, Official Superstring theory web site], retrieved 14 August 2008
Atiyah, Michael, Beauty in Mathematics (video, 3m14s), retrieved 14 August 2008
Atiyah, Michael, The nature of space (Online lecture), retrieved 14 August 2008
Batra, Amba (8 November 2003), Maths guru with Einstein's dream prefers chalk to mouse. (Interview with Atiyah.), Delhi newsline, archived from the original on 8 February 2009, retrieved 14 August 2008
Michael Atiyah at the Mathematics Genealogy Project
Halim, Hala (1998), "Michael Atiyah:Euclid and Victoria", Al-Ahram Weekly On-line (391), archived from the original on 16 August 2004, retrieved 26 August 2008
Meek, James (21 April 2004), "Interview with Michael Atiyah", The Guardian, London, retrieved 14 August 2008
Sir Michael Atiyah FRS, Isaac Newton Institute, retrieved 14 August 2008
"Atiyah and Singer receive 2004 Abel prize" (PDF), Notices of the American Mathematical Society, 51 (6): 650–651, 2006, retrieved 14 August 2008
Raussen, Martin; Skau, Christian (24 May 2004), Interview with Michael Atiyah and Isadore Singer, retrieved 14 August 2008
Photos of Michael Francis Atiyah, Oberwolfach photo collection, retrieved 14 August 2008
Wade, Mike (21 April 2009), Maths and the bomb: Sir Michael Atiyah at 80, London: Timesonline, retrieved 12 May 2010
List of works of Michael Atiyah from Celebratio Mathematica
Academic offices
Preceded by
George Porter
President of the Royal Society
1990–1995
Succeeded by
Sir Aaron Klug
Preceded by
Sir Andrew Huxley
Master of Trinity College, Cambridge
1990–1997
Succeeded by
Amartya Sen
Preceded by
The Lord Porter of Luddenham
Chancellor of the University of Leicester
1995–2005
Succeeded by
Sir Peter Williams
Preceded by
Lord Sutherland of Houndwood
President of the Royal Society of Edinburgh
2005–2008
Succeeded by
David Wilson, Baron Wilson of Tillyorn
Awards and achievements
Preceded by
Robin Hill
Copley Medal
1988
Succeeded by
César Milstein
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黎曼猜想的证明
    

Sir Michael Francis Atiyah
Sir Michael Francis Atiyah

Monday, September 24, 2018 –

Sir Michael Francis Atiyah

Sir Michael Francis Atiyah Heidelberg Laureate Forum
阿蒂亚

袁萌  陈启清   2月18日
附件2：
黎曼猜想未被完全证明！
英数学“教父”公开证明过程
黎曼猜想未被完全证明！英数学“教父”公开证明过程
来源：DeepTech深科技
著名数学家迈克尔•阿提亚（Sir Michael Francis Atiyah）爵士终于向全世界展示了他“证明黎曼猜想”的过程。
就在刚刚，著名数学家迈克尔•阿提亚（Sir Michael Francis Atiyah）爵士终于向全世界展示了他“证明黎曼猜想”的过程。
就在于海德堡举办的
论坛上。密探，阿蒂亚爵士作为第二位演讲嘉宾上场，成功让会议主办方的直播视屏流崩溃。阿蒂亚爵士开始演讲大概10分钟之后，主办方才在其官方推特(31.23, 0.27, 0.87%)账号上恢复直播。

在阿蒂亚 45 分钟的演讲过程中，前 20 多分钟，这位数学家主要分享了一些数学史的内容，并解释了黎曼假说的本质及其与素数的相关性。而其中一些展示也显示了阿蒂亚爵士对此次证明信心满满，在一张片子中，他写道：“如果你证明了质数定理， 你将不朽“。

图 | 海德堡论坛上，阿蒂亚前半部分对数学史进行了一些介绍
之后，他才提到了黎曼猜想的主要证明过程。
在海德堡论坛上，阿蒂亚爵士提出了对黎曼猜想证明方法的一个简单思路。其灵感来源于阿蒂亚爵士在 2018 年 ICM 上提出精细结构常数（Fine structure constant）的推演，这是一个物理学上长期存在的数学问题。这一推演过程结合了冯•诺依曼的算子理论及希策布鲁赫创立并证明的代数簇黎曼-罗赫定理，还应用了Todd函数参与计算，而这个函数将是证明黎曼猜想的核心。阿蒂亚爵士称，他关于精细结构常数的相关论文已投稿至 Royal Society。介于这篇文章目前还未经过同行审议，一些学者对他的推演过程存疑，同样，也有学者对此次黎曼猜想的证明过程质疑。当然，还一些学者认为，阿蒂亚爵士的思路或为后续黎曼猜想证明提供了一种新思路。

而在提到关于证明黎曼猜想的具体细节时，阿蒂亚爵士并未做出证明的全部工作，其思路基于一个物理上未被完全证明的常数，而更多是就自己未尽的事业向数学界的后辈们提出了四点建议。首先，要运用如今最强大的工具，这里的工具不仅限于数学工具，还包括超级计算机甚至是量子计算机，还有其他领域例如物理学界、逻辑学界等的工具；其次，借鉴其他著名猜想，不管是证明了的还是未证明的；其三，需要判断哪些难题是能够高效率完成计算的；最后，老爷子觉得RH已经是很难的一个猜想了，证明出其一部分就已经很了不起了，他希望数学界的后辈们能权衡一下，哪些黎曼猜想的方面是我们有时间完成的，做出决定就无悔地去做吧！

图 | 海德堡论坛上阿蒂亚爵士的一些证明过程细节
图 | 阿蒂亚爵士对未来数学家的一些建议
这一结果似乎并不出乎意料，其实早在一些论坛上，学者对此次黎曼猜想的证明就不看好。在演讲正式开始前，一份谷歌(1113.65, -8.02, -0.71%)云盘中的预印本文件已在网上疯传。预印本文章仅有5页纸，证明过程如他提交给海德堡论坛的摘要所说十分简单。一些网友表示，在去年的国际数学大会（ICM）上，阿蒂亚爵士就曾出现令人尴尬的场面。甚至有人表示了对阿蒂亚爵士处境的隐隐担忧。也有一些网友调侃，用 20 美元打赌此次证明是错的。
迈克尔•阿蒂亚爵士 1929 年出生于英国伦敦，今年已是 89 岁高龄。
这位英国数学“教父”拥有诸多头衔：双料得主、伦敦皇家学会前主席、爱丁堡皇家学会前主席、剑桥大学三一学院前院长，还受封为爵士。他的早期工作主要集中在代数几何领域。上世纪 60 年代他与艾沙道尔•辛格合作，创立阿蒂亚-辛格指标定理，还和德国拓扑学家弗里德里希•希策布鲁赫共同创立拓扑 K 理论，证明了自己的定理。该定理将分析学与拓扑学联系起来，在复数几何、泛函分析及理论物理中均有深远的应用，被公认为 20 世纪最重要的数学成果之一。因此他获得了 1966 年的菲尔兹奖和 2004 年的阿贝尔奖。
图 | 阿蒂亚 （左） 与希策布鲁赫 （右）
黎曼猜想由数学家波恩哈德•黎曼于 1859 年在一篇题为《论小于给定数值的素数个数》的论文提出关于黎曼函数零点分布的猜想。论文所研究的是经典的质数分布问题。
质数是除了 1 和自身以外不能被其他正整数整除的数，从某种意义上说，质数在数论中的地位类似于物理世界中构筑万物的原子。在实际运用中，质数可以在密码学、安全认证等领域发挥作用。如现通用的 RSA 加密算法，加密计算的第一步是产生两个大质数，对极大整数做因数分解的难度决定了加密的可靠性。故寻找大质数、探寻质数分布的规律，一直是数学界关注的问题。
黎曼发表的论文中，一个重大的成果是发现质数蕴藏在一个特殊函数之中，而该函数的零点对质数分布的细致规律有着决定性的影响。这一函数如今被称为黎曼ζ函数。
也就是说，黎曼发现，质数在自然数中的分布并不是毫无规律可循，而是其分布与黎曼ζ函数紧密相关。
德国数学家戴维•希尔伯特在第二届国际数学家大会上提出了 20 世纪数学家应当努力解决的 23 个数学问题，其中便包括黎曼猜想，它也是克雷数学研究所悬赏的世界七大数学难题之一。

许多数学家都为证明黎曼猜想做出贡献。1932 年，德国数学家西格尔从黎曼的手稿里找到了关键的证据，遗留下的公式重现天日，西格尔也获得了菲尔兹奖。人工智能之父图灵计算出了函数的 1104 个非平凡零点，开启了计算机辅助计算的接力赛。1975 年，美国麻省理工学院的莱文森引入了独特的方法，证明黎曼函数临界线上的零点占全部零点的比例达到了 34.74%。一年后中国数学家楼世拓、姚琦证明了比例达到 35%。很快，荷兰数学家利用计算机检验了 2 亿个黎曼函数的零点，证明猜想很可能是对的。80 年代初，苏联数学家宣布利用电脑成功检验了一个与黎曼猜想有关的数学问题。最新的成果是一法国团队将零点计算出了前 10 万亿个而没有发现反例。
似乎人们离证明成功越来越近了，但阿蒂亚爵士这一次宣布证明黎曼猜想还是受到了不少质疑，有人认为他上了年纪，已是“廉颇老矣”，有人认为他之前发表的论文已经显出颓势。
不过，老来依旧致力于解决大问题的他，在几年前为爱丁堡皇家学会主持“科学之美”的会议期间接受采访时就很“佛系“地表示，人们可能会评价他是“大数学家晚节不保”，甚至自己儿子都泼过冷水。但他已得到数学界最重要的奖项，自感没有什么可以再失去，尽情思考就行了。
目前，已有上千项基于黎曼猜想的推论，这其中已不乏一些十分惊艳的想法。如所有自然数的和，即 1 + 2 + 3 + 4 + …，通过黎曼 ζ 函数的解析延拓可产生看似荒谬结果 -1/12，这一结果在量子力学及弦论等领域中已有所应用。可以说，黎曼猜想的证明是无数数学家和物理学家都高度关注的议题。
但从这次会议来看，Atiyah 实际上并没有完全给出黎曼猜想的证明，他的工作似乎集中在试图推导出精细结构常数上，而证明黎曼猜想只是个意外的惊喜。
当然，肯定很多人对于此次仍缺乏大量细节的所谓“证明过程”感到不满意，但实际上，对于一部分吃瓜群众来说，我们其实感受到了这位已经89岁的数学家的探索精神和幽默感——“证明黎曼猜想，你会名声大噪。但如果你已经出名了， 你就会冒着沦落声名狼藉下场的危险。”
同时，此次海德堡论坛上对黎曼猜想的证明尝试，也掀起了社会各界的广泛关注。可以说，阿蒂亚爵士此次的演讲引发了一次空前的科普盛世。虽然对黎曼猜想的解释仍需进一步完善，但其一生中对数学做出的贡献，以及此次讲座中的一些思考，都是对未来科学家进一步探索未知的一种激励。
阿蒂亚爵士在 89 岁高龄仍然站在探索数学问题的第一线，其对数学的热爱也可见一斑。在进行海德堡论坛上的黎曼猜想证明前，他曾表示，“我已从事数学研究 70 年，已无法停下。”用毕生信念追求理想，并为达成理想锲而不舍，这种精神也值得我们学习。
无论如何，对黎曼猜想的证明涉及了关于代数几何、代数数论及代数拓扑等多学科的审视，即使此次证明失败，阿蒂亚爵士思考过程本身也在推动这些分支学科进行更深入的交叉。正如德国物理学家量子力学创始人普朗克曾说过，“科学是内在的统一体，它被分解为单独的部分，不是由于事物的本质，而是由于人类认识能力的局限性。”对黎曼猜想的新思考，正在促进科学家们打破学科间壁垒造成的认知局限。
-End-
就在刚刚，著名数学家迈克尔•阿提亚（Sir Michael Francis Atiyah）爵士终于向全世界展示了他“证明黎曼猜想”的过程。
就在于海德堡举办的 Heidelberg Laureate Forum 上，阿蒂亚爵士作为第二位演讲嘉宾上场，成功让会议主办方的直播视屏流崩溃。阿蒂亚爵士开始演讲大概10分钟之后，主办方才在其官方推特账号上恢复直播。

在阿蒂亚 45 分钟的演讲过程中，前 20 多分钟，这位数学家主要分享了一些数学史的内容，并解释了黎曼假说的本质及其与素数的相关性。而其中一些展示也显示了阿蒂亚爵士对此次证明信心满满，在一张片子中，他写道：“如果你证明了质数定理， 你将不朽“。

图 | 海德堡论坛上，阿蒂亚前半部分对数学史进行了一些介绍
之后，他才提到了黎曼猜想的主要证明过程。
在海德堡论坛上，阿蒂亚爵士提出了对黎曼猜想证明方法的一个简单思路。其灵感来源于阿蒂亚爵士在 2018 年 ICM 上提出精细结构常数（Fine structure constant）的推演，这是一个物理学上长期存在的数学问题。这一推演过程结合了冯•诺依曼的算子理论及希策布鲁赫创立并证明的代数簇黎曼-罗赫定理，还应用了Todd函数参与计算，而这个函数将是证明黎曼猜想的核心。阿蒂亚爵士称，他关于精细结构常数的相关论文已投稿至 Royal Society。介于这篇文章目前还未经过同行审议，一些学者对他的推演过程存疑，同样，也有学者对此次黎曼猜想的证明过程质疑。当然，还一些学者认为，阿蒂亚爵士的思路或为后续黎曼猜想证明提供了一种新思路。

而在提到关于证明黎曼猜想的具体细节时，阿蒂亚爵士并未做出证明的全部工作，其思路基于一个物理上未被完全证明的常数，而更多是就自己未尽的事业向数学界的后辈们提出了四点建议。首先，要运用如今最强大的工具，这里的工具不仅限于数学工具，还包括超级计算机甚至是量子计算机，还有其他领域例如物理学界、逻辑学界等的工具；其次，借鉴其他著名猜想，不管是证明了的还是未证明的；其三，需要判断哪些难题是能够高效率完成计算的；最后，老爷子觉得RH已经是很难的一个猜想了，证明出其一部分就已经很了不起了，他希望数学界的后辈们能权衡一下，哪些黎曼猜想的方面是我们有时间完成的，做出决定就无悔地去做吧！

图 | 海德堡论坛上阿蒂亚爵士的一些证明过程细节
图 | 阿蒂亚爵士对未来数学家的一些建议
这一结果似乎并不出乎意料，其实早在一些论坛上，学者对此次黎曼猜想的证明就不看好。在演讲正式开始前，一份谷歌云盘中的预印本文件已在网上疯传。预印本文章仅有5页纸，证明过程如他提交给海德堡论坛的摘要所说十分简单。一些网友表示，在去年的国际数学大会（ICM）上，阿蒂亚爵士就曾出现令人尴尬的场面。甚至有人表示了对阿蒂亚爵士处境的隐隐担忧。也有一些网友调侃，用 20 美元打赌此次证明是错的。
迈克尔•阿蒂亚爵士 1929 年出生于英国伦敦，今年已是 89 岁高龄。
这位英国数学“教父”拥有诸多头衔：双料得主、伦敦皇家学会前主席、爱丁堡皇家学会前主席、剑桥大学三一学院前院长，还受封为爵士。他的早期工作主要集中在代数几何领域。上世纪 60 年代他与艾沙道尔•辛格合作，创立阿蒂亚-辛格指标定理，还和德国拓扑学家弗里德里希•希策布鲁赫共同创立拓扑 K 理论，证明了自己的定理。该定理将分析学与拓扑学联系起来，在复数几何、泛函分析及理论物理中均有深远的应用，被公认为 20 世纪最重要的数学成果之一。因此他获得了 1966 年的菲尔兹奖和 2004 年的阿贝尔奖。
图 | 阿蒂亚 （左） 与希策布鲁赫 （右）
黎曼猜想由数学家波恩哈德•黎曼于 1859 年在一篇题为《论小于给定数值的素数个数》的论文提出关于黎曼函数零点分布的猜想。论文所研究的是经典的质数分布问题。
质数是除了 1 和自身以外不能被其他正整数整除的数，从某种意义上说，质数在数论中的地位类似于物理世界中构筑万物的原子。在实际运用中，质数可以在密码学、安全认证等领域发挥作用。如现通用的 RSA 加密算法，加密计算的第一步是产生两个大质数，对极大整数做因数分解的难度决定了加密的可靠性。故寻找大质数、探寻质数分布的规律，一直是数学界关注的问题。
黎曼发表的论文中，一个重大的成果是发现质数蕴藏在一个特殊函数之中，而该函数的零点对质数分布的细致规律有着决定性的影响。这一函数如今被称为黎曼ζ函数。
也就是说，黎曼发现，质数在自然数中的分布并不是毫无规律可循，而是其分布与黎曼ζ函数紧密相关。
德国数学家戴维•希尔伯特在第二届国际数学家大会上提出了 20 世纪数学家应当努力解决的 23 个数学问题，其中便包括黎曼猜想，它也是克雷数学研究所悬赏的世界七大数学难题之一。

许多数学家都为证明黎曼猜想做出贡献。1932 年，德国数学家西格尔从黎曼的手稿里找到了关键的证据，遗留下的公式重现天日，西格尔也获得了菲尔兹奖。人工智能之父图灵计算出了函数的 1104 个非平凡零点，开启了计算机辅助计算的接力赛。1975 年，美国麻省理工学院的莱文森引入了独特的方法，证明黎曼函数临界线上的零点占全部零点的比例达到了 34.74%。一年后中国数学家楼世拓、姚琦证明了比例达到 35%。很快，荷兰数学家利用计算机检验了 2 亿个黎曼函数的零点，证明猜想很可能是对的。80 年代初，苏联数学家宣布利用电脑成功检验了一个与黎曼猜想有关的数学问题。最新的成果是一法国团队将零点计算出了前 10 万亿个而没有发现反例。
似乎人们离证明成功越来越近了，但阿蒂亚爵士这一次宣布证明黎曼猜想还是受到了不少质疑，有人认为他上了年纪，已是“廉颇老矣”，有人认为他之前发表的论文已经显出颓势。
不过，老来依旧致力于解决大问题的他，在几年前为爱丁堡皇家学会主持“科学之美”的会议期间接受采访时就很“佛系“地表示，人们可能会评价他是“大数学家晚节不保”，甚至自己儿子都泼过冷水。但他已得到数学界最重要的奖项，自感没有什么可以再失去，尽情思考就行了。
目前，已有上千项基于黎曼猜想的推论，这其中已不乏一些十分惊艳的想法。如所有自然数的和，即 1 + 2 + 3 + 4 + …，通过黎曼 ζ 函数的解析延拓可产生看似荒谬结果 -1/12，这一结果在量子力学及弦论等领域中已有所应用。可以说，黎曼猜想的证明是无数数学家和物理学家都高度关注的议题。
但从这次会议来看，Atiyah 实际上并没有完全给出黎曼猜想的证明，他的工作似乎集中在试图推导出精细结构常数上，而证明黎曼猜想只是个意外的惊喜。
当然，肯定很多人对于此次仍缺乏大量细节的所谓“证明过程”感到不满意，但实际上，对于一部分吃瓜群众来说，我们其实感受到了这位已经89岁的数学家的探索精神和幽默感——“证明黎曼猜想，你会名声大噪。但如果你已经出名了， 你就会冒着沦落声名狼藉下场的危险。”
同时，此次海德堡论坛上对黎曼猜想的证明尝试，也掀起了社会各界的广泛关注。可以说，阿蒂亚爵士此次的演讲引发了一次空前的科普盛世。虽然对黎曼猜想的解释仍需进一步完善，但其一生中对数学做出的贡献，以及此次讲座中的一些思考，都是对未来科学家进一步探索未知的一种激励。
阿蒂亚爵士在 89 岁高龄仍然站在探索数学问题的第一线，其对数学的热爱也可见一斑。在进行海德堡论坛上的黎曼猜想证明前，他曾表示，“我已从事数学研究 70 年，已无法停下。”用毕生信念追求理想，并为达成理想锲而不舍，这种精神也值得我们学习。
无论如何，对黎曼猜想的证明涉及了关于代数几何、代数数论及代数拓扑等多学科的审视，即使此次证明失败，阿蒂亚爵士思考过程本身也在推动这些分支学科进行更深入的交叉。正如德国物理学家量子力学创始人普朗克曾说过，“科学是内在的统一体，它被分解为单独的部分，不是由于事物的本质，而是由于人类认识能力的局限性。”对黎曼猜想的新思考，正在促进科学家们打破学科间壁垒造成的认知局限。