Michael Atiyah - Knowledge

723: 936: 1779:, 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 10023: 2029: 599: 1327:(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. 3651: 7950: 1212: 8300: 7960: 1772:

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.

1301: 2299:. 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. 25: 2091: 1752: 1379: 1987: 1892: 1262:'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 2130:−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 2017:) 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 167: 2302:

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

2113:

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

3836:

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

1771:

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

2012:

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

2184:

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

1107:

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

2165:

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

2055:

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

1374:

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.

3837:

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.

2114:

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 principal bundle over a compact 4-dimensional Riemannian manifold (the

1675:"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." 2426:

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

2174:, 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. 1979:, 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. 1730:

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

1349:

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

1918:, 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 2106:, 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 2239:

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

1670:, 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. 2255:

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

2134:. 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 2188:

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

2075:

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.

2329:

Many of the papers in the 6th volume of his collected works are surveys, obituaries, and general talks. Atiyah continued to publish subsequently, including several surveys, a popular book, and another paper with

1196:

The seven volumes of Atiyah's collected papers include most of his work, except for his commutative algebra textbook; the first five volumes are divided thematically and the sixth and seventh arranged by date.

5717: 2440:

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.

799:

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.

1250:'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 1634: 1739:; 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 2315: 2244:

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.

3711: 2102:

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

1246:, 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 5743: 10127: 2138:

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

6160: 1475: 6244: 1394:

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).

6376: 2306:

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.

10197: 10177: 10137: 9802: 2303:

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.

2110:, 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. 3867:

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

3802: 10187: 10107: 2216:

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

2013:

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

10242: 10207: 10182: 6085: 1788:

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

10247: 8937: 1555: 4869: 1942:

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

6490:. This describes the original proof of the index theorem. (Atiyah and Singer never published their original proof themselves, but only improved versions of it.) 5709: 2072:. 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. 1362:

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

4447: 4357: 2193:

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

2492:

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.

10202: 10132: 10112: 1849:

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

4328: 1853:

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

3301:

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 (

9795: 6000: 156: 10172: 10142: 8334: 6879:

Connes, Alain; Kouneiher, Joseph (2019). "Sir Michael Atiyah, a Knight Mathematician : A tribute to Michael Atiyah, an inspiration and a friend".

3336:. This studies families of real (rather than complex) elliptic operators, when one can sometimes squeeze out a little extra information. Reprinted in ( 46: 39: 2415: 1342:. 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 10072: 4153: 4281: 1983:

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.

940: 825: 810: 2147:

The mathematical problems that have been solved or techniques that have arisen out of physics in the past have been the lifeblood of mathematics.

10122: 5739: 2261: 2364:, inspired by Witten's work and Segal's definition of a conformal field theory. His book “The Geometry and Physics of Knots” describes the new 10162: 9788: 9386: 7118: 2276:

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.

1351: 10152: 10057: 9811: 8930: 7996: 7014: 6240: 6166: 1968:

As an application of the equivariant index theorem, Atiyah and Hirzebruch showed that manifolds with effective circle actions have vanishing

2185:

twistor space into its square. This allowed him to find an explicit formula for the conformally invariant Green's function on a 4-manifold.

10237: 10067: 10062: 8584: 6798: 6422: 6034: 5951: 4242: 2094:

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

6647: 10097: 10092: 9837: 3703: 152: 7458: 6691: 4184: 3746: 2018: 814: 5973: 1961:'s resolution of singularities to answer this affirmatively. An ingenious and elementary solution was found at about the same time by 8576: 8362: 3874:

Lily Atiyah died on 13 March 2018 at the age of 90 while Sir Michael Atiyah died less than a year later on 11 January 2019, aged 89.

1854: 1716: 1335: 673: 148: 6571: 6115: 10227: 10212: 8923: 8608: 3806: 2103: 2037: 1000: 629: 144: 2064:

and was used by Atiyah and Schmid to give a geometric construction, using square integrable harmonic spinors, of Harish-Chandra's

10217: 10117: 9423: 8699: 8485: 8343: 6623: 4514: 4012: 3916: 3883: 2032:

The lacunas discussed by Petrovsky, Atiyah, Bott and Gårding are similar to the spaces between shockwaves of a supersonic object.

1850: 1823: 1776: 1391: 735: 275: 1223:

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

10192: 8370: 8327: 6991: 1743:; this new proof used an idea that he used to give a particularly short and easy proof of Bott's original periodicity theorem. 669: 10167: 8600: 6581: 6547: 6484: 6458: 6329: 4251: 3750: 2989: 2954: 2924: 2895: 2864: 2817: 2786: 2759: 2732: 2705: 2678: 2647: 2481:

and G. Segal he returned to his earlier interest of K-theory, describing some twisted forms of K-theory with applications in

2115: 283: 89: 6722: 6081: 8434: 8418: 833: 610: 61: 6371:. On page 120 Gel'fand suggests that the index of an elliptic operator should be expressible in terms of topological data. 3532:

Atiyah, Michael F.; Schmid, Wilfried (1977), "A geometric construction of the discrete series for semisimple Lie groups",

6765: 6669: 5902: 5846: 2432: 1723:

for integral cohomology is false. The Hodge conjecture for rational cohomology is, as of 2008, a major unsolved problem.

1652: 1275: 1159: 956: 291: 267: 255: 6061: 4440: 4364: 4090: 9379: 8592: 8493: 8208: 7111: 4878: 2322: 1571: 1226:

As an undergraduate Atiyah was interested in classical projective geometry, and wrote his first paper: a short note on

6531: 5873: 5819: 2376:

in terms of topological quantum field theories, and his paper with L. Jeffrey explains Witten's Lagrangian giving the

68: 8878: 8771: 8410: 8320: 7989: 6929: 6368: 6271: 3066: 1980: 771: 108: 3404:

Atiyah, Michael F.; Bott, Raoul (1968), "A Lefschetz Fixed Point Formula for Elliptic Complexes: II. Applications",

9872: 6218: 3810: 2361: 1868:

Instead of just one elliptic operator, one can consider a family of elliptic operators parameterized by some space

1865:, and they used this approach to give proofs of various generalizations in a sequence of papers from 1968 to 1971. 3363:. This states a theorem calculating the Lefschetz number of an endomorphism of an elliptic complex. Reprinted in ( 10102: 9942: 8870: 8426: 8043: 8021: 8013: 6274:; Mann, B. M.; Milgram, R. J. (1993), "The topology of instanton moduli spaces. I. The Atiyah–Jones conjecture", 3778: 3723: 3719: 2273: 1763: 1012: 964: 896: 575: 287: 6187: 1972:. (Lichnerowicz showed that if a manifold has a metric of positive scalar curvature then the Â-genus vanishes.) 1502:'s thesis, that extension was complicated. However a simpler and more general proof was produced by introducing 1423: 75: 10052: 9567: 9439: 9063: 8779: 8656: 8274: 8162: 7605: 7451: 7226: 5898: 3774: 3715: 3699: 3344:

Atiyah, Michael F.; Bott, Raoul (1966), "A Lefschetz Fixed Point Formula for Elliptic Differential Operators",

2170:

on 4-dimensional Euclidean space. Donaldson went on to use the other moduli spaces studied by Atiyah to define

2065: 1931: 1876:, rather than an integer. If the operators in the family are real, then the index lies in the real K theory of 1807: 1503: 853: 829: 775: 665: 6006: 1842:

of a spin manifold, and Atiyah suggested that this integrality could be explained if it were the index of the

10077: 9822: 9372: 8442: 8386: 8061: 7926: 7104: 6715: 4416: 3950: 3856: 3845: 3822: 3786: 3667: 2563:, Arbeitsgemeinschaft für Forschung des Landes Nordrhein-Westfalen, Heft 200, Cologne: Westdeutscher Verlag, 1907: 1343: 818: 636:(deceased) and Joe, and a sister, Selma (deceased). Atiyah went to primary school at the Diocesan school in 10222: 10082: 9922: 9857: 8226: 8178: 8172: 8031: 7982: 6973: 3830: 3794: 2914: 1827: 972: 727: 704: 641: 401: 136: 57: 9719: 8902: 8303: 4161: 3818: 3798: 3691: 3268:. This paper shows how to convert from the K-theory version to a version using cohomology. Reprinted in ( 2504: 2194: 2048:, who found topological conditions describing which regions were lacunas. In collaboration with Bott and 1930:, commuting with the elliptic operator, then one can replace ordinary K-theory in the index theorem with 1914:

in terms of a sum over the fixed points of the endomorphism. As special cases their formula included the

791: 755: 722: 279: 140: 4275: 3435:, paper 62). These give the proofs and some applications of the results announced in the previous paper. 895:, which, informally speaking, describes ways in which spaces can be twisted. His best known result, the 10087: 9519: 9223: 7595: 7056: 6981: 6853: 6746: 3790: 3782: 3770: 3735: 3136:

Atiyah, Michael F.; Singer, Isadore M. (1963), "The Index of Elliptic Operators on Compact Manifolds",

2848: 2631: 2265: 2085: 1954: 1919: 1371: 983:

problem), Jürgen Berndt (projective planes), Roger Bielawski (Berry–Robbins problem), Howard Donnelly (

845: 685: 606: 271: 225: 10157: 10147: 9952: 9882: 9591: 8747: 8731: 8256: 7953: 7444: 7238: 6925: 6819: 5924: 3695: 3371:

Atiyah, Michael F.; Bott, Raoul (1967), "A Lefschetz Fixed Point Formula for Elliptic Complexes: I",

2516: 2419: 1263: 661: 2142:

and wrote up a leisurely account of this classification of instantons on Euclidean space as a book.

9047: 8787: 8525: 8232: 7197: 4141: 4017: 2496: 2021:. This resulted in a series of papers on spectral asymmetry, which were later unexpectedly used in 1819: 1727: 1410: 1339: 1155: 3754: 2161:. Donaldson showed that the moduli space of (degree 1) instantons over a compact simply connected 782:, from 1963 to 1969. He took up a three-year professorship at the Institute for Advanced Study in 9877: 9551: 9303: 9247: 9039: 9015: 8109: 8103: 7071: 7046: 4042: 3963: 3852: 3440: 2512: 2225: 2028: 849: 841: 416: 411: 406: 263: 35: 6653: 6027: 4007: 899:, was proved with Singer in 1963 and is used in counting the number of independent solutions to 259: 9912: 9415: 8552: 8509: 8238: 7747: 7578: 7042: 6873: 6771: 2508: 2257: 1915: 1259: 1190: 935: 912: 837: 763: 5667:

What is the current understanding regarding complex structures on the 6-sphere? (MathOverflow)

4302: 1814:. He noticed the homotopy invariance of the index, and asked for a formula for it by means of 790:

Research Professor and professorial fellow of St Catherine's College. He was president of the

10012: 9897: 9655: 9575: 9175: 8127: 7840: 6381: 6276: 4031: 3762: 3406: 3373: 3311: 3243: 3204: 3165: 2342: 1815: 900: 783: 7380: 6697: 6241:"A tribute to former President of the Royal Society Sir Michael Atiyah OM FRS (1929 - 2019)" 4190: 3985: 2295:; this has a natural Riemannian metric, and a key point is that this metric is complete and 1522:. It was shown that under suitable conditions the completion of the equivariant K theory of 82: 10232: 10047: 10042: 9957: 9932: 9907: 9751: 9743: 9527: 9215: 8640: 8632: 8220: 8097: 7298: 6898: 6557: 6468: 6410: 6347: 6313: 6189:

American University of Beirut establishes the Michael Atiyah Chair in Mathematical Sciences

5979: 4323: 3814: 3727: 3633: 3606: 3584: 3561: 3541: 3520: 3498: 3475: 3455: 3120: 3111:

Atiyah, Michael F. (1976), "Elliptic operators, discrete groups and von Neumann algebras",

3046: 2964: 2905: 2874: 2827: 2796: 2769: 2742: 2715: 2688: 2657: 2614: 2593: 2568: 2551: 2349: 2209: 2098:. These are abelian monopoles; the non-abelian ones studied by Atiyah are more complicated. 2069: 1831: 1780: 1533: 1383: 1331: 1087: 968: 916: 888: 864: 759: 739: 579: 396: 9695: 5710:"Skepticism surrounds renowned mathematician's attempted proof of 160-year-old hypothesis" 8: 9937: 9832: 9127: 8755: 8648: 8624: 8121: 6587: 6107: 4003: 3766: 3758: 2482: 2377: 2241: 2177: 2171: 2131: 2057: 2022: 1414: 1239: 1216: 1099: 1008: 904: 779: 693: 602: 9780: 6902: 6351: 3637: 3588: 3545: 3502: 3459: 2532:

This subsection lists all books written by Atiyah; it omits a few books that he edited.

1278:

for 3-folds. Atiyah's flop can also be used to show that the universal marked family of

9343: 9095: 8894: 8811: 8763: 7963: 7916: 7835: 7796: 7779: 7742: 7363: 6914: 6888: 6518: 6498: 6414: 6301: 5645: 4690: 3623: 3565: 3479: 3423: 3390: 3328: 3293: 3260: 3241:

Atiyah, Michael F.; Singer, Isadore M. (1968b), "The Index of Elliptic Operators III",

3221: 3182: 3029: 2841: 2500: 2478: 2474:. These papers seem to be the first time that Atiyah worked on exceptional Lie groups. 2428: 2408: 2388: 2292: 2288: 2060:; this index is in general real rather than integer valued. This version is called the 2014: 1998: 1363: 1206: 1178: 1036: 892: 7262: 6631: 4508: 3906: 3731: 2515:. Again, the proof did not hold up and the hypothesis remains one of the six unsolved 2247:

Duistermaat and Heckman found a striking formula, saying that the push-forward of the

1768:

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

951:

Atiyah collaborated with many mathematicians. His three main collaborations were with

432: 10022: 9583: 9471: 9351: 9199: 9111: 9023: 8819: 8739: 8299: 8214: 8139: 8115: 8091: 7959: 7877: 7845: 7791: 7764: 7710: 7644: 7522: 7144: 6918: 6577: 6543: 6480: 6454: 6418: 6398: 6364: 6325: 6293: 4037: 3973: 3569: 3483: 3276:

Atiyah, Michael F.; Singer, Isadore M. (1971), "The Index of Elliptic Operators IV",

3163:

Atiyah, Michael F.; Singer, Isadore M. (1968a), "The Index of Elliptic Operators I",

3129: 3062: 3009: 2985: 2950: 2920: 2891: 2860: 2813: 2782: 2755: 2728: 2701: 2674: 2643: 2280: 2248: 2181: 2135: 2107: 1740: 1736: 1732: 1402: 1247: 1123: 988: 960: 653: 467: 295: 251: 9263: 7850: 7551: 7088: 6617: 6522: 6359: 3358: 3309:

Atiyah, Michael F.; Singer, Isadore M. (1971), "The Index of Elliptic Operators V",

3202:

Atiyah, Michael F.; Segal, Graeme B. (1968), "The Index of Elliptic Operators: II",

3150: 3033: 1950: 1170: 9735: 9679: 9535: 9447: 9167: 8999: 8803: 8707: 8378: 8133: 8049: 7931: 7904: 7813: 7774: 7698: 7671: 7649: 7610: 7600: 7517: 7500: 7421: 7368: 7339: 7209: 7185: 6906: 6790: 6573:

The Founders of Index Theory: Reminiscences of Atiyah, Bott, Hirzebruch, and Singer

6510: 6446: 6390: 6355: 6285: 4337: 4234: 3739: 3592: 3549: 3506: 3463: 3415: 3382: 3353: 3320: 3285: 3252: 3213: 3174: 3145: 3093: 3054: 3021: 2977: 2852: 2635: 2539: 2489: 2357: 2335: 2201: 2167: 2158: 2041: 1958: 1911: 1885: 1720: 1709: 1662:, and pointed out that many of the deep results on cobordism of manifolds found by 1355: 1320:, including his book on K-theory are reprinted in volume 2 of his collected works. 1235: 1186: 1135: 1068: 848:(2005–2008). From 1997 until his death in 2019 he was an honorary professor in the 711: 657: 543: 487: 444: 247: 229: 8312: 5895:"Benjamin Franklin Medal for Distinguished Achievement in the Sciences Recipients" 5894: 4049: 3058: 9967: 9947: 9927: 9917: 9631: 9543: 9455: 9395: 9335: 9159: 9031: 8983: 8715: 8691: 8196: 7894: 7867: 7823: 7808: 7688: 7622: 7409: 7397: 7351: 7303: 6730: 6693:

Maths guru with Einstein's dream prefers chalk to mouse. (Interview with Atiyah.)

6613:

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

6553: 6464: 6442: 6406: 6309: 6065: 5788: 4186:

Maths guru with Einstein's dream prefers chalk to mouse. (Interview with Atiyah.)

3841: 3687: 3602: 3557: 3516: 3471: 3229:. This reformulates the result as a sort of Lefschetz fixed point theorem, using 3116: 2960: 2901: 2870: 2823: 2792: 2765: 2738: 2711: 2684: 2653: 2610: 2589: 2585: 2564: 2547: 2229: 2213: 2045: 1488: 1283: 1151: 1119: 1076: 1055:), Alexander Shapiro (Clifford algebras), L. Smith (homotopy groups of spheres), 1044: 880: 502: 462: 350: 299: 8915: 2049: 996: 10007: 9759: 9607: 9511: 9279: 9239: 9207: 9143: 9079: 8975: 8568: 8286: 8085: 8055: 7889: 7769: 7720: 7676: 7654: 7639: 7274: 7161: 6607: 6320:

Barth, Wolf P.; Hulek, Klaus; Peters, Chris A.M.; Van de Ven, Antonius (2004),

3675: 2809: 2467: 2451: 2365: 2190: 1843: 1811: 1755: 1658:

He defined new generalized homology and cohomology theories called bordism and

1359: 1271: 1255: 1231: 1227: 1143: 1131: 1091: 1056: 1024: 980: 976: 868: 633: 497: 482: 449: 319: 6869: 6775: 6675: 6612: 6450: 6058: 5840: 2296: 1880:. This gives a little extra information, as the map from the real K theory of 1324: 1305: 598: 10036: 9987: 9962: 9867: 9852: 9842: 9727: 9687: 9647: 9559: 9407: 9319: 9287: 9151: 9135: 9103: 9007: 8991: 8835: 8827: 8795: 8723: 8672: 8616: 8477: 8347: 8280: 8244: 8190: 8151: 8079: 7899: 7872: 7752: 7737: 7725: 7715: 7693: 7627: 7583: 7488: 7385: 7034: 7006: 6953: 6945: 6402: 6297: 4096: 3977: 3683: 3655: 3098: 2856: 2373: 2369: 2318: 2002: 1910:

for elliptic operators, giving the Lefschetz number of an endomorphism of an

1701: 1667: 1485: 1370:

that Hirzebruch had noticed a few years before. The original solution of the

1309: 1251: 1243: 1166: 1139: 1127: 1080: 1060: 1028: 1004: 884: 876: 787: 677: 625: 567: 525: 512: 492: 472: 332: 303: 6711: 6603: 4412: 3946: 2949:, Oxford Science Publications, The Clarendon Press Oxford University Press, 2890:, Oxford Science Publications, The Clarendon Press Oxford University Press, 2781:, Oxford Science Publications, The Clarendon Press Oxford University Press, 2754:, Oxford Science Publications, The Clarendon Press Oxford University Press, 2727:, Oxford Science Publications, The Clarendon Press Oxford University Press, 2700:, Oxford Science Publications, The Clarendon Press Oxford University Press, 2673:, Oxford Science Publications, The Clarendon Press Oxford University Press, 1719:

to complex analytic embeddings, and in a related paper they showed that the

9982: 9972: 9902: 9887: 9862: 9827: 9767: 9711: 9703: 9599: 9503: 9311: 9295: 9191: 9119: 9087: 9071: 8959: 8946: 8560: 8533: 8517: 8501: 8461: 8402: 8202: 8067: 8037: 7666: 7505: 7483: 7467: 7315: 7214: 7078: 6835: 6561: 6494: 6479:, Annals of Mathematics Studies, vol. 57, S.l.: Princeton Univ Press, 5867: 5813: 5766: 4358:"Edward Witten – Adventures in physics and math (Kyoto Prize lecture 2014)" 4342: 3707: 3663: 3081: 3053:, Proceedings of Symposia in Pure Mathematics, vol. 3, pp. 7–38, 2396: 2331: 2314: 2221: 2205: 1962: 1896: 1835: 1499: 1182: 1174: 1048: 1020: 992: 920: 872: 681: 583: 507: 477: 356: 338: 7539: 3113:

Colloque "Analyse et Topologie" en l'Honneur de Henri Cartan (Orsay, 1974)

2639: 2348:

Several of his papers from around this time study the connections between

2180:

for linear partial differential equations can often be found by using the

1663: 1185:, and that his favourite mathematicians from before the 20th century were 10002: 9997: 9892: 9639: 9623: 9495: 9479: 9463: 9431: 9327: 9271: 9055: 8664: 8469: 8394: 8250: 8184: 8145: 7862: 7818: 7556: 7286: 7250: 7024: 3679: 3650: 3115:, Asterisque, vol. 32–33, Soc. Math. France, Paris, pp. 43–72, 2455: 2404: 2353: 2284: 2236: 2157:

Atiyah's work on instanton moduli spaces was used in Donaldson's work on

1993:, who worked with Atiyah on fixed point formulas and several other topics 1705: 1147: 1072: 1032: 383: 344: 166: 6214: 5771:

Proceedings of International Conference of Mathematicians (Moscow, 1966)

4067: 2546:, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 2052:, Atiyah wrote three papers updating and generalizing Petrovsky's work. 1953:

and Gel'fand, about whether complex powers of analytic functions define

9977: 9847: 9671: 9615: 9364: 9231: 9183: 8967: 8886: 8851: 8268: 8073: 7573: 7534: 7327: 7127: 7096: 6963: 6514: 6394: 6377:"On the cohomology groups of moduli spaces of vector bundles on curves" 6305: 3671: 3597: 3553: 3511: 3467: 3427: 3394: 3332: 3297: 3264: 3225: 3191: 3186: 3025: 2252: 2162: 2139: 2044:: regions where they vanish identically. These were studied in 1945 by 1990: 1976: 1751: 1700:, defined as the group of stable fiber homotopy equivalence classes of 1527: 1367: 1279: 1064: 1040: 984: 952: 924: 860: 649: 587: 362: 7974: 2287:. His book with Hitchin gives a detailed description of their work on 1969: 1839: 1211: 9487: 7173: 6910: 6193: 4695: 3712:

Benjamin Franklin Medal for Distinguished Achievement in the Sciences

2459: 2400: 2392: 2269: 1858: 1659: 1491: 1378: 908: 767: 617: 315: 311: 194: 6532:"Sir Michael Atiyah: a great mathematician of the twentieth century" 6289: 6137: 5687: 5665: 3628: 3419: 3386: 3324: 3289: 3256: 3217: 3178: 1300: 660:

and some future leaders of Arab nations. He returned to England and

24: 7921: 7436: 6893: 5650: 4303:"James Clerk Maxwell Foundation Annual Report and Summary Accounts" 3868: 3230: 2930: 2463: 2384: 2009: 1986: 1900: 1862: 1735:

he analysed the relation of Bott periodicity to the periodicity of

1317: 1295: 1095: 1052: 1016: 747: 743: 689: 637: 571: 307: 5740:"Riemann hypothesis likely remains unsolved despite claimed proof" 2488:

In October 2016, he claimed a short proof of the non-existence of

824:

Within the United Kingdom, he was involved in the creation of the

2981: 2232:

to reproduce and extending the results of Harder and Narasimhan.

2217: 1686: 1162:

1960, Graeme Segal 1967, David Tall 1966, and Graham White 1982.

1023:(topological Lagrangians), John D. S. Jones (Yang–Mills theory), 6217:. University of Oxford Mathematical Institute. 11 January 2019. 5930: 4324:"Friedrich Ernst Peter Hirzebruch 17 October 1927 – 27 May 2012" 3662:

In 1966, when he was thirty-seven years old, he was awarded the

2268:, and this idea was later developed much further by his student 1891: 1165:

Other contemporary mathematicians who influenced Atiyah include

813:

from 1997 to 2002. He also contributed to the foundation of the

6363:. Reprinted in volume 1 of his collected works, p. 65–75, 3826: 2972:

Atiyah, Michael F.; Iagolnitzer, Daniel; Chong, Chitat (2015),

2422:), who asked if there is a map from the configuration space of 2095: 2090: 1643:

to be a point: the left hand side reduced to the completion of

1267: 944: 751: 621: 426: 3908:

Some applications of topological methods in algebraic geometry

3489:

Atiyah, Michael F.; Bott, R.; Patodi, V. K. (1975), "Errata",

698:

Some Applications of Topological Methods in Algebraic Geometry

555: 433:

Some Applications of Topological Methods in Algebraic Geometry

5644:

Atiyah, Michael (2016). "The Non-Existent Complex 6-Sphere".

3981: 3965:

Differentiable manifolds : the space of harmonic spinors

3855:

and the Michael Atiyah Chair in Mathematical Sciences at the

2291:. The main theme of the book is a study of a moduli space of 645: 3488: 2584:, Lecture Notes in Mathematics, Vol. 401, Berlin, New York: 828:

in Cambridge and was its first director (1990–1996). He was

10128:

Fellows of the Academy of Medical Sciences (United Kingdom)

5887: 3718:

in 1993, the Jawaharlal Nehru Birth Centenary Medal of the

2916:

Siamo tutti matematici (Italian: We are all mathematicians)

2555:. A classic textbook covering standard commutative algebra. 2264:. Atiyah showed that the moment map was closely related to 1639:

The original result then followed as a corollary by taking

558: 546: 6319: 4652: 3439:

Atiyah, Michael F.; Bott, Raoul; Patodi, Vijay K. (1973),

2345:

from the point of view of topology and the index theorem.

1872:. In this case the index is an element of the K theory of 1323:

The simplest nontrivial example of a vector bundle is the

730:

in Princeton, where Atiyah was professor from 1969 to 1972

696:

and was awarded a doctorate in 1955 for a thesis entitled

566:; 22 April 1929 – 11 January 2019) was a British-Lebanese 9810: 6269: 5774: 5336: 3670:

and the Atiyah–Singer theorem, for which he also won the

1181:. Atiyah said that the mathematician he most admired was 817:, the Association of European Academies (ALLEA), and the 233: 127: 5789:"2004: Sir Michael Francis Atiyah and Isadore M. Singer" 4142:"Distinguished mathematician and supporter of Humanism." 3575:

Atiyah, Michael F.; Schmid, Wilfried (1979), "Erratum",

3574: 975:

in Princeton in 1955. His other collaborators included;

887:. Together with Hirzebruch, he laid the foundations for 628:. His mother was Scottish and his father was a Lebanese 10198:

Members of the Norwegian Academy of Science and Letters

6630:, Official Superstring theory web site], archived from 1846:(which was rediscovered by Atiyah and Singer in 1961). 10178:

Foreign associates of the National Academy of Sciences

10138:

Foreign fellows of the Indian National Science Academy

6108:"Heriot-Watt University Edinburgh: Honorary Graduates" 5689:

Atiyah's May 2018 paper on the 6-sphere (MathOverflow)

3155:. An announcement of the index theorem. Reprinted in ( 2971: 6338:

Gel'fand, Israel M. (1960), "On elliptic equations",

3666:, for his work in developing K-theory, a generalized 2166:

4-manifolds, such as the existence of non-equivalent

1938:

this gives the index theorem, and for a finite group

1574: 1536: 1426: 10188:

Honorary Fellows of the Royal Academy of Engineering

10108:

Fellows of the American Academy of Arts and Sciences

6233: 6059:

List of Fellows of the American Mathematical Society

4329:

Biographical Memoirs of Fellows of the Royal Society

3678:

in 2004. Among other prizes he has received are the

3620:

Edinburgh Lectures on Geometry, Analysis and Physics

2671:

Collected works. Vol. 1 Early papers: general papers

552: 8342: 6076: 6074: 5222: 2395:, and giving a conjecture for the structure of the 971:on topological K-theory, all of whom he met at the 549: 10243:Royal Electrical and Mechanical Engineers soldiers 10208:People associated with the University of Leicester 10183:Foreign members of the Russian Academy of Sciences 4030: 2840: 2197:, and was later proved by several mathematicians. 1628: 1549: 1469: 1330:Topological K-theory was discovered by Atiyah and 1094:theory, stimulated important interactions between 1039:(spectral asymmetry), A. N. Pressley (convexity), 710:During his time at Cambridge, he was president of 8945: 4001: 1758:(in 1977), who worked with Atiyah on index theory 1629:{\displaystyle K_{G}(X)^{\wedge }\cong K(X_{G}).} 1498:compact Lie groups by incorporating results from 1334:who were inspired by Grothendieck's proof of the 1200: 10248:Members of the Royal Swedish Academy of Sciences 10034: 6821:Interview with Michael Atiyah and Isadore Singer 6818:Raussen, Martin; Skau, Christian (24 May 2004), 6530:Yau, Shing-Tung; Chan, Raymond H., eds. (1999), 6499:"The representation ring of a compact Lie group" 6374: 6071: 5347: 5147: 4783: 4441:"Edward Witten – Adventures in physics and math" 4092:Victoria College: educating the elite, 1902−1956 3438: 3044: 2283:, and studied their scattering using an idea of 2118:). For example, the dimension of the space of SU 2001:gave a new proof of the index theorem using the 1494:. Although soon the result could be extended to 1150:1968, Michael Murray 1983, Peter Newstead 1966, 1083:(M-theory, topological quantum field theories). 991:(instantons), Johan L. Dupont (singularities of 915:, which are responsible for some corrections in 826:Isaac Newton Institute for Mathematical Sciences 811:Pugwash Conferences on Science and World Affairs 734:Atiyah spent the academic year 1955–1956 at the 640:, Sudan (1934–1941), and to secondary school at 6878: 4534: 3190:. This gives a proof using K-theory instead of 3084:(1969), "Equivariant K-Theory and Completion", 2628:The geometry and dynamics of magnetic monopoles 2537: 2025:, in particular in Witten's work on anomalies. 1949:Atiyah solved a problem asked independently by 1818:. Some of the motivating examples included the 903:. Some of his more recent work was inspired by 742:, where he was a research fellow and assistant 4158:Archimedeans: Previous Committees and Officers 3658:, where Atiyah was president from 1990 to 1995 2414:Several papers were inspired by a question of 1118:Atiyah's students included Peter Braam 1987, 947:, where Atiyah supervised many of his students 9796: 9380: 8931: 8328: 7990: 7452: 7112: 6624:"Sir Michael Atiyah on math, physics and fun" 6207: 5442: 5028: 3240: 3162: 2625: 1126:1967, Howard Fegan 1977, Eric Grunwald 1977, 1102:, most notably in the work of Edward Witten. 859:Atiyah's mathematical collaborators included 652:(1941–1945); the school was also attended by 10203:People educated at Manchester Grammar School 10133:Fellows of the Australian Academy of Science 10113:Fellows of the American Mathematical Society 6881:Notices of the American Mathematical Society 6855:Maths and the bomb: Sir Michael Atiyah at 80 6799:Notices of the American Mathematical Society 6337: 5193: 5006: 4995: 4243:Notices of the American Mathematical Society 3531: 3441:"On the heat equation and the index theorem" 3308: 3275: 3135: 3010:"Characters and cohomology of finite groups" 2321:, whose work on invariants of manifolds and 624:, England, the son of Jean (née Levens) and 6982:President of the Royal Society of Edinburgh 6817: 6791:"Atiyah and Singer receive 2004 Abel prize" 5577: 4805: 4235:"Atiyah and Singer receive 2004 Abel prize" 3942: 3940: 3938: 3936: 3934: 3704:King Faisal International Prize for Science 3201: 3079: 2626:Atiyah, Michael F.; Hitchin, Nigel (1988), 2079: 1906:With Bott, Atiyah found an analogue of the 1746: 593: 9803: 9789: 9387: 9373: 8938: 8924: 8335: 8321: 7997: 7983: 7459: 7445: 7119: 7105: 6618:Mathematical descendants of Michael Atiyah 6477:Seminar on the Atiyah–Singer Index Theorem 4839: 4837: 4720: 4718: 3997: 3995: 3403: 3370: 3343: 3132:on the rationality of the L-Betti numbers. 2387:with Nick Manton, finding a relation with 1470:{\displaystyle K(BG)\cong R(G)^{\wedge }.} 815:InterAcademy Panel on International Issues 165: 16:British-Lebanese mathematician (1929–2019) 10173:Members of the French Academy of Sciences 10143:Fellows of the Royal Society of Edinburgh 6892: 6130: 6033:. American Academy of Arts and Sciences. 5769:(1968), "L'oeuvre de Michael F. Atiyah", 5649: 4877:, The Clay Math Institute, archived from 4694: 4341: 3627: 3596: 3510: 3357: 3149: 3097: 2974:Fields Medallists' Lectures (3rd Edition) 2808:, Advanced Book Classics (2nd ed.), 2309: 2104:non-linear partial differential equations 2038:hyperbolic partial differential equations 1480:The same year they proved the result for 1308:is the simplest non-trivial example of a 1266:, a special birational transformation of 674:Royal Electrical and Mechanical Engineers 109:Learn how and when to remove this message 10073:Academics of the University of Edinburgh 9394: 7126: 3931: 3807:Norwegian Academy of Science and Letters 3649: 2313: 2260:, which was a consequence of well-known 2089: 2027: 1985: 1890: 1783:, which follows from the index theorem. 1750: 1377: 1299: 1210: 934: 721: 597: 8004: 6938:Professional and academic associations 6529: 6436: 6028:"Book of Members, 1780-2010: Chapter A" 5583: 5477: 5465: 5453: 5430: 5418: 5406: 5394: 5382: 5370: 5358: 5324: 5312: 5300: 5288: 5276: 5264: 5252: 5240: 5228: 5216: 5204: 5170: 5158: 5135: 5123: 5111: 5099: 5087: 5075: 5063: 5051: 5039: 4983: 4972: 4961: 4949: 4937: 4925: 4913: 4901: 4855: 4843: 4834: 4816: 4760: 4748: 4736: 4724: 4715: 4709: 4674: 4640: 4628: 4616: 4604: 4592: 4580: 4568: 4556: 4545: 4495: 4471: 4400: 4277:Royal Society of Edinburgh announcement 4129: 4013:MacTutor History of Mathematics Archive 3992: 3961: 3915:(PhD thesis). University of Cambridge. 3884:List of presidents of the Royal Society 3745:He was elected a foreign member of the 3611: 3525: 3432: 3399: 3364: 3337: 3302: 3269: 3234: 3195: 3156: 3125: 3104: 3073: 3038: 2832: 2776: 2752:Collected works. Vol. 4 Index theory: 2 2749: 2725:Collected works. Vol. 3 Index theory: 1 2722: 2695: 2668: 2619: 2609:, Scuola Normale Superiore Pisa, Pisa, 2598: 2573: 1354:, describing when a map from a complex 1289: 786:after which he returned to Oxford as a 736:Institute for Advanced Study, Princeton 609:, where Atiyah was a student and later 10123:Fellows of Pembroke College, Cambridge 10035: 6992:David Wilson, Baron Wilson of Tillyorn 6474: 6375:Harder, G.; Narasimhan, M. S. (1975), 6243:. The Royal Society. 11 January 2019. 5926:Jawaharlal Nehru Birth Centenary Medal 5905:from the original on 24 September 2012 5765: 5746:from the original on 24 September 2018 5720:from the original on 26 September 2018 5643: 5631: 5619: 5607: 5595: 5571: 5559: 5547: 5536: 5524: 5512: 5501: 5490: 5182: 5017: 4828: 4772: 4685: 4663: 4517:from the original on 14 September 2008 4483: 4427: 4388: 4318: 4257:from the original on 10 September 2008 4221: 4217: 4215: 4213: 4211: 4209: 4207: 4045: ed.). A & C Black. 4023: 3972:(DPhil thesis). University of Oxford. 3904: 3900: 3898: 3825:. He was also appointed as a Honorary 3738:in 2010 and the Grand Officier of the 3617: 3110: 3007: 2944: 2919:, Roma: Di Renzo Editore, p. 96, 2912: 2885: 2879: 2838: 2803: 2779:Collected works. Vol. 5 Gauge theories 2662: 2604: 2579: 2558: 1708:in a series of papers, leading to the 1704:; this was later studied in detail by 1397:Atiyah showed that for a finite group 943:(now the Department of Statistics) in 758:(1958–1961). In 1961, he moved to the 717: 45:Please improve this article by adding 10163:Masters of Trinity College, Cambridge 9812:Masters of Trinity College, Cambridge 9784: 9368: 8919: 8316: 7978: 7440: 7100: 6720: 6689: 6493: 6020: 5574:, papers 163, 164, 165, 166, 167, 168 4794: 4689:(2000). "K-Theory Past and Present". 4284:from the original on 20 November 2008 4182: 4125: 4123: 4121: 4119: 4117: 4115: 4113: 4041:. Vol. 2014 (online edition via 3919:from the original on 18 November 2017 3821:. In 2012, he became a fellow of the 3751:American Academy of Arts and Sciences 3645: 3051:Vector bundles and homogeneous spaces 2831:. First edition (1967) reprinted as ( 2582:Elliptic operators and compact groups 2341:One paper is a detailed study of the 1899:(in 1982), who worked with Atiyah on 1530:to the ordinary K-theory of a space, 1219:, the subject of Atiyah's first paper 10153:Institute for Advanced Study faculty 10058:Alumni of Trinity College, Cambridge 7466: 7015:Master of Trinity College, Cambridge 6851: 6744: 6723:"Michael Atiyah:Euclid and Victoria" 6671:The nature of space (Online lecture) 6649:Beauty in Mathematics (video, 3m14s) 6576:, International Press, p. 358, 6503:Inst. Hautes Études Sci. Publ. Math. 6247:from the original on 11 January 2019 6221:from the original on 11 January 2019 4061: 4059: 3014:Inst. Hautes Études Sci. Publ. Math. 2036:The fundamental solutions of linear 834:Master of Trinity College, Cambridge 616:Atiyah was born on 22 April 1929 in 18: 10238:Victoria College, Alexandria alumni 10068:21st-century British mathematicians 10063:20th-century British mathematicians 6604:Michael Atiyah tells his life story 6569: 4867: 4453:from the original on 23 August 2016 4350: 4204: 4154:"[Presidents Archimedeans]" 3895: 3851:The Michael Atiyah building at the 2544:Introduction to commutative algebra 2433:Atiyah conjecture on configurations 2272:. Witten shortly after applied the 1775:Several deep theorems, such as the 1392:Atiyah–Hirzebruch spectral sequence 1047:(discrete series representations), 276:Atiyah–Hirzebruch spectral sequence 256:Atiyah conjecture on configurations 13: 10098:English people of Scottish descent 10093:British people of Lebanese descent 6667: 6645: 6441:, Universitext, Berlin, New York: 6118:from the original on 18 April 2016 4110: 4088: 3710:of the Royal Society in 1988, the 3000: 2947:Collected works. Vol. 7. 2002-2013 2323:topological quantum field theories 2019:Atiyah–Patodi–Singer eta invariant 1838:had proved the integrality of the 1358:to a sphere has a cross section. ( 844:(1995–2005), and president of the 14: 10259: 6930:National Portrait Gallery, London 6597: 6542:(1), International Press: 1–332, 6324:, Berlin: Springer, p. 334, 4056: 3844:in 1983 and made a member of the 2843:The geometry and physics of knots 2360:. He introduced the concept of a 1855:Grothendieck–Riemann–Roch theorem 1717:Grothendieck–Riemann–Roch theorem 1386:(right), the creators of K-theory 1336:Grothendieck–Riemann–Roch theorem 1001:hyperbolic differential equations 930: 10021: 8298: 7958: 7949: 7948: 6837:Photos of Michael Francis Atiyah 6696:, Delhi newsline, archived from 6439:Introduction to the Mori program 6180: 6153: 6100: 6088:from the original on 8 June 2016 6052: 6040:from the original on 10 May 2011 5993: 5966: 5944: 5917: 5876:from the original on 9 June 2008 5869:Copley archive winners 1989–1900 5860: 5849:from the original on 31 May 2008 5833: 5822:from the original on 9 June 2008 5806: 5781: 5758: 5732: 5702: 5680: 5658: 5637: 5625: 5613: 5601: 5589: 5565: 5553: 5541: 5530: 5518: 5506: 5495: 5483: 5471: 5459: 5447: 5436: 5424: 4189:, Delhi newsline, archived from 3905:Atiyah, Michael Francis (1955). 3862: 3811:Royal Spanish Academy of Science 3128:, paper 89). Formulation of the 3086:Journal of Differential Geometry 2698:Collected works. Vol. 2 K-theory 2499:, he claimed to have solved the 2362:topological quantum field theory 2204:described the cohomology of the 1715:With Hirzebruch he extended the 1011:(Integrals of the second kind), 668:studies (1945–1947) and did his 574:. His contributions include the 542: 23: 10228:Savilian Professors of Geometry 10213:Presidents of the Royal Society 6870:List of works of Michael Atiyah 6747:"Interview with Michael Atiyah" 6690:Batra, Amba (8 November 2003), 6360:10.1070/rm1960v015n03ABEH004094 5815:Royal archive winners 1989–1950 5412: 5400: 5388: 5376: 5364: 5352: 5341: 5330: 5318: 5306: 5294: 5282: 5270: 5258: 5246: 5234: 5210: 5198: 5187: 5176: 5164: 5152: 5141: 5129: 5117: 5105: 5093: 5081: 5069: 5057: 5045: 5033: 5022: 5011: 5000: 4989: 4977: 4966: 4955: 4943: 4931: 4919: 4907: 4895: 4861: 4849: 4822: 4810: 4799: 4788: 4777: 4766: 4754: 4742: 4730: 4703: 4679: 4668: 4657: 4646: 4634: 4622: 4610: 4598: 4586: 4574: 4562: 4550: 4539: 4528: 4501: 4489: 4477: 4465: 4433: 4421: 4406: 4394: 4382: 4312: 4295: 4268: 4227: 4183:Batra, Amba (8 November 2003), 4176: 4146: 4032:"ATIYAH, Sir Michael (Francis)" 3779:Indian National Science Academy 3720:Indian National Science Academy 3359:10.1090/S0002-9904-1966-11483-0 3151:10.1090/S0002-9904-1963-10957-X 2522: 2066:discrete series representations 1861:theory of the first proof with 1851:Hirzebruch–Riemann–Roch theorem 1824:Hirzebruch–Riemann–Roch theorem 1808:elliptic differential operators 1777:Hirzebruch–Riemann–Roch theorem 1653:Atiyah–Segal completion theorem 1514:-vector bundles over a compact 1270:that was later heavily used in 957:Atiyah–Bott fixed-point theorem 746:(1957–1958), then a university 292:Atiyah–Segal completion theorem 268:Atiyah–Bott fixed-point theorem 10218:Recipients of the Copley Medal 10118:Fellows of New College, Oxford 6954:President of the Royal Society 6840:, Oberwolfach photo collection 6729:, no. 391, archived from 6142:, Charles University in Prague 5899:American Philosophical Society 5148:Atiyah, Bott & Patodi 1973 4135: 4082: 3955: 3775:American Philosophical Society 3716:American Philosophical Society 3700:Accademia Nazionale dei Lincei 1620: 1607: 1592: 1585: 1455: 1448: 1439: 1430: 1201:Algebraic geometry (1952–1958) 854:James Clerk Maxwell Foundation 830:President of the Royal Society 776:Savilian Professor of Geometry 1: 10193:Members of the Order of Merit 6745:Meek, James (21 April 2004), 6716:Mathematics Genealogy Project 6570:Yau, Shing-Tung, ed. (2005), 4510:Interview with Michael Atiyah 4417:Mathematics Genealogy Project 3951:Mathematics Genealogy Project 3889: 3857:American University of Beirut 3823:American Mathematical Society 3787:Australian Academy of Science 3668:Lefschetz fixed-point theorem 2607:Geometry of Yang–Mills fields 2462:he described the dynamics of 2116:Atiyah–Hitchin–Singer theorem 1908:Lefschetz fixed-point formula 1344:generalized cohomology theory 819:European Mathematical Society 778:and a professorial fellow of 284:Atiyah–Hitchin–Singer theorem 47:secondary or tertiary sources 10168:Fellows of the Royal Society 7035:The Lord Porter of Luddenham 6974:Lord Sutherland of Houndwood 6852:Wade, Mike (21 April 2009), 6068:, retrieved 3 November 2012. 5348:Harder & Narasimhan 1975 4784:Atiyah & Hirzebruch 1961 4050:UK public library membership 3831:Royal Academy of Engineering 3803:Venezuela Academy of Science 3795:Ukrainian Academy of Science 3747:National Academy of Sciences 2777:Atiyah, Michael F. (1988e), 2750:Atiyah, Michael F. (1988d), 2723:Atiyah, Michael F. (1988c), 2696:Atiyah, Michael F. (1988b), 2669:Atiyah, Michael F. (1988a), 2519:in mathematics, as of 2024. 1828:Hirzebruch signature theorem 1230:. He started research under 1158:1985, Brian Sanderson 1963, 973:Institute for Advanced Study 959:and many other topics, with 871:, and his students included 809:Atiyah was president of the 728:Institute for Advanced Study 705:British Humanist Association 402:Institute for Advanced Study 7: 8304:University of Oxford portal 6926:Portraits of Michael Atiyah 6475:Palais, Richard S. (1965), 6162:The Michael Atiyah building 4535:Atiyah & Macdonald 1969 3877: 3819:Moscow Mathematical Society 3799:Georgian Academy of Science 3692:London Mathematical Society 3008:Atiyah, Michael F. (1961), 2913:Atiyah, Michael F. (2007), 2886:Atiyah, Michael F. (2004), 2839:Atiyah, Michael F. (1990), 2804:Atiyah, Michael F. (1989), 2605:Atiyah, Michael F. (1979), 2580:Atiyah, Michael F. (1974), 2559:Atiyah, Michael F. (1970), 2274:Duistermaat–Heckman formula 1965:, and discussed by Atiyah. 1822:and its generalization the 1764:Atiyah–Singer index theorem 965:Atiyah–Singer index theorem 897:Atiyah–Singer index theorem 852:. He was a Trustee of the 792:London Mathematical Society 756:Pembroke College, Cambridge 703:Atiyah was a member of the 576:Atiyah–Singer index theorem 522:Other notable students 288:Atiyah–Singer index theorem 10: 10264: 9224:Subrahmanyan Chandrasekhar 6262: 6215:"Michael Atiyah 1929-2019" 3962:Hitchin, Nigel J. (1972). 3791:Russian Academy of Science 3783:Chinese Academy of Science 3771:Royal Society of Edinburgh 3736:French Academy of Sciences 2849:Cambridge University Press 2632:Princeton University Press 2561:Vector fields on manifolds 2279:With Hitchin he worked on 2266:geometric invariant theory 2086:Gauge theory (mathematics) 2083: 1761: 1372:Hopf invariant one problem 1293: 1204: 846:Royal Society of Edinburgh 686:Trinity College, Cambridge 607:Trinity College, Cambridge 538:Sir Michael Francis Atiyah 226:Trinity College, Cambridge 10019: 9953:William Hepworth Thompson 9818: 9402: 8954: 8862: 8748:Frederick Gowland Hopkins 8732:Charles Scott Sherrington 8683: 8544: 8453: 8354: 8296: 8257:Edward Charles Titchmarsh 8161: 8020: 8010: 7944: 7474: 7135: 7085: 7076: 7068: 7063: 7053: 7039: 7031: 7021: 7011: 7003: 6998: 6988: 6978: 6970: 6960: 6950: 6942: 6937: 6451:10.1007/978-1-4757-5602-9 5443:Atiyah & Hitchin 1988 5029:Atiyah & Singer 1968a 4513:, superstringtheory.com, 3696:Antonio Feltrinelli Prize 3059:10.1090/pspum/003/0139181 2630:, M. B. Porter Lectures, 2517:Millennium Prize Problems 2497:Heidelberg Laureate Forum 2132:invariants of 4-manifolds 1888:is not always injective. 1130:1972, Lisa Jeffrey 1991, 662:Manchester Grammar School 531: 521: 455: 443: 425: 389: 379: 372: 325: 243: 218: 202: 176: 164: 125: 9048:Cyril Norman Hinshelwood 8788:Cyril Norman Hinshelwood 8233:Henry John Stephen Smith 7198:S. R. Srinivasa Varadhan 7064:Awards and achievements 6322:Compact Complex Surfaces 5489:Michael Atiyah 5194:Atiyah & Schmid 1977 5007:Atiyah & Singer 1963 4018:University of St Andrews 3618:Atiyah, Michael (2010), 2945:Atiyah, Michael (2014), 2857:10.1017/CBO9780511623868 2527: 2505:Hilbert's eighth problem 2325:was influenced by Atiyah 2080:Gauge theory (1977–1985) 1926:on the compact manifold 1895:Atiyah's former student 1747:Index theory (1963–1984) 1728:Bott periodicity theorem 594:Early life and education 9304:Frederick Charles Frank 9016:John Edensor Littlewood 8609:Benjamin Collins Brodie 8585:Marquess of Northampton 7047:University of Leicester 6727:Al-Ahram Weekly On-line 6437:Matsuki, Kenji (2002), 5777:, Moscow, pp. 9–14 5764:Fields medal citation: 4806:Atiyah & Segal 1969 4043:Oxford University Press 3871:with his wife Maj-Lis. 3853:University of Leicester 3740:French Légion d'honneur 2888:Collected works. Vol. 6 2513:fine-structure constant 2195:Atiyah–Jones conjecture 1510:equivalence classes of 1409:, is isomorphic to the 1338:and Bott's work on the 891:, an important tool in 850:University of Edinburgh 842:University of Leicester 774:(1961–1963). He became 632:. He had two brothers, 417:University of Cambridge 412:University of Edinburgh 407:University of Leicester 280:Atiyah–Jones conjecture 272:Atiyah–Floer conjecture 10103:Differential geometers 9943:Christopher Wordsworth 9416:James Joseph Sylvester 8553:William Hyde Wollaston 8239:James Joseph Sylvester 8012:Chairs established by 6874:Celebratio Mathematica 6772:Isaac Newton Institute 6767:Sir Michael Atiyah FRS 5953:2008 President's medal 5842:Sir Michael Atiyah FRS 5480:, papers 119, 120, 121 4343:10.1098/rsbm.2014.0010 3859:were named after him. 3659: 3138:Bull. Amer. Math. Soc. 3099:10.4310/jdg/1214428815 2442: 2326: 2310:Later work (1986–2019) 2258:equivariant cohomology 2149: 2099: 2033: 1994: 1916:Weyl character formula 1903: 1816:topological invariants 1806:The index problem for 1798: 1759: 1696:) of a finite complex 1677: 1630: 1551: 1471: 1401:, the K theory of its 1387: 1313: 1260:Alexander Grothendieck 1220: 1191:William Rowan Hamilton 1110: 1086:His later research on 948: 941:Mathematical Institute 901:differential equations 801: 772:St Catherine's College 731: 688:(1949–1955). He was a 684:studies took place at 613: 181:Michael Francis Atiyah 171:Michael Atiyah in 2007 34:relies excessively on 10053:People from Hampstead 9576:George Neville Watson 9176:Robert Burns Woodward 8657:George Gabriel Stokes 6858:, London: Timesonline 6628:superstringtheory.com 6382:Mathematische Annalen 6277:Annals of Mathematics 6064:5 August 2013 at the 5716:. 24 September 2018. 3763:Royal Swedish Academy 3755:Académie des Sciences 3653: 3407:Annals of Mathematics 3374:Annals of Mathematics 3312:Annals of Mathematics 3278:Annals of Mathematics 3244:Annals of Mathematics 3205:Annals of Mathematics 3166:Annals of Mathematics 3047:Hirzebruch, Friedrich 2640:10.1515/9781400859306 2438: 2431:, and introduced the 2420:Berry–Robbins problem 2343:Dedekind eta function 2317: 2262:localization theorems 2235:An old result due to 2210:stable vector bundles 2145: 2093: 2070:semisimple Lie groups 2031: 1989: 1934:. For trivial groups 1894: 1810:was posed in 1959 by 1786: 1754: 1673: 1631: 1552: 1550:{\displaystyle X_{G}} 1472: 1381: 1303: 1214: 1160:Rolph Schwarzenberger 1105: 938: 919:. He was awarded the 797: 725: 601: 582:. He was awarded the 10078:Abel Prize laureates 9958:Henry Montagu Butler 9933:Thomas Postlethwaite 9752:John Griggs Thompson 9744:Keith William Morton 9528:Godfrey Harold Hardy 9396:De Morgan Medallists 9216:Rodney Robert Porter 8641:William Spottiswoode 8633:Joseph Dalton Hooker 8486:Earl of Macclesfield 8251:Godfrey Harold Hardy 7299:John Forbes Nash Jr. 6721:Halim, Hala (1998), 6634:on 14 September 2008 6009:on 24 September 2011 5845:, Newton institute, 4871:The Hodge conjecture 4250:(6): 650–651, 2006, 4066:Atiyah, Joe (2007), 4004:Robertson, Edmund F. 3913:repository.cam.ac.uk 3815:Accademia dei Lincei 3728:Institute of Physics 3654:The premises of the 3346:Bull. Am. Math. Soc. 3231:equivariant K-theory 3080:Atiyah, Michael F.; 3045:Atiyah, Michael F.; 2976:, World Scientific, 2847:, Lezioni Lincee. , 2538:Atiyah, Michael F.; 2378:Donaldson invariants 2350:quantum field theory 2242:symplectic manifolds 2226:Yang–Mills equations 2172:Donaldson invariants 1932:equivariant K-theory 1901:equivariant K-theory 1820:Riemann–Roch theorem 1572: 1557:, which fibred over 1534: 1504:equivariant K-theory 1424: 1384:Friedrich Hirzebruch 1332:Friedrich Hirzebruch 1290:K-theory (1959–1974) 1088:gauge field theories 1053:equivariant K-theory 989:Vladimir G. Drinfeld 969:Friedrich Hirzebruch 917:quantum field theory 889:topological K-theory 865:Friedrich Hirzebruch 760:University of Oxford 740:Cambridge University 580:topological K-theory 397:University of Oxford 10223:Royal Medal winners 10083:Algebraic geometers 9938:William Lort Mansel 9833:John Christopherson 9128:Nevill Francis Mott 8756:William Henry Bragg 8649:Thomas Henry Huxley 8625:George Biddell Airy 8163:Savilian Professors 8022:Savilian Professors 8005:Savilian Professors 6903:2019arXiv191007851C 6352:1960RuMaS..15..113G 6270:Boyer, Charles P.; 6139:Honorary Doctorates 5173:, papers 84, 85, 86 5066:, papers 61, 62, 63 4002:O'Connor, John J.; 3767:Royal Irish Academy 3759:Akademie Leopoldina 3638:2010arXiv1009.4827A 3589:1979InMat..54..189A 3546:1977InMat..42....1A 3503:1975InMat..28..277A 3460:1973InMat..19..279A 3130:Atiyah "Conjecture" 2483:theoretical physics 2477:In his papers with 2122:instantons of rank 2058:von Neumann algebra 2023:theoretical physics 1382:Michael Atiyah and 1364:Steenrod operations 1340:periodicity theorem 1217:twisted cubic curve 1009:William V. D. Hodge 905:theoretical physics 794:from 1974 to 1976. 780:New College, Oxford 738:, then returned to 718:Career and research 694:William V. D. Hodge 264:Atiyah–Bott formula 213:Edinburgh, Scotland 9344:John Maynard Smith 9096:Tadeusz Reichstein 8895:Venki Ramakrishnan 8844:Sir Michael Atiyah 8812:Alan Lloyd Hodgkin 8764:Henry Hallett Dale 8363:Viscount Brouncker 7964:Mathematics portal 7364:Hillel Furstenberg 7057:Sir Peter Williams 6806:(6): 650–651, 2006 6700:on 8 February 2009 6590:on 7 February 2006 6515:10.1007/BF02684592 6395:10.1007/BF01357141 6169:on 9 February 2009 5975:La Grande Medaille 4474:, paper 12, p. 233 4193:on 8 February 2009 3986:uk.bl.ethos.459281 3840:Atiyah was made a 3660: 3646:Awards and honours 3598:10.1007/BF01408936 3554:10.1007/BF01389783 3512:10.1007/BF01425562 3468:10.1007/BF01425417 3026:10.1007/BF02698718 2933:on 14 January 2019 2501:Riemann hypothesis 2490:complex structures 2409:projective 3-space 2389:magnetic monopoles 2327: 2293:magnetic monopoles 2289:magnetic monopoles 2281:magnetic monopoles 2100: 2034: 2015:signature operator 1995: 1904: 1760: 1741:elliptic operators 1685:He introduced the 1655:for more details. 1626: 1547: 1467: 1388: 1316:Atiyah's works on 1314: 1221: 1207:Algebraic geometry 1179:Jean-Michel Bismut 1043:(vector bundles), 949: 893:algebraic topology 732: 630:Orthodox Christian 614: 10088:British humanists 10030: 10029: 9778: 9777: 9696:Albrecht Fröhlich 9656:C. Ambrose Rogers 9584:A. S. Besicovitch 9472:J. W. L. Glaisher 9362: 9361: 9200:Peter D. Mitchell 9112:Alexander R. Todd 9024:Macfarlane Burnet 8947:Copley Medallists 8913: 8912: 8740:Ernest Rutherford 8371:Joseph Williamson 8310: 8309: 8215:Abraham Robertson 8140:George Efstathiou 8116:Charles Pritchard 8092:Abraham Robertson 7972: 7971: 7434: 7433: 7145:Jean-Pierre Serre 7095: 7094: 7086:Succeeded by 7054:Succeeded by 7022:Succeeded by 7007:Sir Andrew Huxley 6999:Academic offices 6989:Succeeded by 6961:Succeeded by 6887:(10): 1660–1685. 6733:on 16 August 2004 6668:Atiyah, Michael, 6646:Atiyah, Michael, 6583:978-1-57146-120-9 6549:978-1-57146-080-6 6486:978-0-691-08031-4 6460:978-0-387-98465-0 6340:Russ. Math. Surv. 6331:978-3-540-00832-3 6280:, Second Series, 6082:"List of Fellows" 5586:, paper 19, p. 19 5562:, papers 141, 142 5433:, papers 115, 116 5337:Boyer et al. 1993 5291:, paper 19, p. 13 4986:, paper 17, p. 76 4884:on 27 August 2008 4868:Deligne, Pierre, 4831:, paper 160, p. 7 4653:Barth et al. 2004 4430:, pp. xi–xxv 4413:Alexander Shapiro 4370:on 23 August 2016 4164:on 26 August 2019 4069:The Atiyah Family 4048:(Subscription or 3970:bodleian.ox.ac.uk 3724:President's Medal 3528:, paper 79, 79a). 3410:, Second Series, 3377:, Second Series, 3315:, Second Series, 3280:, Second Series, 3247:, Second Series, 3208:, Second Series, 2991:978-981-4696-18-0 2956:978-0-19-968926-2 2926:978-88-8323-157-5 2897:978-0-19-853099-2 2866:978-0-521-39521-2 2819:978-0-201-09394-0 2788:978-0-19-853279-8 2761:978-0-19-853278-1 2734:978-0-19-853277-4 2707:978-0-19-853276-7 2680:978-0-19-853275-0 2649:978-0-691-08480-0 2540:Macdonald, Ian G. 2249:Liouville measure 2182:Fourier transform 2178:Green's functions 2168:smooth structures 2136:ADHM construction 2108:Penrose transform 2042:Petrovsky lacunas 1997:Atiyah, Bott and 1781:Rochlin's theorem 1737:Clifford algebras 1647:and the right to 1403:classifying space 1248:Solomon Lefschetz 1124:K. David Elworthy 1079:(instantons) and 1069:Stiefel manifolds 961:Isadore M. Singer 762:, where he was a 676:(1947–1949). His 656:displaced by the 654:European nobility 535: 534: 468:K. David Elworthy 456:Doctoral students 374:Scientific career 296:ADHM construction 252:Atiyah conjecture 197:, London, England 119: 118: 111: 93: 10255: 10158:Knights Bachelor 10148:Fields Medalists 10025: 9805: 9798: 9791: 9782: 9781: 9771: 9763: 9755: 9747: 9739: 9736:Bryan John Birch 9731: 9723: 9715: 9707: 9699: 9691: 9683: 9680:J. W. S. Cassels 9675: 9667: 9659: 9651: 9643: 9635: 9627: 9619: 9611: 9603: 9595: 9592:E. C. Titchmarsh 9587: 9579: 9571: 9563: 9555: 9552:J. E. Littlewood 9547: 9539: 9536:Bertrand Russell 9531: 9523: 9515: 9507: 9499: 9491: 9483: 9475: 9467: 9459: 9451: 9448:William Burnside 9443: 9435: 9427: 9419: 9411: 9389: 9382: 9375: 9366: 9365: 9355: 9347: 9339: 9331: 9323: 9315: 9307: 9299: 9291: 9283: 9275: 9267: 9259: 9251: 9243: 9235: 9227: 9219: 9211: 9203: 9195: 9187: 9179: 9171: 9168:Frederick Sanger 9163: 9155: 9147: 9139: 9131: 9123: 9115: 9107: 9099: 9091: 9083: 9075: 9067: 9059: 9051: 9043: 9035: 9027: 9019: 9011: 9003: 9000:Patrick Blackett 8995: 8987: 8979: 8971: 8963: 8940: 8933: 8926: 8917: 8916: 8906: 8898: 8890: 8882: 8874: 8855: 8847: 8839: 8831: 8823: 8815: 8807: 8804:Patrick Blackett 8799: 8791: 8783: 8775: 8767: 8759: 8751: 8743: 8735: 8727: 8719: 8711: 8708:Archibald Geikie 8703: 8695: 8676: 8668: 8660: 8652: 8644: 8636: 8628: 8620: 8612: 8604: 8596: 8588: 8580: 8572: 8564: 8556: 8537: 8529: 8521: 8513: 8505: 8497: 8489: 8481: 8473: 8465: 8446: 8438: 8430: 8427:Robert Southwell 8422: 8419:Earl of Pembroke 8414: 8406: 8398: 8390: 8382: 8379:Christopher Wren 8374: 8366: 8337: 8330: 8323: 8314: 8313: 8302: 8134:Donald Blackwell 8050:Christopher Wren 8014:Sir Henry Savile 7999: 7992: 7985: 7976: 7975: 7962: 7952: 7951: 7468:Fields Medalists 7461: 7454: 7447: 7438: 7437: 7422:Michel Talagrand 7369:Grigory Margulis 7340:Robert Langlands 7210:John G. Thompson 7186:Lennart Carleson 7121: 7114: 7107: 7098: 7097: 7069:Preceded by 7032:Preceded by 7004:Preceded by 6971:Preceded by 6943:Preceded by 6935: 6934: 6922: 6911:10.1090/noti1981 6896: 6866: 6865: 6863: 6848: 6847: 6845: 6831: 6830: 6828: 6814: 6813: 6811: 6795: 6786: 6785: 6783: 6774:, archived from 6761: 6760: 6758: 6741: 6740: 6738: 6708: 6707: 6705: 6686: 6685: 6683: 6678:on 3 August 2020 6674:, archived from 6664: 6663: 6661: 6652:, archived from 6642: 6641: 6639: 6591: 6586:, archived from 6565: 6564:on 8 August 2008 6560:, archived from 6525: 6495:Segal, Graeme B. 6489: 6471: 6433: 6432: 6430: 6421:, archived from 6362: 6334: 6316: 6272:Hurtubise, J. C. 6257: 6256: 6254: 6252: 6237: 6231: 6230: 6228: 6226: 6211: 6205: 6204: 6203: 6201: 6192:, archived from 6184: 6178: 6177: 6176: 6174: 6165:, archived from 6157: 6151: 6150: 6149: 6147: 6134: 6128: 6127: 6125: 6123: 6104: 6098: 6097: 6095: 6093: 6078: 6069: 6056: 6050: 6049: 6047: 6045: 6039: 6032: 6024: 6018: 6017: 6016: 6014: 6005:, archived from 6002:Legion d'honneur 5997: 5991: 5990: 5989: 5987: 5982:on 1 August 2010 5978:, archived from 5970: 5964: 5963: 5962: 5960: 5948: 5942: 5941: 5940: 5938: 5929:, archived from 5921: 5915: 5914: 5912: 5910: 5891: 5885: 5884: 5883: 5881: 5864: 5858: 5857: 5856: 5854: 5837: 5831: 5830: 5829: 5827: 5810: 5804: 5803: 5801: 5799: 5793:www.abelprize.no 5785: 5779: 5778: 5773:, Izdatyel'stvo 5762: 5756: 5755: 5753: 5751: 5736: 5730: 5729: 5727: 5725: 5706: 5700: 5699: 5698: 5696: 5684: 5678: 5677: 5676: 5674: 5662: 5656: 5655: 5653: 5641: 5635: 5629: 5623: 5617: 5611: 5605: 5599: 5593: 5587: 5581: 5575: 5569: 5563: 5557: 5551: 5545: 5539: 5534: 5528: 5522: 5516: 5510: 5504: 5499: 5493: 5487: 5481: 5475: 5469: 5463: 5457: 5451: 5445: 5440: 5434: 5428: 5422: 5416: 5410: 5404: 5398: 5392: 5386: 5380: 5374: 5368: 5362: 5361:, papers 104–105 5356: 5350: 5345: 5339: 5334: 5328: 5322: 5316: 5310: 5304: 5298: 5292: 5286: 5280: 5274: 5268: 5262: 5256: 5250: 5244: 5238: 5232: 5226: 5220: 5214: 5208: 5202: 5196: 5191: 5185: 5180: 5174: 5168: 5162: 5156: 5150: 5145: 5139: 5133: 5127: 5121: 5115: 5109: 5103: 5097: 5091: 5085: 5079: 5073: 5067: 5061: 5055: 5049: 5043: 5037: 5031: 5026: 5020: 5015: 5009: 5004: 4998: 4993: 4987: 4981: 4975: 4970: 4964: 4959: 4953: 4947: 4941: 4935: 4929: 4923: 4917: 4911: 4905: 4899: 4893: 4892: 4891: 4889: 4883: 4876: 4865: 4859: 4853: 4847: 4841: 4832: 4826: 4820: 4814: 4808: 4803: 4797: 4792: 4786: 4781: 4775: 4770: 4764: 4758: 4752: 4746: 4740: 4734: 4728: 4722: 4713: 4707: 4701: 4700: 4698: 4683: 4677: 4672: 4666: 4661: 4655: 4650: 4644: 4638: 4632: 4626: 4620: 4614: 4608: 4602: 4596: 4590: 4584: 4578: 4572: 4566: 4560: 4554: 4548: 4543: 4537: 4532: 4526: 4525: 4524: 4522: 4505: 4499: 4493: 4487: 4481: 4475: 4469: 4463: 4462: 4460: 4458: 4452: 4445: 4437: 4431: 4425: 4419: 4410: 4404: 4398: 4392: 4386: 4380: 4379: 4377: 4375: 4369: 4363:. Archived from 4362: 4354: 4348: 4347: 4345: 4316: 4310: 4309: 4307: 4299: 4293: 4292: 4291: 4289: 4272: 4266: 4265: 4264: 4262: 4256: 4239: 4231: 4225: 4219: 4202: 4201: 4200: 4198: 4180: 4174: 4173: 4171: 4169: 4160:. Archived from 4150: 4144: 4139: 4133: 4127: 4108: 4107: 4106: 4104: 4099:on 16 April 2008 4095:, archived from 4086: 4080: 4079: 4078: 4076: 4063: 4054: 4053: 4046: 4034: 4027: 4021: 4020: 4008:"Michael Atiyah" 3999: 3990: 3989: 3959: 3953: 3944: 3929: 3928: 3926: 3924: 3902: 3640: 3631: 3610:. Reprinted in ( 3609: 3600: 3572: 3523: 3514: 3486: 3445: 3431:. Reprinted in ( 3430: 3402:, paper 61))and 3397: 3362: 3361: 3335: 3300: 3267: 3233:. Reprinted in ( 3228: 3194:. Reprinted in ( 3189: 3154: 3153: 3124:. Reprinted in ( 3123: 3103:. Reprinted in ( 3102: 3101: 3082:Segal, Graeme B. 3072:. Reprinted in ( 3071: 3037:. Reprinted in ( 3036: 2994: 2967: 2941: 2940: 2938: 2929:, archived from 2908: 2878:. Reprinted as ( 2877: 2846: 2830: 2799: 2772: 2745: 2718: 2691: 2661:. Reprinted as ( 2660: 2618:. Reprinted as ( 2617: 2597:. Reprinted as ( 2596: 2572:. Reprinted as ( 2571: 2554: 2509:by contradiction 2468:manifolds with G 2446: 2416:Jonathan Robbins 2358:Donaldson theory 2336:twisted K-theory 2214:Riemann surfaces 2202:M. S. Narasimhan 2159:Donaldson theory 2153: 2062:L index theorem, 1912:elliptic complex 1886:complex K-theory 1857:, replacing the 1802: 1721:Hodge conjecture 1710:Adams conjecture 1681: 1635: 1633: 1632: 1627: 1619: 1618: 1600: 1599: 1584: 1583: 1556: 1554: 1553: 1548: 1546: 1545: 1476: 1474: 1473: 1468: 1463: 1462: 1356:Stiefel manifold 1187:Bernhard Riemann 1136:Peter Kronheimer 1114: 1063:(lambda rings), 1005:Nigel J. Hitchin 923:in 1966 and the 907:, in particular 805: 712:The Archimedeans 670:national service 658:Second World War 642:Victoria College 586:in 1966 and the 578:and co-founding 570:specialising in 565: 564: 561: 560: 557: 554: 551: 548: 488:Peter Kronheimer 445:Doctoral advisor 439: 248:Atiyah algebroid 209: 190: 188: 169: 159: 123: 122: 114: 107: 103: 100: 94: 92: 58:"Michael Atiyah" 51: 27: 19: 10263: 10262: 10258: 10257: 10256: 10254: 10253: 10252: 10033: 10032: 10031: 10026: 10017: 9968:G. M. Trevelyan 9948:William Whewell 9928:John Hinchliffe 9918:Richard Bentley 9883:John Arrowsmith 9858:John Richardson 9838:Robert Beaumont 9814: 9809: 9779: 9774: 9766: 9758: 9750: 9742: 9734: 9726: 9718: 9710: 9702: 9694: 9686: 9678: 9670: 9662: 9654: 9646: 9638: 9632:Mary Cartwright 9630: 9622: 9614: 9606: 9598: 9590: 9582: 9574: 9566: 9558: 9550: 9544:E. T. Whittaker 9542: 9534: 9526: 9518: 9510: 9502: 9494: 9486: 9478: 9470: 9462: 9456:A. G. Greenhill 9454: 9446: 9438: 9430: 9422: 9414: 9406: 9398: 9393: 9363: 9358: 9350: 9342: 9336:James Lighthill 9334: 9326: 9318: 9310: 9302: 9296:James D. Watson 9294: 9286: 9278: 9270: 9262: 9254: 9246: 9238: 9230: 9222: 9214: 9206: 9198: 9190: 9182: 9174: 9166: 9160:Dorothy Hodgkin 9158: 9150: 9142: 9134: 9126: 9118: 9110: 9102: 9094: 9086: 9078: 9070: 9062: 9054: 9046: 9038: 9032:Harold Jeffreys 9030: 9022: 9014: 9006: 8998: 8990: 8984:E. T. Whittaker 8982: 8974: 8966: 8958: 8950: 8944: 8914: 8909: 8901: 8893: 8885: 8877: 8869: 8858: 8850: 8842: 8834: 8826: 8818: 8810: 8802: 8794: 8786: 8778: 8772:Robert Robinson 8770: 8762: 8754: 8746: 8738: 8730: 8722: 8716:William Crookes 8714: 8706: 8698: 8692:William Huggins 8690: 8679: 8671: 8663: 8655: 8647: 8639: 8631: 8623: 8615: 8607: 8601:Lord Wrottesley 8599: 8591: 8583: 8575: 8567: 8559: 8551: 8540: 8532: 8524: 8516: 8508: 8500: 8492: 8484: 8476: 8468: 8460: 8449: 8441: 8435:Charles Montagu 8433: 8425: 8417: 8411:Earl of Carbery 8409: 8401: 8393: 8385: 8377: 8369: 8361: 8350: 8341: 8311: 8306: 8292: 8197:Nathaniel Bliss 8164: 8157: 8032:John Bainbridge 8023: 8016: 8006: 8003: 7973: 7968: 7940: 7939: 7936: 7909: 7882: 7855: 7828: 7801: 7784: 7757: 7730: 7703: 7681: 7659: 7632: 7615: 7588: 7561: 7544: 7527: 7510: 7493: 7470: 7465: 7435: 7430: 7429: 7426: 7414: 7410:Luis Caffarelli 7402: 7398:Dennis Sullivan 7390: 7373: 7356: 7352:Karen Uhlenbeck 7344: 7332: 7320: 7308: 7304:Louis Nirenberg 7291: 7279: 7267: 7263:Endre Szemerédi 7255: 7243: 7231: 7219: 7202: 7190: 7178: 7166: 7149: 7131: 7125: 7091: 7082: 7074: 7059: 7050: 7037: 7027: 7018: 7009: 6994: 6985: 6976: 6966: 6957: 6948: 6861: 6859: 6843: 6841: 6834: 6826: 6824: 6809: 6807: 6793: 6789: 6781: 6779: 6764: 6756: 6754: 6736: 6734: 6703: 6701: 6681: 6679: 6659: 6657: 6637: 6635: 6622: 6600: 6595: 6584: 6550: 6487: 6461: 6443:Springer-Verlag 6428: 6426: 6425:on 5 March 2016 6332: 6290:10.2307/2946532 6265: 6260: 6250: 6248: 6239: 6238: 6234: 6224: 6222: 6213: 6212: 6208: 6199: 6197: 6196:on 3 April 2008 6186: 6185: 6181: 6172: 6170: 6159: 6158: 6154: 6145: 6143: 6136: 6135: 6131: 6121: 6119: 6106: 6105: 6101: 6091: 6089: 6080: 6079: 6072: 6066:Wayback Machine 6057: 6053: 6043: 6041: 6037: 6030: 6026: 6025: 6021: 6012: 6010: 5999: 5998: 5994: 5985: 5983: 5972: 5971: 5967: 5958: 5956: 5950: 5949: 5945: 5936: 5934: 5933:on 10 July 2012 5923: 5922: 5918: 5908: 5906: 5893: 5892: 5888: 5879: 5877: 5866: 5865: 5861: 5852: 5850: 5839: 5838: 5834: 5825: 5823: 5812: 5811: 5807: 5797: 5795: 5787: 5786: 5782: 5763: 5759: 5749: 5747: 5738: 5737: 5733: 5723: 5721: 5708: 5707: 5703: 5694: 5692: 5686: 5685: 5681: 5672: 5670: 5664: 5663: 5659: 5642: 5638: 5630: 5626: 5618: 5614: 5606: 5602: 5594: 5590: 5582: 5578: 5570: 5566: 5558: 5554: 5546: 5542: 5535: 5531: 5523: 5519: 5511: 5507: 5500: 5496: 5488: 5484: 5476: 5472: 5464: 5460: 5452: 5448: 5441: 5437: 5429: 5425: 5417: 5413: 5405: 5401: 5393: 5389: 5381: 5377: 5369: 5365: 5357: 5353: 5346: 5342: 5335: 5331: 5323: 5319: 5311: 5307: 5299: 5295: 5287: 5283: 5275: 5271: 5263: 5259: 5251: 5247: 5243:, papers 94, 97 5239: 5235: 5227: 5223: 5219:, papers 92, 93 5215: 5211: 5203: 5199: 5192: 5188: 5181: 5177: 5169: 5165: 5157: 5153: 5146: 5142: 5134: 5130: 5122: 5118: 5110: 5106: 5098: 5094: 5086: 5082: 5074: 5070: 5062: 5058: 5050: 5046: 5038: 5034: 5027: 5023: 5016: 5012: 5005: 5001: 4994: 4990: 4982: 4978: 4971: 4967: 4960: 4956: 4948: 4944: 4936: 4932: 4924: 4920: 4912: 4908: 4900: 4896: 4887: 4885: 4881: 4874: 4866: 4862: 4854: 4850: 4842: 4835: 4827: 4823: 4815: 4811: 4804: 4800: 4793: 4789: 4782: 4778: 4771: 4767: 4759: 4755: 4747: 4743: 4735: 4731: 4723: 4716: 4708: 4704: 4687:Atiyah, Michael 4684: 4680: 4673: 4669: 4662: 4658: 4651: 4647: 4639: 4635: 4627: 4623: 4615: 4611: 4603: 4599: 4591: 4587: 4579: 4575: 4567: 4563: 4555: 4551: 4544: 4540: 4533: 4529: 4520: 4518: 4507: 4506: 4502: 4494: 4490: 4482: 4478: 4470: 4466: 4456: 4454: 4450: 4443: 4439: 4438: 4434: 4426: 4422: 4411: 4407: 4399: 4395: 4387: 4383: 4373: 4371: 4367: 4360: 4356: 4355: 4351: 4320:Atiyah, Michael 4317: 4313: 4305: 4301: 4300: 4296: 4287: 4285: 4274: 4273: 4269: 4260: 4258: 4254: 4237: 4233: 4232: 4228: 4220: 4205: 4196: 4194: 4181: 4177: 4167: 4165: 4152: 4151: 4147: 4140: 4136: 4128: 4111: 4102: 4100: 4089:Raafat, Samir, 4087: 4083: 4074: 4072: 4065: 4064: 4057: 4047: 4029: 4028: 4024: 4000: 3993: 3960: 3956: 3945: 3932: 3922: 3920: 3903: 3896: 3892: 3880: 3865: 3842:Knight Bachelor 3732:Grande Médaille 3688:De Morgan Medal 3648: 3643: 3443: 3420:10.2307/1970721 3398:(reprinted in ( 3387:10.2307/1970694 3325:10.2307/1970757 3290:10.2307/1970756 3257:10.2307/1970717 3218:10.2307/1970716 3179:10.2307/1970715 3069: 3003: 3001:Selected papers 2998: 2992: 2957: 2936: 2934: 2927: 2898: 2867: 2820: 2789: 2762: 2735: 2708: 2681: 2650: 2586:Springer-Verlag 2530: 2525: 2471: 2448: 2444: 2429:Nahm's equation 2366:knot invariants 2312: 2230:Riemann surface 2191:homology groups 2155: 2151: 2121: 2088: 2082: 2046:I. G. Petrovsky 1999:Vijay K. Patodi 1804: 1800: 1766: 1749: 1683: 1679: 1614: 1610: 1595: 1591: 1579: 1575: 1573: 1570: 1569: 1541: 1537: 1535: 1532: 1531: 1458: 1454: 1425: 1422: 1421: 1298: 1292: 1240:sheaf-theoretic 1238:for 1954 for a 1209: 1203: 1152:Ian R. Porteous 1120:Simon Donaldson 1116: 1112: 1090:, particularly 1077:Richard S. Ward 1045:Wilfried Schmid 1037:Vijay K. Patodi 1013:Michael Hopkins 933: 881:Simon Donaldson 807: 803: 720: 596: 545: 541: 517: 503:Ian R. Porteous 463:Simon Donaldson 437: 421: 368: 351:De Morgan Medal 318: 314: 310: 306: 302: 300:Fredholm module 298: 294: 290: 286: 282: 278: 274: 270: 266: 262: 258: 254: 250: 239: 214: 211: 207: 206:11 January 2019 198: 192: 186: 184: 183: 182: 172: 160: 135: 133: 130: 115: 104: 98: 95: 52: 50: 44: 40:primary sources 28: 17: 12: 11: 5: 10261: 10251: 10250: 10245: 10240: 10235: 10230: 10225: 10220: 10215: 10210: 10205: 10200: 10195: 10190: 10185: 10180: 10175: 10170: 10165: 10160: 10155: 10150: 10145: 10140: 10135: 10130: 10125: 10120: 10115: 10110: 10105: 10100: 10095: 10090: 10085: 10080: 10075: 10070: 10065: 10060: 10055: 10050: 10045: 10028: 10027: 10020: 10018: 10016: 10015: 10010: 10008:Gregory Winter 10005: 10000: 9995: 9993:Michael Atiyah 9990: 9985: 9980: 9975: 9970: 9965: 9960: 9955: 9950: 9945: 9940: 9935: 9930: 9925: 9920: 9915: 9910: 9905: 9900: 9895: 9890: 9885: 9880: 9875: 9870: 9865: 9860: 9855: 9850: 9845: 9840: 9835: 9830: 9825: 9819: 9816: 9815: 9808: 9807: 9800: 9793: 9785: 9776: 9775: 9773: 9772: 9764: 9760:Timothy Gowers 9756: 9748: 9740: 9732: 9724: 9716: 9708: 9700: 9692: 9684: 9676: 9668: 9664:Michael Atiyah 9660: 9652: 9644: 9636: 9628: 9620: 9612: 9608:W. V. D. Hodge 9604: 9596: 9588: 9580: 9572: 9568:Sydney Chapman 9564: 9556: 9548: 9540: 9532: 9524: 9516: 9512:P. A. MacMahon 9508: 9500: 9492: 9484: 9476: 9468: 9460: 9452: 9444: 9436: 9428: 9420: 9412: 9403: 9400: 9399: 9392: 9391: 9384: 9377: 9369: 9360: 9359: 9357: 9356: 9352:Alan Battersby 9348: 9340: 9332: 9324: 9316: 9308: 9300: 9292: 9284: 9280:Sydney Brenner 9276: 9268: 9264:César Milstein 9260: 9256:Michael Atiyah 9252: 9244: 9240:Rudolf Peierls 9236: 9228: 9220: 9212: 9208:John Cornforth 9204: 9196: 9188: 9180: 9172: 9164: 9156: 9148: 9144:W. V. D. Hodge 9140: 9132: 9124: 9116: 9108: 9100: 9092: 9084: 9080:Lawrence Bragg 9076: 9068: 9064:Sydney Chapman 9060: 9052: 9044: 9036: 9028: 9020: 9012: 9004: 8996: 8988: 8980: 8976:Albert Kluyver 8972: 8964: 8955: 8952: 8951: 8943: 8942: 8935: 8928: 8920: 8911: 8910: 8908: 8907: 8899: 8891: 8887:Sir Paul Nurse 8883: 8875: 8866: 8864: 8860: 8859: 8857: 8856: 8852:Sir Aaron Klug 8848: 8840: 8832: 8824: 8816: 8808: 8800: 8792: 8784: 8776: 8768: 8760: 8752: 8744: 8736: 8728: 8720: 8712: 8704: 8696: 8687: 8685: 8681: 8680: 8678: 8677: 8669: 8661: 8653: 8645: 8637: 8629: 8621: 8613: 8605: 8597: 8589: 8581: 8577:Duke of Sussex 8573: 8569:Davies Gilbert 8565: 8557: 8548: 8546: 8542: 8541: 8539: 8538: 8530: 8522: 8514: 8506: 8498: 8494:Earl of Morton 8490: 8482: 8474: 8466: 8457: 8455: 8451: 8450: 8448: 8447: 8439: 8431: 8423: 8415: 8407: 8399: 8391: 8383: 8375: 8367: 8358: 8356: 8352: 8351: 8340: 8339: 8332: 8325: 8317: 8308: 8307: 8297: 8294: 8293: 8291: 8290: 8287:Frances Kirwan 8284: 8278: 8275:Richard Taylor 8272: 8266: 8263:Michael Atiyah 8260: 8254: 8248: 8242: 8236: 8230: 8224: 8221:Stephen Rigaud 8218: 8212: 8206: 8200: 8194: 8188: 8182: 8176: 8169: 8167: 8159: 8158: 8156: 8155: 8149: 8143: 8137: 8131: 8128:Harry Plaskett 8125: 8122:Herbert Turner 8119: 8113: 8110:William Donkin 8107: 8104:George Johnson 8101: 8098:Stephen Rigaud 8095: 8089: 8086:Thomas Hornsby 8083: 8077: 8071: 8065: 8059: 8056:Edward Bernard 8053: 8047: 8041: 8035: 8028: 8026: 8018: 8017: 8011: 8008: 8007: 8002: 8001: 7994: 7987: 7979: 7970: 7969: 7967: 7966: 7956: 7945: 7942: 7941: 7938: 7937: 7935: 7934: 7929: 7924: 7919: 7912: 7910: 7908: 7907: 7902: 7897: 7892: 7885: 7883: 7881: 7880: 7875: 7870: 7865: 7858: 7856: 7854: 7853: 7848: 7843: 7838: 7831: 7829: 7827: 7826: 7821: 7816: 7811: 7804: 7802: 7800: 7799: 7794: 7787: 7785: 7783: 7782: 7777: 7772: 7767: 7760: 7758: 7756: 7755: 7750: 7745: 7740: 7733: 7731: 7729: 7728: 7723: 7718: 7713: 7706: 7704: 7702: 7701: 7696: 7691: 7684: 7682: 7680: 7679: 7674: 7669: 7662: 7660: 7658: 7657: 7652: 7647: 7642: 7635: 7633: 7631: 7630: 7625: 7618: 7616: 7614: 7613: 7608: 7603: 7598: 7591: 7589: 7587: 7586: 7581: 7576: 7571: 7564: 7562: 7560: 7559: 7554: 7547: 7545: 7543: 7542: 7537: 7530: 7528: 7526: 7525: 7520: 7513: 7511: 7509: 7508: 7503: 7496: 7494: 7492: 7491: 7486: 7479: 7476: 7475: 7472: 7471: 7464: 7463: 7456: 7449: 7441: 7432: 7431: 7428: 7427: 7425: 7424: 7417: 7415: 7413: 7412: 7405: 7403: 7401: 7400: 7393: 7391: 7389: 7388: 7383: 7376: 7374: 7372: 7371: 7366: 7359: 7357: 7355: 7354: 7347: 7345: 7343: 7342: 7335: 7333: 7331: 7330: 7323: 7321: 7319: 7318: 7311: 7309: 7307: 7306: 7301: 7294: 7292: 7290: 7289: 7282: 7280: 7278: 7277: 7275:Pierre Deligne 7270: 7268: 7266: 7265: 7258: 7256: 7254: 7253: 7246: 7244: 7242: 7241: 7234: 7232: 7230: 7229: 7227:Mikhail Gromov 7222: 7220: 7218: 7217: 7212: 7205: 7203: 7201: 7200: 7193: 7191: 7189: 7188: 7181: 7179: 7177: 7176: 7169: 7167: 7165: 7164: 7162:Isadore Singer 7159: 7157:Michael Atiyah 7152: 7150: 7148: 7147: 7140: 7137: 7136: 7133: 7132: 7124: 7123: 7116: 7109: 7101: 7093: 7092: 7089:César Milstein 7087: 7084: 7075: 7070: 7066: 7065: 7061: 7060: 7055: 7052: 7038: 7033: 7029: 7028: 7023: 7020: 7010: 7005: 7001: 7000: 6996: 6995: 6990: 6987: 6977: 6972: 6968: 6967: 6964:Sir Aaron Klug 6962: 6959: 6949: 6944: 6940: 6939: 6933: 6932: 6923: 6876: 6867: 6849: 6832: 6815: 6787: 6778:on 31 May 2008 6762: 6742: 6718: 6712:Michael Atiyah 6709: 6687: 6665: 6656:on 26 May 2011 6643: 6620: 6615: 6610: 6608:Web of Stories 6599: 6598:External links 6596: 6594: 6593: 6582: 6567: 6548: 6536:Asian J. Math. 6527: 6491: 6485: 6472: 6459: 6434: 6389:(3): 215–248, 6372: 6346:(3): 113–123, 6335: 6330: 6317: 6284:(3): 561–609, 6266: 6264: 6261: 6259: 6258: 6232: 6206: 6179: 6152: 6129: 6099: 6070: 6051: 6019: 5992: 5965: 5943: 5916: 5886: 5859: 5832: 5805: 5780: 5757: 5731: 5714:Science | AAAS 5701: 5679: 5657: 5636: 5624: 5612: 5600: 5588: 5576: 5564: 5552: 5540: 5529: 5517: 5505: 5494: 5482: 5470: 5458: 5446: 5435: 5423: 5411: 5399: 5387: 5375: 5363: 5351: 5340: 5329: 5317: 5305: 5293: 5281: 5269: 5257: 5245: 5233: 5221: 5209: 5197: 5186: 5175: 5163: 5161:, papers 80–83 5151: 5140: 5128: 5116: 5104: 5092: 5080: 5068: 5056: 5044: 5032: 5021: 5010: 4999: 4988: 4976: 4965: 4954: 4942: 4930: 4918: 4906: 4894: 4860: 4848: 4833: 4821: 4809: 4798: 4787: 4776: 4765: 4753: 4751:, papers 30,31 4741: 4729: 4714: 4702: 4678: 4667: 4656: 4645: 4633: 4621: 4609: 4597: 4585: 4573: 4561: 4549: 4538: 4527: 4500: 4488: 4476: 4464: 4432: 4420: 4405: 4393: 4381: 4349: 4311: 4294: 4267: 4226: 4203: 4175: 4145: 4134: 4109: 4081: 4055: 4022: 3991: 3954: 3947:Michael Atiyah 3930: 3893: 3891: 3888: 3887: 3886: 3879: 3876: 3864: 3861: 3846:Order of Merit 3676:Isadore Singer 3647: 3644: 3642: 3641: 3615: 3583:(2): 189–192, 3529: 3524:Reprinted in ( 3497:(3): 277–280, 3454:(4): 279–330, 3436: 3414:(3): 451–491, 3381:(2): 374–407, 3368: 3341: 3319:(1): 139–149, 3306: 3284:(1): 119–138, 3273: 3251:(3): 546–604, 3238: 3212:(3): 531–545, 3199: 3173:(3): 484–530, 3160: 3144:(3): 322–433, 3133: 3108: 3077: 3067: 3042: 3004: 3002: 2999: 2997: 2996: 2990: 2969: 2955: 2942: 2925: 2910: 2896: 2883: 2865: 2836: 2818: 2810:Addison-Wesley 2801: 2787: 2774: 2760: 2747: 2733: 2720: 2706: 2693: 2679: 2666: 2648: 2623: 2602: 2577: 2556: 2534: 2529: 2526: 2524: 2521: 2469: 2452:Juan Maldacena 2445:Michael Atiyah 2437: 2311: 2308: 2152:Michael Atiyah 2144: 2119: 2084:Main article: 2081: 2078: 1957:. Atiyah used 1844:Dirac operator 1801:Michael Atiyah 1785: 1762:Main article: 1756:Isadore Singer 1748: 1745: 1702:sphere bundles 1680:Michael Atiyah 1672: 1637: 1636: 1625: 1622: 1617: 1613: 1609: 1606: 1603: 1598: 1594: 1590: 1587: 1582: 1578: 1544: 1540: 1478: 1477: 1466: 1461: 1457: 1453: 1450: 1447: 1444: 1441: 1438: 1435: 1432: 1429: 1415:character ring 1294:Main article: 1291: 1288: 1276:minimal models 1272:Shigefumi Mori 1256:elliptic curve 1252:vector bundles 1244:ruled surfaces 1232:W. V. D. Hodge 1228:twisted cubics 1205:Main article: 1202: 1199: 1171:Lars Hörmander 1144:George Lusztig 1132:Frances Kirwan 1113:Michael Atiyah 1104: 1057:Paul Sutcliffe 1033:Nick S. Manton 1031:(instantons), 1025:Juan Maldacena 981:Hopf invariant 977:J. Frank Adams 932: 931:Collaborations 929: 869:Isadore Singer 804:Michael Atiyah 796: 719: 716: 595: 592: 533: 532: 529: 528: 523: 519: 518: 516: 515: 510: 505: 500: 498:George Lusztig 495: 490: 485: 483:Frances Kirwan 480: 475: 470: 465: 459: 457: 453: 452: 450:W. V. D. Hodge 447: 441: 440: 429: 423: 422: 420: 419: 414: 409: 404: 399: 393: 391: 387: 386: 381: 377: 376: 370: 369: 367: 366: 360: 354: 348: 342: 336: 329: 327: 323: 322: 320:Toric manifold 245: 244:Known for 241: 240: 238: 237: 222: 220: 216: 215: 212: 210:(aged 89) 204: 200: 199: 193: 180: 178: 174: 173: 170: 162: 161: 134: 132:Michael Atiyah 131: 126: 117: 116: 31: 29: 22: 15: 9: 6: 4: 3: 2: 10260: 10249: 10246: 10244: 10241: 10239: 10236: 10234: 10231: 10229: 10226: 10224: 10221: 10219: 10216: 10214: 10211: 10209: 10206: 10204: 10201: 10199: 10196: 10194: 10191: 10189: 10186: 10184: 10181: 10179: 10176: 10174: 10171: 10169: 10166: 10164: 10161: 10159: 10156: 10154: 10151: 10149: 10146: 10144: 10141: 10139: 10136: 10134: 10131: 10129: 10126: 10124: 10121: 10119: 10116: 10114: 10111: 10109: 10106: 10104: 10101: 10099: 10096: 10094: 10091: 10089: 10086: 10084: 10081: 10079: 10076: 10074: 10071: 10069: 10066: 10064: 10061: 10059: 10056: 10054: 10051: 10049: 10046: 10044: 10041: 10040: 10038: 10024: 10014: 10011: 10009: 10006: 10004: 10001: 9999: 9996: 9994: 9991: 9989: 9988:Andrew Huxley 9986: 9984: 9981: 9979: 9976: 9974: 9971: 9969: 9966: 9964: 9963:J. J. Thomson 9961: 9959: 9956: 9954: 9951: 9949: 9946: 9944: 9941: 9939: 9936: 9934: 9931: 9929: 9926: 9924: 9921: 9919: 9916: 9914: 9911: 9909: 9906: 9904: 9901: 9899: 9896: 9894: 9891: 9889: 9886: 9884: 9881: 9879: 9876: 9874: 9873:Thomas Comber 9871: 9869: 9868:Samuel Brooke 9866: 9864: 9861: 9859: 9856: 9854: 9853:Thomas Nevile 9851: 9849: 9846: 9844: 9843:John Whitgift 9841: 9839: 9836: 9834: 9831: 9829: 9826: 9824: 9821: 9820: 9817: 9813: 9806: 9801: 9799: 9794: 9792: 9787: 9786: 9783: 9769: 9765: 9761: 9757: 9753: 9749: 9745: 9741: 9737: 9733: 9729: 9728:Roger Penrose 9725: 9721: 9717: 9713: 9709: 9705: 9701: 9697: 9693: 9689: 9688:D. G. Kendall 9685: 9681: 9677: 9673: 9669: 9665: 9661: 9657: 9653: 9649: 9648:Graham Higman 9645: 9641: 9637: 9633: 9629: 9625: 9621: 9617: 9613: 9609: 9605: 9601: 9597: 9593: 9589: 9585: 9581: 9577: 9573: 9569: 9565: 9561: 9560:Louis Mordell 9557: 9553: 9549: 9545: 9541: 9537: 9533: 9529: 9525: 9521: 9520:A. E. H. Love 9517: 9513: 9509: 9505: 9501: 9497: 9493: 9489: 9485: 9481: 9477: 9473: 9469: 9465: 9461: 9457: 9453: 9449: 9445: 9441: 9437: 9433: 9429: 9425: 9424:Lord Rayleigh 9421: 9417: 9413: 9409: 9408:Arthur Cayley 9405: 9404: 9401: 9397: 9390: 9385: 9383: 9378: 9376: 9371: 9370: 9367: 9353: 9349: 9345: 9341: 9337: 9333: 9329: 9325: 9321: 9320:Alan Cottrell 9317: 9313: 9309: 9305: 9301: 9297: 9293: 9289: 9288:George Porter 9285: 9281: 9277: 9273: 9269: 9265: 9261: 9257: 9253: 9249: 9245: 9241: 9237: 9233: 9229: 9225: 9221: 9217: 9213: 9209: 9205: 9201: 9197: 9193: 9189: 9185: 9181: 9177: 9173: 9169: 9165: 9161: 9157: 9153: 9152:Francis Crick 9149: 9145: 9141: 9137: 9136:Andrew Huxley 9133: 9129: 9125: 9121: 9117: 9113: 9109: 9105: 9104:Peter Medawar 9101: 9097: 9093: 9089: 9085: 9081: 9077: 9073: 9069: 9065: 9061: 9057: 9053: 9049: 9045: 9041: 9037: 9033: 9029: 9025: 9021: 9017: 9013: 9009: 9008:Howard Florey 9005: 9001: 8997: 8993: 8992:Ronald Fisher 8989: 8985: 8981: 8977: 8973: 8969: 8965: 8961: 8957: 8956: 8953: 8948: 8941: 8936: 8934: 8929: 8927: 8922: 8921: 8918: 8904: 8900: 8896: 8892: 8888: 8884: 8880: 8876: 8872: 8868: 8867: 8865: 8861: 8853: 8849: 8845: 8841: 8837: 8836:George Porter 8833: 8829: 8828:Andrew Huxley 8825: 8821: 8817: 8813: 8809: 8805: 8801: 8797: 8796:Howard Florey 8793: 8789: 8785: 8781: 8777: 8773: 8769: 8765: 8761: 8757: 8753: 8749: 8745: 8741: 8737: 8733: 8729: 8725: 8724:J. J. Thomson 8721: 8717: 8713: 8709: 8705: 8701: 8700:Lord Rayleigh 8697: 8693: 8689: 8688: 8686: 8682: 8674: 8673:Joseph Lister 8670: 8666: 8662: 8658: 8654: 8650: 8646: 8642: 8638: 8634: 8630: 8626: 8622: 8618: 8617:Edward Sabine 8614: 8610: 8606: 8602: 8598: 8594: 8593:Earl of Rosse 8590: 8586: 8582: 8578: 8574: 8570: 8566: 8562: 8558: 8554: 8550: 8549: 8547: 8543: 8535: 8531: 8527: 8523: 8519: 8515: 8511: 8507: 8503: 8499: 8495: 8491: 8487: 8483: 8479: 8478:Martin Folkes 8475: 8471: 8467: 8463: 8459: 8458: 8456: 8452: 8444: 8440: 8436: 8432: 8428: 8424: 8420: 8416: 8412: 8408: 8404: 8400: 8396: 8392: 8388: 8384: 8380: 8376: 8372: 8368: 8364: 8360: 8359: 8357: 8353: 8349: 8348:Royal Society 8345: 8338: 8333: 8331: 8326: 8324: 8319: 8318: 8315: 8305: 8301: 8295: 8288: 8285: 8282: 8281:Nigel Hitchin 8279: 8276: 8273: 8270: 8267: 8264: 8261: 8258: 8255: 8252: 8249: 8246: 8245:William Esson 8243: 8240: 8237: 8234: 8231: 8228: 8225: 8222: 8219: 8216: 8213: 8210: 8207: 8204: 8201: 8198: 8195: 8192: 8191:Edmond Halley 8189: 8186: 8183: 8180: 8177: 8174: 8171: 8170: 8168: 8166: 8160: 8153: 8152:Steven Balbus 8150: 8147: 8144: 8141: 8138: 8135: 8132: 8129: 8126: 8123: 8120: 8117: 8114: 8111: 8108: 8105: 8102: 8099: 8096: 8093: 8090: 8087: 8084: 8081: 8080:James Bradley 8078: 8075: 8072: 8069: 8066: 8063: 8062:David Gregory 8060: 8057: 8054: 8051: 8048: 8045: 8042: 8039: 8036: 8033: 8030: 8029: 8027: 8025: 8019: 8015: 8009: 8000: 7995: 7993: 7988: 7986: 7981: 7980: 7977: 7965: 7961: 7957: 7955: 7947: 7946: 7943: 7933: 7930: 7928: 7925: 7923: 7920: 7918: 7917:Duminil-Copin 7914: 7913: 7911: 7906: 7903: 7901: 7898: 7896: 7893: 7891: 7887: 7886: 7884: 7879: 7876: 7874: 7871: 7869: 7866: 7864: 7860: 7859: 7857: 7852: 7849: 7847: 7844: 7842: 7839: 7837: 7836:Lindenstrauss 7833: 7832: 7830: 7825: 7822: 7820: 7817: 7815: 7812: 7810: 7806: 7805: 7803: 7798: 7795: 7793: 7789: 7788: 7786: 7781: 7778: 7776: 7773: 7771: 7768: 7766: 7762: 7761: 7759: 7754: 7751: 7749: 7746: 7744: 7741: 7739: 7735: 7734: 7732: 7727: 7724: 7722: 7719: 7717: 7714: 7712: 7708: 7707: 7705: 7700: 7697: 7695: 7692: 7690: 7686: 7685: 7683: 7678: 7675: 7673: 7670: 7668: 7664: 7663: 7661: 7656: 7653: 7651: 7648: 7646: 7643: 7641: 7637: 7636: 7634: 7629: 7626: 7624: 7620: 7619: 7617: 7612: 7609: 7607: 7604: 7602: 7599: 7597: 7593: 7592: 7590: 7585: 7582: 7580: 7577: 7575: 7572: 7570: 7566: 7565: 7563: 7558: 7555: 7553: 7549: 7548: 7546: 7541: 7538: 7536: 7532: 7531: 7529: 7524: 7521: 7519: 7515: 7514: 7512: 7507: 7504: 7502: 7498: 7497: 7495: 7490: 7487: 7485: 7481: 7480: 7478: 7477: 7473: 7469: 7462: 7457: 7455: 7450: 7448: 7443: 7442: 7439: 7423: 7419: 7418: 7416: 7411: 7407: 7406: 7404: 7399: 7395: 7394: 7392: 7387: 7386:Avi Wigderson 7384: 7382: 7381:László Lovász 7378: 7377: 7375: 7370: 7367: 7365: 7361: 7360: 7358: 7353: 7349: 7348: 7346: 7341: 7337: 7336: 7334: 7329: 7325: 7324: 7322: 7317: 7313: 7312: 7310: 7305: 7302: 7300: 7296: 7295: 7293: 7288: 7284: 7283: 7281: 7276: 7272: 7271: 7269: 7264: 7260: 7259: 7257: 7252: 7248: 7247: 7245: 7240: 7236: 7235: 7233: 7228: 7224: 7223: 7221: 7216: 7213: 7211: 7207: 7206: 7204: 7199: 7195: 7194: 7192: 7187: 7183: 7182: 7180: 7175: 7171: 7170: 7168: 7163: 7160: 7158: 7154: 7153: 7151: 7146: 7142: 7141: 7139: 7138: 7134: 7129: 7122: 7117: 7115: 7110: 7108: 7103: 7102: 7099: 7090: 7081: 7080: 7073: 7067: 7062: 7058: 7049: 7048: 7044: 7036: 7030: 7026: 7017: 7016: 7008: 7002: 6997: 6993: 6984: 6983: 6975: 6969: 6965: 6956: 6955: 6947: 6946:George Porter 6941: 6936: 6931: 6927: 6924: 6920: 6916: 6912: 6908: 6904: 6900: 6895: 6890: 6886: 6882: 6877: 6875: 6871: 6868: 6857: 6856: 6850: 6839: 6838: 6833: 6823: 6822: 6816: 6805: 6801: 6800: 6792: 6788: 6777: 6773: 6769: 6768: 6763: 6752: 6748: 6743: 6732: 6728: 6724: 6719: 6717: 6713: 6710: 6699: 6695: 6694: 6688: 6677: 6673: 6672: 6666: 6655: 6651: 6650: 6644: 6633: 6629: 6625: 6621: 6619: 6616: 6614: 6611: 6609: 6605: 6602: 6601: 6589: 6585: 6579: 6575: 6574: 6568: 6563: 6559: 6555: 6551: 6545: 6541: 6537: 6533: 6528: 6524: 6520: 6516: 6512: 6508: 6504: 6500: 6496: 6492: 6488: 6482: 6478: 6473: 6470: 6466: 6462: 6456: 6452: 6448: 6444: 6440: 6435: 6424: 6420: 6416: 6412: 6408: 6404: 6400: 6396: 6392: 6388: 6384: 6383: 6378: 6373: 6370: 6369:0-387-13619-3 6366: 6361: 6357: 6353: 6349: 6345: 6341: 6336: 6333: 6327: 6323: 6318: 6315: 6311: 6307: 6303: 6299: 6295: 6291: 6287: 6283: 6279: 6278: 6273: 6268: 6267: 6246: 6242: 6236: 6220: 6216: 6210: 6195: 6191: 6190: 6183: 6168: 6164: 6163: 6156: 6141: 6140: 6133: 6117: 6113: 6112:www1.hw.ac.uk 6109: 6103: 6087: 6083: 6077: 6075: 6067: 6063: 6060: 6055: 6036: 6029: 6023: 6008: 6004: 6003: 5996: 5981: 5977: 5976: 5969: 5955: 5954: 5947: 5932: 5928: 5927: 5920: 5904: 5900: 5896: 5890: 5875: 5871: 5870: 5863: 5848: 5844: 5843: 5836: 5821: 5817: 5816: 5809: 5794: 5790: 5784: 5776: 5772: 5768: 5767:Cartan, Henri 5761: 5745: 5741: 5735: 5719: 5715: 5711: 5705: 5691: 5690: 5683: 5669: 5668: 5661: 5652: 5647: 5640: 5633: 5628: 5621: 5616: 5609: 5604: 5597: 5592: 5585: 5580: 5573: 5568: 5561: 5556: 5549: 5544: 5538: 5533: 5526: 5521: 5514: 5509: 5503: 5498: 5492: 5486: 5479: 5474: 5467: 5462: 5455: 5450: 5444: 5439: 5432: 5427: 5420: 5415: 5408: 5403: 5396: 5391: 5384: 5379: 5372: 5367: 5360: 5355: 5349: 5344: 5338: 5333: 5326: 5321: 5314: 5309: 5302: 5297: 5290: 5285: 5278: 5273: 5266: 5261: 5254: 5249: 5242: 5237: 5230: 5225: 5218: 5213: 5206: 5201: 5195: 5190: 5184: 5179: 5172: 5167: 5160: 5155: 5149: 5144: 5137: 5132: 5125: 5120: 5113: 5108: 5101: 5096: 5089: 5084: 5077: 5072: 5065: 5060: 5053: 5048: 5041: 5036: 5030: 5025: 5019: 5014: 5008: 5003: 4997: 4996:Gel'fand 1960 4992: 4985: 4980: 4974: 4969: 4963: 4958: 4951: 4946: 4939: 4934: 4927: 4922: 4915: 4910: 4903: 4898: 4880: 4873: 4872: 4864: 4857: 4852: 4845: 4840: 4838: 4830: 4825: 4818: 4813: 4807: 4802: 4796: 4791: 4785: 4780: 4774: 4769: 4762: 4757: 4750: 4745: 4738: 4733: 4726: 4721: 4719: 4711: 4706: 4697: 4692: 4688: 4682: 4676: 4671: 4665: 4660: 4654: 4649: 4642: 4637: 4630: 4625: 4618: 4613: 4606: 4601: 4595:, papers 3, 4 4594: 4589: 4582: 4577: 4570: 4565: 4558: 4553: 4547: 4542: 4536: 4531: 4516: 4512: 4511: 4504: 4498:, p. 307 4497: 4492: 4485: 4480: 4473: 4468: 4449: 4442: 4436: 4429: 4424: 4418: 4414: 4409: 4402: 4397: 4390: 4385: 4366: 4359: 4353: 4344: 4339: 4335: 4331: 4330: 4325: 4321: 4315: 4304: 4298: 4283: 4279: 4278: 4271: 4253: 4249: 4245: 4244: 4236: 4230: 4223: 4218: 4216: 4214: 4212: 4210: 4208: 4192: 4188: 4187: 4179: 4163: 4159: 4155: 4149: 4143: 4138: 4131: 4126: 4124: 4122: 4120: 4118: 4116: 4114: 4098: 4094: 4093: 4085: 4071: 4070: 4062: 4060: 4051: 4044: 4040: 4039: 4033: 4026: 4019: 4015: 4014: 4009: 4005: 3998: 3996: 3987: 3983: 3979: 3975: 3971: 3967: 3966: 3958: 3952: 3948: 3943: 3941: 3939: 3937: 3935: 3918: 3914: 3910: 3909: 3901: 3899: 3894: 3885: 3882: 3881: 3875: 3872: 3870: 3863:Personal life 3860: 3858: 3854: 3849: 3847: 3843: 3838: 3834: 3832: 3828: 3824: 3820: 3816: 3812: 3808: 3804: 3800: 3796: 3792: 3788: 3784: 3780: 3776: 3772: 3768: 3764: 3760: 3756: 3752: 3748: 3743: 3741: 3737: 3733: 3730:in 2008, the 3729: 3725: 3722:in 1993, the 3721: 3717: 3713: 3709: 3706:in 1987, the 3705: 3702:in 1981, the 3701: 3697: 3694:in 1980, the 3693: 3689: 3686:in 1968, the 3685: 3684:Royal Society 3681: 3677: 3674:jointly with 3673: 3669: 3665: 3657: 3656:Royal Society 3652: 3639: 3635: 3630: 3625: 3621: 3616: 3613: 3608: 3604: 3599: 3594: 3590: 3586: 3582: 3578: 3577:Invent. Math. 3571: 3567: 3563: 3559: 3555: 3551: 3547: 3543: 3539: 3535: 3534:Invent. Math. 3530: 3527: 3522: 3518: 3513: 3508: 3504: 3500: 3496: 3492: 3491:Invent. Math. 3485: 3481: 3477: 3473: 3469: 3465: 3461: 3457: 3453: 3449: 3448:Invent. Math. 3442: 3437: 3434: 3429: 3425: 3421: 3417: 3413: 3409: 3408: 3401: 3396: 3392: 3388: 3384: 3380: 3376: 3375: 3369: 3366: 3360: 3355: 3352:(2): 245–50, 3351: 3347: 3342: 3339: 3334: 3330: 3326: 3322: 3318: 3314: 3313: 3307: 3304: 3299: 3295: 3291: 3287: 3283: 3279: 3274: 3271: 3266: 3262: 3258: 3254: 3250: 3246: 3245: 3239: 3236: 3232: 3227: 3223: 3219: 3215: 3211: 3207: 3206: 3200: 3197: 3193: 3188: 3184: 3180: 3176: 3172: 3168: 3167: 3161: 3158: 3152: 3147: 3143: 3139: 3134: 3131: 3127: 3122: 3118: 3114: 3109: 3106: 3100: 3095: 3092:(1–2): 1–18, 3091: 3087: 3083: 3078: 3075: 3070: 3068:9780821814031 3064: 3060: 3056: 3052: 3048: 3043: 3040: 3035: 3031: 3027: 3023: 3019: 3015: 3011: 3006: 3005: 2993: 2987: 2983: 2979: 2975: 2970: 2966: 2962: 2958: 2952: 2948: 2943: 2932: 2928: 2922: 2918: 2917: 2911: 2907: 2903: 2899: 2893: 2889: 2884: 2881: 2876: 2872: 2868: 2862: 2858: 2854: 2850: 2845: 2844: 2837: 2834: 2829: 2825: 2821: 2815: 2811: 2807: 2802: 2798: 2794: 2790: 2784: 2780: 2775: 2771: 2767: 2763: 2757: 2753: 2748: 2744: 2740: 2736: 2730: 2726: 2721: 2717: 2713: 2709: 2703: 2699: 2694: 2690: 2686: 2682: 2676: 2672: 2667: 2664: 2659: 2655: 2651: 2645: 2641: 2637: 2633: 2629: 2624: 2621: 2616: 2612: 2608: 2603: 2600: 2595: 2591: 2587: 2583: 2578: 2575: 2570: 2566: 2562: 2557: 2553: 2549: 2545: 2541: 2536: 2535: 2533: 2520: 2518: 2514: 2510: 2506: 2502: 2498: 2493: 2491: 2486: 2484: 2480: 2475: 2473: 2465: 2461: 2457: 2453: 2447: 2441: 2436: 2434: 2430: 2425: 2421: 2417: 2412: 2410: 2406: 2403:as a certain 2402: 2398: 2394: 2390: 2386: 2381: 2379: 2375: 2374:Edward Witten 2371: 2370:Vaughan Jones 2367: 2363: 2359: 2355: 2351: 2346: 2344: 2339: 2337: 2333: 2324: 2320: 2319:Edward Witten 2316: 2307: 2304: 2300: 2298: 2294: 2290: 2286: 2282: 2277: 2275: 2271: 2267: 2263: 2259: 2254: 2250: 2245: 2243: 2238: 2233: 2231: 2227: 2223: 2219: 2215: 2211: 2207: 2206:moduli spaces 2203: 2198: 2196: 2192: 2186: 2183: 2179: 2175: 2173: 2169: 2164: 2160: 2154: 2148: 2143: 2141: 2137: 2133: 2129: 2125: 2117: 2111: 2109: 2105: 2097: 2092: 2087: 2077: 2073: 2071: 2067: 2063: 2059: 2053: 2051: 2047: 2043: 2039: 2030: 2026: 2024: 2020: 2016: 2011: 2006: 2004: 2003:heat equation 2000: 1992: 1988: 1984: 1982: 1978: 1973: 1971: 1966: 1964: 1960: 1956: 1955:distributions 1952: 1947: 1945: 1941: 1937: 1933: 1929: 1925: 1921: 1917: 1913: 1909: 1902: 1898: 1893: 1889: 1887: 1883: 1879: 1875: 1871: 1866: 1864: 1860: 1856: 1852: 1847: 1845: 1841: 1837: 1833: 1829: 1825: 1821: 1817: 1813: 1809: 1803: 1797: 1796:non-trivial. 1795: 1791: 1784: 1782: 1778: 1773: 1769: 1765: 1757: 1753: 1744: 1742: 1738: 1734: 1729: 1724: 1722: 1718: 1713: 1711: 1707: 1703: 1699: 1695: 1691: 1688: 1682: 1676: 1671: 1669: 1668:C. T. C. Wall 1665: 1661: 1656: 1654: 1650: 1646: 1642: 1623: 1615: 1611: 1604: 1601: 1596: 1588: 1580: 1576: 1568: 1567: 1566: 1564: 1560: 1542: 1538: 1529: 1525: 1521: 1517: 1513: 1509: 1505: 1501: 1497: 1493: 1490: 1487: 1483: 1464: 1459: 1451: 1445: 1442: 1436: 1433: 1427: 1420: 1419: 1418: 1416: 1412: 1408: 1404: 1400: 1395: 1393: 1385: 1380: 1376: 1373: 1369: 1365: 1361: 1357: 1353: 1347: 1345: 1341: 1337: 1333: 1328: 1326: 1321: 1319: 1311: 1310:vector bundle 1307: 1302: 1297: 1287: 1285: 1281: 1277: 1273: 1269: 1265: 1261: 1257: 1253: 1249: 1245: 1241: 1237: 1236:Smith's prize 1233: 1229: 1224: 1218: 1213: 1208: 1198: 1194: 1192: 1188: 1184: 1180: 1176: 1172: 1168: 1167:Roger Penrose 1163: 1161: 1157: 1153: 1149: 1145: 1141: 1140:Ruth Lawrence 1137: 1133: 1129: 1128:Nigel Hitchin 1125: 1121: 1115: 1109: 1103: 1101: 1097: 1093: 1089: 1084: 1082: 1081:Edward Witten 1078: 1074: 1070: 1066: 1062: 1061:David O. Tall 1059:(polyhedra), 1058: 1054: 1050: 1046: 1042: 1038: 1035:(Skyrmions), 1034: 1030: 1029:Yuri I. Manin 1026: 1022: 1018: 1014: 1010: 1007:(monopoles), 1006: 1002: 998: 994: 993:vector fields 990: 986: 982: 978: 974: 970: 966: 962: 958: 954: 946: 942: 937: 928: 926: 922: 918: 914: 910: 906: 902: 898: 894: 890: 886: 885:Edward Witten 882: 878: 877:Nigel Hitchin 874: 870: 866: 862: 857: 855: 851: 847: 843: 839: 836:(1990–1997), 835: 832:(1990–1995), 831: 827: 822: 820: 816: 812: 806: 800: 795: 793: 789: 788:Royal Society 785: 781: 777: 773: 769: 765: 761: 757: 753: 750:and tutorial 749: 745: 741: 737: 729: 724: 715: 713: 708: 706: 701: 699: 695: 691: 687: 683: 679: 678:undergraduate 675: 671: 667: 663: 659: 655: 651: 647: 643: 639: 635: 631: 627: 626:Edward Atiyah 623: 619: 612: 608: 604: 600: 591: 589: 585: 581: 577: 573: 569: 568:mathematician 563: 539: 530: 527: 526:Edward Witten 524: 520: 514: 513:David O. Tall 511: 509: 506: 504: 501: 499: 496: 494: 493:Ruth Lawrence 491: 489: 486: 484: 481: 479: 476: 474: 473:Nigel Hitchin 471: 469: 466: 464: 461: 460: 458: 454: 451: 448: 446: 442: 435: 434: 430: 428: 424: 418: 415: 413: 410: 408: 405: 403: 400: 398: 395: 394: 392: 388: 385: 382: 378: 375: 371: 364: 361: 358: 355: 352: 349: 346: 343: 340: 337: 334: 333:Berwick Prize 331: 330: 328: 324: 321: 317: 313: 309: 305: 304:Eta invariant 301: 297: 293: 289: 285: 281: 277: 273: 269: 265: 261: 257: 253: 249: 246: 242: 235: 231: 227: 224: 223: 221: 217: 205: 201: 196: 191:22 April 1929 179: 175: 168: 163: 158: 154: 150: 146: 142: 138: 129: 124: 121: 113: 110: 102: 91: 88: 84: 81: 77: 74: 70: 67: 63: 60: – 59: 55: 54:Find sources: 48: 42: 41: 37: 32:This article 30: 26: 21: 20: 10013:Sally Davies 9992: 9983:Alan Hodgkin 9973:Edgar Adrian 9923:Robert Smith 9913:John Montagu 9903:Isaac Barrow 9898:John Pearson 9888:John Wilkins 9863:Leonard Mawe 9828:William Bill 9768:Andrew Wiles 9712:R. A. Rankin 9704:W. K. Hayman 9663: 9600:G. I. Taylor 9504:E. W. Hobson 9312:Frank Fenner 9255: 9192:Derek Barton 9120:Norman Pirie 9088:Bernard Katz 9072:Alan Hodgkin 8960:David Keilin 8903:Adrian Smith 8863:21st century 8843: 8684:20th century 8561:Humphry Davy 8545:19th century 8534:Joseph Banks 8526:John Pringle 8518:James Burrow 8502:James Burrow 8462:Isaac Newton 8454:18th century 8403:Samuel Pepys 8387:John Hoskyns 8355:17th century 8289: (2017) 8283: (1997) 8277: (1995) 8271: (1969) 8265: (1963) 8262: 8259: (1931) 8253: (1919) 8247: (1897) 8241: (1883) 8235: (1861) 8229: (1827) 8227:Baden Powell 8223: (1810) 8217: (1797) 8211: (1766) 8205: (1765) 8203:Joseph Betts 8199: (1742) 8193: (1704) 8187: (1649) 8181: (1631) 8179:Peter Turner 8175: (1619) 8173:Henry Briggs 8154: (2012) 8148: (1999) 8142: (1994) 8136: (1960) 8130: (1932) 8124: (1893) 8118: (1870) 8112: (1842) 8106: (1839) 8100: (1827) 8094: (1810) 8088: (1763) 8082: (1721) 8076: (1712) 8070: (1709) 8068:John Caswell 8064: (1691) 8058: (1673) 8052: (1661) 8046: (1649) 8040: (1642) 8038:John Greaves 8034: (1620) 8024:of Astronomy 7579:Grothendieck 7568: 7316:Andrew Wiles 7215:Jacques Tits 7156: 7079:Copley Medal 7077: 7040: 7012: 6979: 6951: 6884: 6880: 6860:, retrieved 6854: 6842:, retrieved 6836: 6825:, retrieved 6820: 6808:, retrieved 6803: 6797: 6780:, retrieved 6776:the original 6766: 6755:, retrieved 6751:The Guardian 6750: 6735:, retrieved 6731:the original 6726: 6702:, retrieved 6698:the original 6692: 6680:, retrieved 6676:the original 6670: 6658:, retrieved 6654:the original 6648: 6636:, retrieved 6632:the original 6627: 6588:the original 6572: 6562:the original 6539: 6535: 6506: 6502: 6476: 6438: 6429:30 September 6427:, retrieved 6423:the original 6386: 6380: 6343: 6339: 6321: 6281: 6275: 6249:. Retrieved 6235: 6223:. Retrieved 6209: 6198:, retrieved 6194:the original 6188: 6182: 6171:, retrieved 6167:the original 6161: 6155: 6144:, retrieved 6138: 6132: 6120:. Retrieved 6111: 6102: 6090:. Retrieved 6054: 6042:. Retrieved 6022: 6013:11 September 6011:, retrieved 6007:the original 6001: 5995: 5984:, retrieved 5980:the original 5974: 5968: 5957:, retrieved 5952: 5946: 5935:, retrieved 5931:the original 5925: 5919: 5907:. Retrieved 5889: 5878:, retrieved 5868: 5862: 5851:, retrieved 5841: 5835: 5824:, retrieved 5814: 5808: 5796:. Retrieved 5792: 5783: 5770: 5760: 5750:24 September 5748:. Retrieved 5734: 5724:26 September 5722:. Retrieved 5713: 5704: 5695:24 September 5693:, retrieved 5688: 5682: 5673:24 September 5671:, retrieved 5666: 5660: 5639: 5627: 5615: 5603: 5591: 5584:Atiyah 1988a 5579: 5567: 5555: 5543: 5532: 5520: 5508: 5497: 5485: 5478:Atiyah 1988e 5473: 5466:Atiyah 1988e 5461: 5454:Atiyah 1988e 5449: 5438: 5431:Atiyah 1988e 5426: 5419:Atiyah 1988e 5414: 5407:Atiyah 1988e 5402: 5395:Atiyah 1988e 5390: 5383:Atiyah 1988e 5378: 5371:Atiyah 1988e 5366: 5359:Atiyah 1988e 5354: 5343: 5332: 5325:Atiyah 1988e 5320: 5313:Atiyah 1988e 5308: 5301:Atiyah 1988e 5296: 5289:Atiyah 1988a 5284: 5277:Atiyah 1988e 5272: 5265:Atiyah 1988e 5260: 5253:Atiyah 1988e 5248: 5241:Atiyah 1988e 5236: 5229:Atiyah 1988e 5224: 5217:Atiyah 1988d 5212: 5205:Atiyah 1988d 5200: 5189: 5178: 5171:Atiyah 1988d 5166: 5159:Atiyah 1988d 5154: 5143: 5136:Atiyah 1988c 5131: 5124:Atiyah 1988c 5119: 5112:Atiyah 1988a 5107: 5100:Atiyah 1988c 5095: 5088:Atiyah 1988c 5083: 5076:Atiyah 1988c 5071: 5064:Atiyah 1988c 5059: 5052:Atiyah 1988c 5047: 5040:Atiyah 1988c 5035: 5024: 5013: 5002: 4991: 4984:Atiyah 1988a 4979: 4973:Atiyah 1988d 4968: 4962:Atiyah 1988c 4957: 4950:Atiyah 1988b 4945: 4938:Atiyah 1988b 4933: 4926:Atiyah 1988b 4921: 4914:Atiyah 1988b 4909: 4902:Atiyah 1988b 4897: 4886:, retrieved 4879:the original 4870: 4863: 4856:Atiyah 1988b 4851: 4844:Atiyah 1988b 4824: 4817:Atiyah 1988b 4812: 4801: 4790: 4779: 4768: 4761:Atiyah 1988b 4756: 4749:Atiyah 1988a 4744: 4737:Atiyah 1988b 4732: 4725:Atiyah 1988b 4710:Atiyah 1988b 4705: 4696:math/0012213 4686: 4681: 4675:Atiyah 1988b 4670: 4659: 4648: 4641:Matsuki 2002 4636: 4629:Atiyah 1988a 4624: 4617:Atiyah 1988a 4612: 4605:Atiyah 1988a 4600: 4593:Atiyah 1988a 4588: 4581:Atiyah 1988a 4576: 4569:Atiyah 1988a 4564: 4557:Atiyah 1988a 4552: 4546:Atiyah 1988a 4541: 4530: 4519:, retrieved 4509: 4503: 4496:Atiyah 1988a 4491: 4486:, p. 10 4479: 4472:Atiyah 1988a 4467: 4455:. Retrieved 4435: 4423: 4408: 4401:Atiyah 1988a 4396: 4384: 4372:. Retrieved 4365:the original 4352: 4333: 4327: 4319: 4314: 4297: 4286:, retrieved 4276: 4270: 4259:, retrieved 4247: 4241: 4229: 4224:, p. ix 4195:, retrieved 4191:the original 4185: 4178: 4166:. Retrieved 4162:the original 4157: 4148: 4137: 4132:, p. xi 4130:Atiyah 1988a 4101:, retrieved 4097:the original 4091: 4084: 4073:, retrieved 4068: 4036: 4025: 4011: 3969: 3964: 3957: 3921:. Retrieved 3912: 3907: 3873: 3866: 3850: 3839: 3835: 3753:(1969), the 3744: 3708:Copley Medal 3664:Fields Medal 3661: 3619: 3614:, paper 90). 3612:Atiyah 1988d 3580: 3576: 3537: 3533: 3526:Atiyah 1988d 3494: 3490: 3451: 3447: 3433:Atiyah 1988c 3411: 3405: 3400:Atiyah 1988c 3378: 3372: 3367:, paper 61). 3365:Atiyah 1988c 3349: 3345: 3340:, paper 68). 3338:Atiyah 1988c 3316: 3310: 3305:, paper 67). 3303:Atiyah 1988c 3281: 3277: 3272:, paper 66). 3270:Atiyah 1988c 3248: 3242: 3237:, paper 65). 3235:Atiyah 1988c 3209: 3203: 3198:, paper 64). 3196:Atiyah 1988c 3170: 3164: 3159:, paper 56). 3157:Atiyah 1988c 3141: 3137: 3126:Atiyah 1988d 3112: 3107:, paper 49). 3105:Atiyah 1988b 3089: 3085: 3076:, paper 28). 3074:Atiyah 1988b 3050: 3041:, paper 29). 3039:Atiyah 1988b 3017: 3013: 2982:10.1142/9652 2973: 2946: 2935:, retrieved 2931:the original 2915: 2887: 2882:, item 136). 2842: 2833:Atiyah 1988b 2805: 2778: 2751: 2724: 2697: 2670: 2665:, item 126). 2627: 2620:Atiyah 1988e 2606: 2599:Atiyah 1988c 2581: 2574:Atiyah 1988b 2560: 2543: 2531: 2523:Bibliography 2495:At the 2018 2494: 2487: 2476: 2449: 2443: 2439: 2423: 2418:(called the 2413: 2397:moduli space 2382: 2347: 2340: 2328: 2305: 2301: 2278: 2246: 2234: 2222:Morse theory 2199: 2187: 2176: 2156: 2150: 2146: 2127: 2123: 2112: 2101: 2074: 2061: 2054: 2050:Lars Gårding 2035: 2007: 1996: 1974: 1967: 1963:J. Bernstein 1948: 1943: 1939: 1935: 1927: 1923: 1920:group action 1905: 1897:Graeme Segal 1881: 1877: 1873: 1869: 1867: 1848: 1805: 1799: 1793: 1789: 1787: 1774: 1770: 1767: 1725: 1714: 1697: 1693: 1689: 1684: 1678: 1674: 1657: 1648: 1644: 1640: 1638: 1562: 1558: 1523: 1519: 1515: 1511: 1507: 1500:Graeme Segal 1495: 1481: 1479: 1406: 1398: 1396: 1389: 1368:Todd classes 1352:James number 1348: 1329: 1322: 1315: 1242:approach to 1234:and won the 1225: 1222: 1195: 1183:Hermann Weyl 1175:Alain Connes 1164: 1117: 1111: 1106: 1085: 1075:(M-theory), 1065:John A. Todd 1049:Graeme Segal 1027:(M-theory), 1021:Lisa Jeffrey 997:Lars Gårding 950: 921:Fields Medal 873:Graeme Segal 858: 823: 808: 802: 798: 768:professorial 733: 709: 702: 697: 682:postgraduate 615: 584:Fields Medal 537: 536: 508:Graeme Segal 478:Lisa Jeffrey 431: 390:Institutions 373: 357:Copley Medal 339:Fields Medal 208:(2019-01-11) 120: 105: 99:January 2022 96: 86: 79: 72: 65: 53: 33: 10233:Topologists 10048:2019 deaths 10043:1929 births 10003:Martin Rees 9998:Amartya Sen 9893:Henry Ferne 9878:Thomas Hill 9823:John Redman 9720:J. A. Green 9640:Kurt Mahler 9624:Philip Hall 9496:W. H. Young 9480:Horace Lamb 9464:H. F. Baker 9432:Felix Klein 9328:Hugh Huxley 9272:Abdus Salam 9056:Paul Fildes 8949:(1951–2000) 8780:Lord Adrian 8665:Lord Kelvin 8470:Hans Sloane 8443:Lord Somers 8395:Cyril Wyche 8185:John Wallis 8165:of Geometry 8146:Joseph Silk 7915:2022 7888:2018 7861:2014 7834:2010 7807:2006 7790:2002 7763:1998 7736:1994 7709:1990 7687:1986 7665:1982 7638:1978 7621:1974 7594:1970 7567:1966 7550:1962 7533:1958 7516:1954 7499:1950 7482:1936 7420:2024 7408:2023 7396:2022 7379:2021 7362:2020 7350:2019 7338:2018 7326:2017 7314:2016 7297:2015 7287:Yakov Sinai 7285:2014 7273:2013 7261:2012 7251:John Milnor 7249:2011 7237:2010 7225:2009 7208:2008 7196:2007 7184:2006 7172:2005 7155:2004 7143:2003 7025:Amartya Sen 6509:: 113–128, 5909:27 November 5634:, paper 173 5632:Atiyah 2004 5622:, paper 172 5620:Atiyah 2004 5610:, paper 170 5608:Atiyah 2004 5598:, paper 169 5596:Atiyah 2004 5572:Atiyah 2004 5560:Atiyah 2004 5550:, paper 139 5548:Atiyah 2004 5537:Atiyah 1990 5527:, paper 132 5525:Atiyah 2004 5515:, paper 127 5513:Atiyah 2004 5502:Atiyah 2007 5468:, paper 117 5456:, paper 118 5421:, paper 124 5409:, paper 110 5397:, paper 109 5385:, paper 108 5373:, paper 106 5327:, paper 102 5315:, paper 101 5303:, paper 112 5183:Atiyah 1976 5078:, p. 3 5018:Palais 1965 4829:Atiyah 2004 4773:Atiyah 1961 4664:Atiyah 1989 4583:, p. 1 4484:Atiyah 2004 4428:Atiyah 2004 4403:, p. 2 4391:, p. 9 4389:Atiyah 2004 4336:: 229–247. 4222:Atiyah 2004 3923:17 November 3680:Royal Medal 3629:1009.4827v1 2880:Atiyah 2004 2835:, item 45). 2663:Atiyah 2004 2622:, item 99). 2601:, item 78). 2576:, item 50). 2456:Cumrun Vafa 2407:of complex 2405:subquotient 2383:He studied 2297:hyperkähler 2285:Nick Manton 2200:Harder and 2140:quaternions 2040:often have 1922:of a group 1706:J. F. Adams 1561:with fibre 1325:Möbius band 1306:Möbius band 1280:K3 surfaces 1274:'s work on 1258:(extending 1148:Jack Morava 1073:Cumrun Vafa 985:L-functions 967:, and with 692:student of 603:Great Court 384:Mathematics 345:Royal Medal 260:Atiyah flop 10037:Categories 9978:Rab Butler 9908:John North 9848:John Still 9672:K. F. Roth 9616:Max Newman 9440:S. Roberts 9248:Robin Hill 9232:Aaron Klug 9184:Max Perutz 9040:Hans Krebs 8968:Paul Dirac 8510:James West 8344:Presidents 8269:Ioan James 8209:John Smith 8074:John Keill 7878:Mirzakhani 7775:Kontsevich 7328:Yves Meyer 7128:Abel Prize 7072:Robin Hill 7051:1995–2005 7043:Chancellor 7019:1990–1997 6986:2005–2008 6958:1990–1995 6894:1910.07851 6251:11 January 6225:11 January 6092:28 October 5986:25 January 5651:1610.09366 5279:, paper 99 5267:, paper 96 5255:, paper 95 5207:, paper 91 5138:, paper 76 5126:, paper 74 5114:, paper 15 5102:, paper 73 5090:, paper 65 5054:, paper 68 5042:, paper 67 4952:, paper 48 4940:, paper 46 4928:, paper 39 4916:, paper 45 4904:, paper 40 4858:, paper 36 4846:, paper 37 4819:, paper 34 4795:Segal 1968 4763:, paper 42 4739:, paper 26 4727:, paper 28 4712:, paper 24 4457:30 October 4374:30 October 4052:required.) 3890:References 3672:Abel Prize 3192:cohomology 2511:using the 2479:M. Hopkins 2393:instantons 2253:moment map 2163:4-manifold 2126:>0 is 8 1991:Raoul Bott 1977:Elmer Rees 1832:Hirzebruch 1826:, and the 1528:isomorphic 1411:completion 1092:Yang–Mills 1041:Elmer Rees 953:Raoul Bott 925:Abel Prize 909:instantons 861:Raoul Bott 838:Chancellor 770:fellow at 650:Alexandria 588:Abel Prize 363:Abel Prize 187:1929-04-22 69:newspapers 36:references 9488:J. Larmor 8879:Lord Rees 8820:Lord Todd 8044:Seth Ward 7932:Viazovska 7905:Venkatesh 7797:Voevodsky 7792:Lafforgue 7765:Borcherds 7689:Donaldson 7645:Fefferman 7552:Hörmander 7239:John Tate 7174:Peter Lax 7130:laureates 6919:204743755 6844:14 August 6827:14 August 6810:14 August 6782:14 August 6757:14 August 6737:26 August 6704:14 August 6682:14 August 6660:14 August 6638:14 August 6419:117851906 6403:0025-5831 6298:0003-486X 6200:14 August 6173:14 August 5959:14 August 5937:14 August 5880:14 August 5853:14 August 5826:14 August 5798:22 August 4888:14 August 4631:, paper 8 4619:, paper 7 4607:, paper 5 4571:, paper 2 4559:, paper 1 4521:14 August 4288:14 August 4261:14 August 4197:14 August 4103:14 August 4075:14 August 4038:Who's Who 3978:500473357 3848:in 1992. 3833:in 1993. 3742:in 2011. 3726:from the 3698:from the 3570:189831012 3484:115700319 3020:: 23–64, 2460:E. Witten 2401:skyrmions 2385:skyrmions 2368:found by 2270:F. Kirwan 1951:Hörmander 1859:cobordism 1664:René Thom 1660:cobordism 1602:≅ 1597:∧ 1492:Lie group 1489:connected 1460:∧ 1443:≅ 1284:Hausdorff 927:in 2004. 913:monopoles 784:Princeton 672:with the 618:Hampstead 590:in 2004. 316:Pin group 312:KR-theory 219:Education 195:Hampstead 8871:Lord May 7954:Category 7868:Bhargava 7814:Perelman 7809:Okounkov 7780:McMullen 7753:Zelmanov 7738:Bourgain 7711:Drinfeld 7699:Freedman 7694:Faltings 7672:Thurston 7650:Margulis 7623:Bombieri 7611:Thompson 7601:Hironaka 7501:Schwartz 6753:, London 6523:55847918 6497:(1968), 6245:Archived 6219:Archived 6116:Archived 6086:Archived 6062:Archived 6044:27 April 6035:Archived 5903:Archived 5874:archived 5847:archived 5820:archived 5744:Archived 5718:Archived 4515:archived 4448:Archived 4322:(2014). 4282:archived 4252:archived 4168:10 April 3917:Archived 3878:See also 3869:Pyrenees 3817:and the 3540:: 1–62, 3049:(1961), 3034:54764252 2806:K-theory 2542:(1969), 2472:holonomy 2464:M-theory 2224:and the 2010:manifold 1981:Horrocks 1959:Hironaka 1863:K-theory 1812:Gel'fand 1318:K-theory 1296:K-theory 1156:John Roe 1108:corner. 1096:geometry 1017:K-theory 939:The old 748:lecturer 744:lecturer 690:doctoral 664:for his 638:Khartoum 572:geometry 308:K-theory 157:HonFREng 8346:of the 7927:Maynard 7900:Scholze 7895:Figalli 7851:Villani 7846:Smirnov 7655:Quillen 7640:Deligne 7628:Mumford 7606:Novikov 7518:Kodaira 7506:Selberg 7489:Douglas 7484:Ahlfors 7045:of the 6928:at the 6899:Bibcode 6714:at the 6558:1701915 6469:1875410 6411:0364254 6348:Bibcode 6314:1217348 6306:2946532 6263:Sources 6122:4 April 4415:at the 4308:. 2019. 3949:at the 3829:of the 3734:of the 3714:of the 3690:of the 3682:of the 3634:Bibcode 3607:0550183 3585:Bibcode 3562:0463358 3542:Bibcode 3521:0650829 3499:Bibcode 3476:0650828 3456:Bibcode 3428:1970721 3395:1970694 3333:1970757 3298:1970756 3265:1970717 3226:1970716 3187:1970715 3121:0420729 2965:3223085 2937:23 July 2906:2160826 2875:1078014 2828:1043170 2797:0951892 2770:0951892 2743:0951892 2716:0951892 2689:0951892 2658:0934202 2615:0554924 2594:0482866 2569:0263102 2552:0242802 2399:of two 2228:over a 2218:R. Bott 2008:If the 1970:Â-genus 1884:to the 1840:Â genus 1792:simple 1733:Shapiro 1687:J-group 1518:-space 1486:compact 1413:of its 1282:is not 1268:3-folds 1100:physics 963:on the 955:on the 840:of the 821:(EMS). 634:Patrick 149:FMedSci 83:scholar 9770:(2019) 9762:(2016) 9754:(2013) 9746:(2010) 9738:(2007) 9730:(2004) 9722:(2001) 9714:(1998) 9706:(1995) 9698:(1992) 9690:(1989) 9682:(1986) 9674:(1983) 9666:(1980) 9658:(1977) 9650:(1974) 9642:(1971) 9634:(1968) 9626:(1965) 9618:(1962) 9610:(1959) 9602:(1956) 9594:(1953) 9586:(1950) 9578:(1947) 9570:(1944) 9562:(1941) 9554:(1938) 9546:(1935) 9538:(1932) 9530:(1929) 9522:(1926) 9514:(1923) 9506:(1920) 9498:(1917) 9490:(1914) 9482:(1911) 9474:(1908) 9466:(1905) 9458:(1902) 9450:(1899) 9442:(1896) 9434:(1893) 9426:(1890) 9418:(1887) 9410:(1884) 9354:(2000) 9346:(1999) 9338:(1998) 9330:(1997) 9322:(1996) 9314:(1995) 9306:(1994) 9298:(1993) 9290:(1992) 9282:(1991) 9274:(1990) 9266:(1989) 9258:(1988) 9250:(1987) 9242:(1986) 9234:(1985) 9226:(1984) 9218:(1983) 9210:(1982) 9202:(1981) 9194:(1980) 9186:(1979) 9178:(1978) 9170:(1977) 9162:(1976) 9154:(1975) 9146:(1974) 9138:(1973) 9130:(1972) 9122:(1971) 9114:(1970) 9106:(1969) 9098:(1968) 9090:(1967) 9082:(1966) 9074:(1965) 9066:(1964) 9058:(1963) 9050:(1962) 9042:(1961) 9034:(1960) 9026:(1959) 9018:(1958) 9010:(1957) 9002:(1956) 8994:(1955) 8986:(1954) 8978:(1953) 8970:(1952) 8962:(1951) 8905:(2020) 8897:(2015) 8889:(2010) 8881:(2005) 8873:(2000) 8854:(1995) 8846:(1990) 8838:(1985) 8830:(1980) 8822:(1975) 8814:(1970) 8806:(1965) 8798:(1960) 8790:(1955) 8782:(1950) 8774:(1945) 8766:(1940) 8758:(1935) 8750:(1930) 8742:(1925) 8734:(1920) 8726:(1915) 8718:(1913) 8710:(1908) 8702:(1905) 8694:(1900) 8675:(1895) 8667:(1890) 8659:(1885) 8651:(1883) 8643:(1878) 8635:(1873) 8627:(1871) 8619:(1861) 8611:(1858) 8603:(1854) 8595:(1848) 8587:(1838) 8579:(1830) 8571:(1827) 8563:(1820) 8555:(1820) 8536:(1778) 8528:(1772) 8520:(1772) 8512:(1768) 8504:(1768) 8496:(1764) 8488:(1752) 8480:(1741) 8472:(1727) 8464:(1703) 8445:(1698) 8437:(1695) 8429:(1690) 8421:(1689) 8413:(1686) 8405:(1684) 8397:(1683) 8389:(1682) 8381:(1680) 8373:(1677) 8365:(1662) 7890:Birkar 7873:Hairer 7824:Werner 7770:Gowers 7748:Yoccoz 7726:Witten 7667:Connes 7569:Atiyah 7557:Milnor 6917: 6862:12 May 6580: 6556: 6546: 6521: 6483: 6467: 6457: 6417: 6409: 6401: 6367: 6328: 6312: 6304: 6296: 3984: 3976: 3827:Fellow 3813:, the 3809:, the 3805:, the 3801:, the 3797:, the 3793:, the 3789:, the 3785:, the 3781:, the 3777:, the 3773:, the 3769:, the 3765:, the 3761:, the 3757:, the 3749:, the 3605: 3568: 3560: 3519: 3482: 3474: 3426: 3393: 3331: 3296: 3263: 3224: 3185: 3119: 3065: 3032: 2988: 2963: 2953: 2923: 2904: 2894: 2873: 2863: 2826: 2816: 2795: 2785: 2768: 2758: 2741: 2731: 2714: 2704: 2687: 2677: 2656: 2646: 2613: 2592: 2567: 2550: 2458:, and 2356:, and 2096:dipole 1651:. See 1254:on an 1154:1961, 1146:1971, 1142:1989, 1138:1986, 1134:1984, 1122:1983, 945:Oxford 883:, and 764:reader 752:fellow 622:London 611:Master 438:(1955) 436: 427:Thesis 380:Fields 365:(2004) 359:(1988) 353:(1980) 347:(1968) 341:(1966) 335:(1961) 326:Awards 85: 78: 71: 64: 56: 7863:Avila 7743:Lions 7716:Jones 7596:Baker 7584:Smale 7574:Cohen 7523:Serre 7083:1988 7013:35th 6980:42nd 6952:57th 6915:S2CID 6889:arXiv 6872:from 6794:(PDF) 6519:S2CID 6415:S2CID 6302:JSTOR 6146:4 May 6038:(PDF) 6031:(PDF) 5646:arXiv 4882:(PDF) 4875:(PDF) 4691:arXiv 4451:(PDF) 4444:(PDF) 4368:(PDF) 4361:(PDF) 4306:(PDF) 4255:(PDF) 4238:(PDF) 3982:EThOS 3624:arXiv 3566:S2CID 3480:S2CID 3444:(PDF) 3424:JSTOR 3391:JSTOR 3329:JSTOR 3294:JSTOR 3261:JSTOR 3222:JSTOR 3183:JSTOR 3030:S2CID 2528:Books 2450:With 2354:knots 2332:Segal 2251:of a 2237:Schur 2220:used 2212:over 1975:With 1836:Borel 1649:K(BG) 1360:Adams 646:Cairo 155: 151: 147: 143: 139: 90:JSTOR 76:books 7721:Mori 7540:Thom 7535:Roth 7041:4th 6864:2010 6846:2008 6829:2008 6812:2008 6784:2008 6759:2008 6739:2008 6706:2008 6684:2008 6662:2008 6640:2008 6578:ISBN 6544:ISBN 6481:ISBN 6455:ISBN 6431:2013 6399:ISSN 6365:ISBN 6326:ISBN 6294:ISSN 6253:2019 6227:2019 6202:2008 6175:2008 6148:2018 6124:2016 6094:2014 6046:2011 6015:2011 5988:2011 5961:2008 5939:2008 5911:2011 5882:2008 5855:2008 5828:2008 5800:2022 5752:2018 5726:2018 5697:2018 5675:2018 5491:2004 4890:2008 4523:2008 4459:2016 4376:2016 4290:2008 4263:2008 4199:2008 4170:2019 4105:2008 4077:2008 3974:OCLC 3925:2017 3063:ISBN 2986:ISBN 2951:ISBN 2939:2008 2921:ISBN 2892:ISBN 2861:ISBN 2814:ISBN 2783:ISBN 2756:ISBN 2729:ISBN 2702:ISBN 2675:ISBN 2644:ISBN 2454:and 2391:and 2372:and 1834:and 1790:both 1726:The 1645:R(G) 1508:i.e. 1484:any 1390:The 1366:and 1264:flop 1189:and 1177:and 1098:and 911:and 867:and 766:and 726:The 680:and 648:and 203:Died 177:Born 145:FRSE 62:news 7922:Huh 7841:Ngô 7819:Tao 7677:Yau 6907:doi 6606:at 6511:doi 6447:doi 6391:doi 6387:212 6356:doi 6286:doi 6282:137 5775:Mir 4338:doi 3593:doi 3550:doi 3507:doi 3464:doi 3416:doi 3383:doi 3354:doi 3321:doi 3286:doi 3253:doi 3214:doi 3175:doi 3146:doi 3094:doi 3055:doi 3022:doi 2978:doi 2853:doi 2636:doi 2466:on 2334:on 2208:of 2068:of 1794:and 1526:is 1496:all 1071:), 1019:), 1003:), 995:), 987:), 754:at 666:HSC 644:in 605:of 234:PhD 153:FAA 141:FRS 128:Sir 38:to 10039:: 6913:. 6905:. 6897:. 6885:66 6883:. 6804:51 6802:, 6796:, 6770:, 6749:, 6725:, 6626:, 6554:MR 6552:, 6538:, 6534:, 6517:, 6507:34 6505:, 6501:, 6465:MR 6463:, 6453:, 6445:, 6413:, 6407:MR 6405:, 6397:, 6385:, 6379:, 6354:, 6344:15 6342:, 6310:MR 6308:, 6300:, 6292:, 6114:. 6110:. 6084:. 6073:^ 5901:. 5897:. 5872:, 5818:, 5791:. 5742:. 5712:. 4836:^ 4717:^ 4446:. 4334:60 4332:. 4326:. 4280:, 4248:51 4246:, 4240:, 4206:^ 4156:. 4112:^ 4058:^ 4035:. 4016:, 4010:, 4006:, 3994:^ 3980:. 3968:. 3933:^ 3911:. 3897:^ 3632:, 3622:, 3603:MR 3601:, 3591:, 3581:54 3579:, 3573:; 3564:, 3558:MR 3556:, 3548:, 3538:42 3536:, 3517:MR 3515:, 3505:, 3495:28 3493:, 3487:; 3478:, 3472:MR 3470:, 3462:, 3452:19 3450:, 3446:, 3422:, 3412:88 3389:, 3379:86 3350:72 3348:, 3327:, 3317:93 3292:, 3282:93 3259:, 3249:87 3220:, 3210:87 3181:, 3171:87 3169:, 3142:69 3140:, 3117:MR 3088:, 3061:, 3028:, 3016:, 3012:, 2984:, 2961:MR 2959:, 2902:MR 2900:, 2871:MR 2869:, 2859:, 2851:, 2824:MR 2822:, 2812:, 2793:MR 2791:, 2766:MR 2764:, 2739:MR 2737:, 2712:MR 2710:, 2685:MR 2683:, 2654:MR 2652:, 2642:, 2634:, 2611:MR 2590:MR 2588:, 2565:MR 2548:MR 2507:, 2503:, 2485:. 2435:. 2411:. 2380:. 2352:, 2338:. 2005:. 1946:. 1830:. 1712:. 1666:, 1565:: 1559:BG 1506:, 1417:: 1407:BG 1405:, 1346:. 1304:A 1286:. 1215:A 1193:. 1173:, 1169:, 879:, 875:, 863:, 856:. 714:. 707:. 700:. 620:, 556:iː 232:, 230:BA 137:OM 49:. 9804:e 9797:t 9790:v 9388:e 9381:t 9374:v 8939:e 8932:t 8925:v 8336:e 8329:t 8322:v 7998:e 7991:t 7984:v 7460:e 7453:t 7446:v 7120:e 7113:t 7106:v 6921:. 6909:: 6901:: 6891:: 6592:. 6566:. 6540:3 6526:. 6513:: 6449:: 6393:: 6358:: 6350:: 6288:: 6255:. 6229:. 6126:. 6096:. 6048:. 5913:. 5802:. 5754:. 5728:. 5654:. 5648:: 5231:. 4699:. 4693:: 4643:. 4461:. 4378:. 4346:. 4340:: 4172:. 3988:. 3927:. 3636:: 3626:: 3595:: 3587:: 3552:: 3544:: 3509:: 3501:: 3466:: 3458:: 3418:: 3385:: 3356:: 3323:: 3288:: 3255:: 3216:: 3177:: 3148:: 3096:: 3090:3 3057:: 3024:: 3018:9 2995:. 2980:: 2968:. 2909:. 2855:: 2800:. 2773:. 2746:. 2719:. 2692:. 2638:: 2470:2 2424:n 2128:k 2124:k 2120:2 1944:G 1940:G 1936:G 1928:X 1924:G 1882:Y 1878:Y 1874:Y 1870:Y 1698:X 1694:X 1692:( 1690:J 1641:X 1624:. 1621:) 1616:G 1612:X 1608:( 1605:K 1593:) 1589:X 1586:( 1581:G 1577:K 1563:X 1543:G 1539:X 1524:X 1520:X 1516:G 1512:G 1482:G 1465:. 1456:) 1452:G 1449:( 1446:R 1440:) 1437:G 1434:B 1431:( 1428:K 1399:G 1312:. 1067:( 1051:( 1015:( 999:( 979:( 562:/ 559:ə 553:t 550:ˈ 547:ə 544:/ 540:( 236:) 228:( 189:) 185:( 112:) 106:( 101:) 97:( 87:· 80:· 73:· 66:· 43:.

Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.