M. F. Atiyah and F. Hirzebruch :
“Riemann–Roch theorems for differentiable manifolds Bull. Am. Math. Soc.
65 : 4
(1959 ),
pp. 276–281 .
MR
110106 Zbl
0142.40901 article
Abstract
People
BibTeX
The Riemann–Roch Theorem for an algebraic variety \( Y \) (see [Hirzebruch 1956]) led to certain divisibility conditions for the Chern classes of \( Y \) . It was natural to ask whether these conditions held more generally for any compact almost complex manifold. This question, and various generalizations of it, were raised in [Hirzebruch 1954] and most of these have since been answered in the affirmative in [Borel and Hirzebruch 1958] and [Milnor 1960].
More recently Grothendieck has obtained [Borel and Serre 1958] a more general Riemann–Roch Theorem for a map \( f: Y\to X \) of algebraic varieties. This reduces to the previous Riemann–Roch Theorem on taking \( X \) to be a point. Grothendieck’s Theorem implies many conditions on characteristic classes, and again it is natural to ask if these conditions hold more generally for almost complex or even differentiable manifolds. The purpose of this note is to enunciate certain differentiable analogues of Grothendieck’s Theorem. These “differentiable Riemann–Roch Theorems” yield, as special cases, the divisibility conditions mentioned above and also certain new homotopy invariance properties of Pontrjagin classes. As an application of the latter we get a new proof (and slight improvement) of the result of Kervaire–Milnor [1960] on the stable \( J \) -homomorphism.
Another differentiable Riemann–Roch Theorem, with applications to embeddability problems of differentiable manifolds, will be found in [Atiyah and Hirzebruch 1959].
The proofs of our theorems rely heavily on the Bott periodicity of the classical groups [Bott 1957, 1958, 1959], and are altogether different from the earlier methods of [Borel and Hirzebruch 1958] and [Milnor 1960], which were based on Thom’s cobordism theory and Adams’ spectral sequence.
@article {key110106m,
AUTHOR = {Atiyah, M. F. and Hirzebruch, F.},
TITLE = {Riemann--{R}och theorems for differentiable
manifolds},
JOURNAL = {Bull. Am. Math. Soc.},
FJOURNAL = {Bulletin of the American Mathematical
Society},
VOLUME = {65},
NUMBER = {4},
YEAR = {1959},
PAGES = {276--281},
DOI = {10.1090/S0002-9904-1959-10344-X},
NOTE = {MR:110106. Zbl:0142.40901.},
ISSN = {0002-9904},
}
M. F. Atiyah and F. Hirzebruch :
“Quelques théorèmes de non-plongement pour les variétés différentiables Some non-immersion theorems for differentiable manifolds ],
Bull. Soc. Math. France
87
(1959 ),
pp. 383–396 .
MR
114231 Zbl
0196.55903 article
Abstract
People
BibTeX
Nous avons montré dans [Atiyah and Hirzebruch 1959] que le théorème de Riemann–Roch [Borel and Serre 1958] a des analogues différentiables. Un exposé de ces résultats a été fait par l’un des auteurs au Séminaire Bourbaki [Hirzebruch 1958/59]. Les théorèmes de Riemann–Roch différentiables fournissent comme cas particulier certaines conditions de divisibilité pour les classes caractéristiques d’une variété différentiable que l’on peut considérer comme des analogues différentiables du théorème de Riemann–Roch de [Hirzebruch 1956].
La plupart de ces conditions de divisibilité ont été prouvées précédement dans [Borel and Hirzebruch 1958], [Borel and Hirzebruch 1960] et [Milnor 1960]. Dans ce qui suit nous démontrons à l’aide des méthodes de [Atiyah and Hirzebruch 1959] que les classes caractéristiques d’une variété différentiable compacte orientée de dimension \( d \) satisfont aux conditions de divisibilité supplémentaires si la variété peut être différentiablement plongée dans un espace euclidien (ou ce qui est équivalent, une sphère) de dimension \( 2d - q \) . Ces conditions de divisibilité «non stables» nous permettent de prouver des théorèmes de non-plongement qui semblent beaucoup plus forts que ceux qui étaient connus avant (3.6). L’outil essentiel est encore le théorème de Bott [Borel and Hirzebruch 1958/59; Bott 1958] qui dit que la \( n \) -ième classe de Chern d’un fibré vectoriel complexe sur la sphère \( S_{2n} \) est divisible par \( (n-1)! \) .
@article {key114231m,
AUTHOR = {Atiyah, Michael F. and Hirzebruch, Friedrich},
TITLE = {Quelques th\'eor\`emes de non-plongement
pour les vari\'et\'es diff\'erentiables
[Some non-immersion theorems for differentiable
manifolds]},
JOURNAL = {Bull. Soc. Math. France},
FJOURNAL = {Bulletin de la Soci\'et\'e Math\'ematique
de France},
VOLUME = {87},
YEAR = {1959},
PAGES = {383--396},
URL = {http://www.numdam.org/item?id=BSMF_1959__87__383_0},
NOTE = {MR:114231. Zbl:0196.55903.},
ISSN = {0037-9484},
}
M. F. Atiyah and F. Hirzebruch :
“Quelques théoremes de non-plongement pour les variétés différentiables ”
[Some non-embedding theorems for differentiable manifolds ],
Colloques Int. Centre Nat. Rech. Sci.
89
(1960 ),
pp. 383–396 .
See also Bull. Soc. Math. France 87 (1959)
Zbl
0108.18202 article
Abstract
People
BibTeX
Nous avons montré dans [Atiyah and Hirzebruch 1959] que le théorème de Riemann–Roch [Borel and Serre 1958] a des analogues différentiables. Un exposé de ces résultats a été fait par l’un des auteurs au Séminaire Bourbaki [Hirzebruch, 1958/59]. Les théorèmes de Riemann–Roch différentiables fournissent comme cas particulier certaines conditions de divisibilité pour les classes caractéristiques d’une variété différentiable que l’on peut considérer comme des analogues différentiables du théorème de Riemann–Roch de [Hirzebruch 1956].
La plupart de ces conditions de divisibilité ont été prouvées précédement dans [Borel and Hirzebruch 1958; 1960] et [Milnor 1960]. Dans ce qui suit nous démontrons à l’aide des méthodes de [Atiyah and Hirzebruch 1959] que les classes caractéristiques d’une variété différentiable compacte orientée de dimension \( d \) satisfont aux conditions de divisibilité supplémentaires si la variété peut être différentiablement plongée dans un espance euclidien (ou ce qui est équivalent, une spère) de dimension \( 2d - q \) . Ces conditions de divisibilité «non stables» nous permettent de prouver des théorèmes de non-plongement qui semblent beaucoup plus forts que ceux qui ’\etaient connus avant (3.6). L’outil essentiel est encore le théorème de Bott ([Borel and Hirzebruch, 1958/59] et [Bott 1958]) qui dit que la \( n \) -ième classe de Chern d’un fibré vectoriel complexe sur la sphère \( S_{2n} \) est divisible par \( (n-1)! \) .
@article {key0108.18202z,
AUTHOR = {Atiyah, Michael F. and Hirzebruch, Friedrich},
TITLE = {Quelques th\'eoremes de non-plongement
pour les vari\'et\'es diff\'erentiables
[Some non-embedding theorems for differentiable
manifolds]},
JOURNAL = {Colloques Int. Centre Nat. Rech. Sci.},
FJOURNAL = {Colloques Internationaux du Centre National
de la Recherche Scientifique},
VOLUME = {89},
YEAR = {1960},
PAGES = {383--396},
NOTE = {See also \textit{Bull. Soc. Math. France}
\textbf{87} (1959). Zbl:0108.18202.},
ISSN = {0366-7634},
}
M. F. Atiyah and F. Hirzebruch :
“Bott periodicity and the parallelizability of the spheres Proc. Camb. Philos. Soc.
57 : 2
(April 1961 ),
pp. 223–226 .
MR
126282 Zbl
0108.35902 article
Abstract
People
BibTeX
The theorems of Bott [1958, 1959a] on the stable homotopy of the classical groups imply that the sphere \( S^n \) is not parallelizable for \( n \neq 1 \) , \( 3, 7 \) . This was shown independently by Kervaire [1958] and Milnor [1958; Bott and Milnor 1958]. Another proof can be found in [Borel and Hirzebruch 1959, §26.11]. The work of J. F. Adams (on the non-existence of elements of Hopf invariant one) implies more strongly that \( S^n \) with any (perhaps extraordinary) differentiable structure is not parallelizable if \( n \neq 1 \) , \( 3, 7 \) . Thus there exist already four proofs for the non-parallelizability of the spheres, the first three mentioned relying on the Bott theory, as given in [Bott 1958, 1959a]. The purpose of this note is to show how the refined form of Bott’s results given in [Bott 1959b] leads to a very simple proof of the non-parallelizability (only for the usual differentiable structures of the spheres). We shall prove in fact the folowing theorem due to Milnor [1958] which implies the non-parallelizability.
There exists a real vector bundle \( \xi \) over the sphere \( S^n \) with \( w_n(\xi) \neq 0 \) only for \( n = 1 \) , \( 2,4 \) or 8
@article {key126282m,
AUTHOR = {Atiyah, M. F. and Hirzebruch, F.},
TITLE = {Bott periodicity and the parallelizability
of the spheres},
JOURNAL = {Proc. Camb. Philos. Soc.},
FJOURNAL = {Proceedings of the Cambridge Philosophical
Society},
VOLUME = {57},
NUMBER = {2},
MONTH = {April},
YEAR = {1961},
PAGES = {223--226},
DOI = {10.1017/S0305004100035088},
NOTE = {MR:126282. Zbl:0108.35902.},
ISSN = {0305-0041},
}
M. F. Atiyah and F. Hirzebruch :
“Vector bundles and homogeneous spaces ,”
pp. 7–38
in
Differential geometry
(Tucson, AZ, 18–19 February 1960 ).
Edited by C. B. Allendoerfer .
Proceedings of Symposia in Pure Mathematics 3 .
American Mathematical Society (Providence, RI ),
1961 .
Republished in Algebraic topology: A student’s guide (1972)
MR
139181 Zbl
0108.17705 incollection
People
BibTeX
@incollection {key139181m,
AUTHOR = {Atiyah, M. F. and Hirzebruch, F.},
TITLE = {Vector bundles and homogeneous spaces},
BOOKTITLE = {Differential geometry},
EDITOR = {Allendoerfer, Carl Barnett},
SERIES = {Proceedings of Symposia in Pure Mathematics},
NUMBER = {3},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {1961},
PAGES = {7--38},
NOTE = {(Tucson, AZ, 18--19 February 1960).
Republished in \textit{Algebraic topology:
A student's guide} (1972). MR:139181.
Zbl:0108.17705.},
ISSN = {1098-3627},
}
M. F. Atiyah and F. Hirzebruch :
“Charakteristische Klassen und Anwendungen Characteristic classes and applications ],
Enseignement Math. (2)
7 : 1
(1961 ),
pp. 188–213 .
MR
154294 Zbl
0104.39801 article
People
BibTeX
@article {key154294m,
AUTHOR = {Atiyah, M. F. and Hirzebruch, F.},
TITLE = {Charakteristische {K}lassen und {A}nwendungen
[Characteristic classes and applications]},
JOURNAL = {Enseignement Math. (2)},
FJOURNAL = {L'Enseignement Math\'ematique. Revue
Internationale. IIe S\'erie},
VOLUME = {7},
NUMBER = {1},
YEAR = {1961},
PAGES = {188--213},
DOI = {10.5169/seals-37131},
NOTE = {MR:154294. Zbl:0104.39801.},
ISSN = {0013-8584},
}
M. F. Atiyah and F. Hirzebruch :
“Cohomologie-Operationen und charakteristische Klassen Cohomology operations and characteristic classes ],
Math. Z.
77 : 1
(1961 ),
pp. 149–187 .
Dedicated to Friedrich Karl Schmidt on the occasion of his sixtieth birthday.
MR
156361 Zbl
0109.16002 article
People
BibTeX
@article {key156361m,
AUTHOR = {Atiyah, M. F. and Hirzebruch, F.},
TITLE = {Cohomologie-{O}perationen und charakteristische
{K}lassen [Cohomology operations and
characteristic classes]},
JOURNAL = {Math. Z.},
FJOURNAL = {Mathematische Zeitschrift},
VOLUME = {77},
NUMBER = {1},
YEAR = {1961},
PAGES = {149--187},
DOI = {10.1007/BF01180171},
NOTE = {Dedicated to Friedrich Karl Schmidt
on the occasion of his sixtieth birthday.
MR:156361. Zbl:0109.16002.},
ISSN = {0025-5874},
}
M. F. Atiyah and F. Hirzebruch :
“Analytic cycles on complex manifolds Topology
1 : 1
(January–March 1962 ),
pp. 25–45 .
MR
145560 Zbl
0108.36401 article
Abstract
People
BibTeX
Let \( X \) be a complex manifold, \( Y \) a closed irreducible \( k \) -dimensional complex analytic subspace of \( X \) . Then \( Y \) defines or “carries” a \( 2k \) -dimensional integral homology class \( y \) of \( X \) , although the precise definition of \( y \) presents technical difficu1ties. A finite formal linear combination \( \sum n_iY_i \) with \( n_i \) integers and \( Y_i \) as above is called a complex analytic cycle, and the corresponding homology class \( \sum n_iY_i \) is called a complex analytic homology class. If an integral cohomology class \( u \) corresponds under Poincaré duality to a complex analytic homology class we shall say that \( u \) is a complex analytic cohomology class . The purpose of this paper is to show that a complex analytic cohomology class \( u \) satisfies certain topological conditions, independent of the complex structure of \( X \) . These conditions are that certain cohomology operations should vanish on \( u \) , for example \( \mathrm{Sq}^3u = 0 \) : they are all torsion conditions. We also produce examples to show that these conditions are not vacuous even in the restricted classes of (a) Stein manifolds and (b) projective algebraic manifolds.
@article {key145560m,
AUTHOR = {Atiyah, M. F. and Hirzebruch, F.},
TITLE = {Analytic cycles on complex manifolds},
JOURNAL = {Topology},
FJOURNAL = {Topology. An International Journal of
Mathematics},
VOLUME = {1},
NUMBER = {1},
MONTH = {January--March},
YEAR = {1962},
PAGES = {25--45},
DOI = {10.1016/0040-9383(62)90094-0},
NOTE = {MR:145560. Zbl:0108.36401.},
ISSN = {0040-9383},
}
M. F. Atiyah and F. Hirzebruch :
“The Riemann–Roch theorem for analytic embeddings Topology
1 : 2
(April–June 1962 ),
pp. 151–166 .
MR
148084 Zbl
0108.36402 article
Abstract
People
BibTeX
In [Borel and Serre 1958] Grothendieck formulated and proved a generalization of the Riemann–Roch theorem which we shall refer to as GRR. This theorem is concerned with a proper morphism \( f:Y\to X \) of algebraic manifolds (any ground field) and reduces to the version (HRR) given in [Hirzebruch 1956] when \( X \) is a point (and the ground field is \( \mathbb{C} \) ). It is not known whether GRR or even HRR holds for arbitrary complex manifolds. However the proof of GRR given in [Borel and Serre 1958] breaks up into two separate cases:
\( f \) is an embedding,\( f \) is a projection \( X\times P_N \to X \) , where \( P_N \) is a projective space,
and the main purpose of this paper is to give a proof of GRR in case (i) for arbitrary complex manifolds. This proof is quite different from, and in many ways simpler than, that of [Borel and Serre 1958] and, for the complex algebraic case, it gives a new proof of GRR.
@article {key148084m,
AUTHOR = {Atiyah, M. F. and Hirzebruch, F.},
TITLE = {The {R}iemann--{R}och theorem for analytic
embeddings},
JOURNAL = {Topology},
FJOURNAL = {Topology. An International Journal of
Mathematics},
VOLUME = {1},
NUMBER = {2},
MONTH = {April--June},
YEAR = {1962},
PAGES = {151--166},
DOI = {10.1016/0040-9383(65)90023-6},
NOTE = {MR:148084. Zbl:0108.36402.},
ISSN = {0040-9383},
}
M. Atiyah and F. Hirzebruch :
“Spin-manifolds and group actions 18–28
in
Essays on topology and related topics (Mémoires dédiés à Georges de Rham)
[Essays on topology and related topics (Memoirs dedicated to Georges de Rham) ]
(Geneva, 26–28 March 1969 ).
Edited by R. Narasimhan and A. Haefliger .
Springer (New York ),
1970 .
MR
278334 Zbl
0193.52401 incollection
People
BibTeX
@incollection {key278334m,
AUTHOR = {Atiyah, Michael and Hirzebruch, Friedrich},
TITLE = {Spin-manifolds and group actions},
BOOKTITLE = {Essays on topology and related topics
({M}\'emoires d\'edi\'es \`a {G}eorges
de {R}ham) [Essays on topology and related
topics ({M}emoirs dedicated to {G}eorges
de {R}ham)]},
EDITOR = {Narasimhan, Raghavan and Haefliger,
Andre},
PUBLISHER = {Springer},
ADDRESS = {New York},
YEAR = {1970},
PAGES = {18--28},
DOI = {10.1007/978-3-642-49197-9_3},
NOTE = {(Geneva, 26--28 March 1969). MR:278334.
Zbl:0193.52401.},
ISBN = {9783642491993},
}
M. Atiyah :
“Eigenvalues of the Dirac operator 251–260
in
Arbeitstagung Bonn 1984
(Max-Planck-Institut für Mathematik, Bonn, 15–22 June 1984 ).
Edited by F. Hirzebruch, J. Schwermer, and S. Suter .
Lecture Notes in Mathematics 1111 .
Springer (Berlin ),
1985 .
MR
797424 Zbl
0568.53022 incollection
Abstract
People
BibTeX
In recent years mathematicians have learnt a great deal from physicists and in particular from the work of Edward Witten. In a recent preprint [1984], Vafa and Witten have proved some striking results about the eigenvalues of the Dirac operator, and this talk will present their results. I shall concentrate entirely on the mathematical parts of their preprint leaving aside the physical interpretation which is their main motivation.
@incollection {key797424m,
AUTHOR = {Atiyah, Michael},
TITLE = {Eigenvalues of the {D}irac operator},
BOOKTITLE = {Arbeitstagung {B}onn 1984},
EDITOR = {Friedrich Hirzebruch and Joachim Schwermer
and Silke Suter},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {1111},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1985},
PAGES = {251--260},
DOI = {10.1007/BFb0084593},
NOTE = {(Max-Planck-Institut f\"ur Mathematik,
Bonn, 15--22 June 1984). MR:797424.
Zbl:0568.53022.},
ISSN = {0075-8434},
ISBN = {9780387151953},
}
M. Atiyah :
“Commentary on the article of Manin 103–109
in
Arbeitstagung Bonn 1984
(Max-Planck-Institut für Mathematik, Bonn, 15–22 June 1984 ).
Edited by F. Hirzebruch, J. Schwermer, and S. Suter .
Lecture Notes in Mathematics 1111 .
Springer (Berlin ),
1985 .
The article is Yu. I. Manin, “New dimensions in geometry,” from the same volume.
MR
797417 Zbl
0595.53071 incollection
Abstract
People
BibTeX
Manin’s stimulating contribution to the 25th Arbeitstagung which, in his absence, I attempted to present, provided me with an opportunity of adding some further reflections of my own. This commentary, which is therefore a very personal response to Manin’s article, consists of very general and speculative remarks about large areas of contemporary mathematics. Such speculations are, for good reason, rarely put down on paper but the record of the 25th Arbeitstagung provides a rather singular occasion where ideas of this type may not be out of place.
@incollection {key797417m,
AUTHOR = {Atiyah, Michael},
TITLE = {Commentary on the article of {M}anin},
BOOKTITLE = {Arbeitstagung {B}onn 1984},
EDITOR = {Friedrich Hirzebruch and Joachim Schwermer
and Silke Suter},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {1111},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1985},
PAGES = {103--109},
DOI = {10.1007/BFb0084586},
NOTE = {(Max-Planck-Institut f\"ur Mathematik,
Bonn, 15--22 June 1984). The article
is {Y}u.\ {I}. {M}anin, ``{N}ew dimensions
in geometry,'' from the same volume.
MR:797417. Zbl:0595.53071.},
ISSN = {0075-8434},
ISBN = {9780387151953},
}
M. Atiyah :
“The European Mathematical Society ,”
pp. 1–5
in
Miscellanea mathematica .
Edited by P. J. Hilton, F. Hirzebruch, and R. Remmert .
Springer (Berlin ),
1991 .
MR
1131114 incollection
People
BibTeX
@incollection {key1131114m,
AUTHOR = {Atiyah, Michael},
TITLE = {The {E}uropean {M}athematical {S}ociety},
BOOKTITLE = {Miscellanea mathematica},
EDITOR = {Peter John Hilton and Friedrich Hirzebruch
and Reinhold Remmert},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1991},
PAGES = {1--5},
NOTE = {MR:1131114.},
ISBN = {9783540541745},
}
M. Atiyah :
“Friedrich Hirzebruch: An appreciation ,”
pp. 1–5
in
Proceedings of the Hirzebruch 65 conference on algebraic geometry
(Ramat Gan, Israel, 2–7 May 1993 ).
Edited by M. Teicher .
Israel Mathematical Conference Proceedings 9 .
Bar-Ilan University (Ramat Gan, Israel ),
1996 .
Zbl
0834.01012 incollection
People
BibTeX
@incollection {key0834.01012z,
AUTHOR = {Atiyah, M.},
TITLE = {Friedrich {H}irzebruch: {A}n appreciation},
BOOKTITLE = {Proceedings of the {H}irzebruch 65 conference
on algebraic geometry},
EDITOR = {Teicher, Mina},
SERIES = {Israel Mathematical Conference Proceedings},
NUMBER = {9},
PUBLISHER = {Bar-Ilan University},
ADDRESS = {Ramat Gan, Israel},
YEAR = {1996},
PAGES = {1--5},
NOTE = {(Ramat Gan, Israel, 2--7 May 1993).
Zbl:0834.01012.},
ISSN = {0792-4119},
ISBN = {9789996281068},
}
M. Atiyah :
“Physics and geometry: A look at the last twenty years ,”
pp. 1–8
in
Algebraic geometry: Hirzebruch 70
(Warsaw, 11–16 May 1998 ).
Edited by P. Pragacz, M. Szurek, and J. Wiśniewski .
Contemporary Mathematics 241 .
American Mathematical Society (Providence, RI ),
1999 .
MR
1718133 Zbl
0945.14026 incollection
Abstract
People
BibTeX
These are notes from the special lecture given by Professor Michael Atiyah during the “Algebraic Geometry Conference: Hirzebruch 70.” The text concerns the interactions between Physics and Geometry in the last two decades, and the role of Professor F. Hirzebruch and his Bonn “Arbeitstagung” in these interactions.
@incollection {key1718133m,
AUTHOR = {Atiyah, Michael},
TITLE = {Physics and geometry: {A} look at the
last twenty years},
BOOKTITLE = {Algebraic geometry: {H}irzebruch 70},
EDITOR = {Pragacz, Piotr and Szurek, Micha{\l}
and Wi\'sniewski, Jaros\l aw},
SERIES = {Contemporary Mathematics},
NUMBER = {241},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {1999},
PAGES = {1--8},
NOTE = {(Warsaw, 11--16 May 1998). MR:1718133.
Zbl:0945.14026.},
ISSN = {0271-4132},
ISBN = {9780821811498},
}
Papers dedicated to Atiyah, Bott, Hirzebruch, and Singer .
Edited by S.-T. Yau .
Surveys in Differential Geometry (supplement to Journal of Differential Geometry) 7 .
International Press (Somerville, MA ),
2000 .
MR
1919419 book
People
BibTeX
@book {key1919419m,
TITLE = {Papers dedicated to {A}tiyah, {B}ott,
{H}irzebruch, and {S}inger},
EDITOR = {Shing-Tung Yau},
SERIES = {Surveys in Differential Geometry (supplement
to {J}ournal of {D}ifferential {G}eometry)},
NUMBER = {7},
PUBLISHER = {International Press},
ADDRESS = {Somerville, MA},
YEAR = {2000},
PAGES = {696},
NOTE = {MR:1919419.},
ISSN = {1052-9233},
ISBN = {9781571461780},
}
M. Atiyah :
“\( K \) -theory past and present411–417
in
Sitzungsberichte der Berliner Mathematischen Gesellschaft, 1997–2000 .
Berliner Mathematischen Gesellschaft ,
2001 .
MR
2091892 Zbl
1061.19500 ArXiv
math/0012213 incollection
Abstract
People
BibTeX
@incollection {key2091892m,
AUTHOR = {Atiyah, Michael},
TITLE = {\$K\$-theory past and present},
BOOKTITLE = {Sitzungsberichte der {B}erliner {M}athematischen
{G}esellschaft, 1997--2000},
PUBLISHER = {Berliner Mathematischen Gesellschaft},
YEAR = {2001},
PAGES = {411--417},
NOTE = {ArXiv:math/0012213. MR:2091892. Zbl:1061.19500.},
}
E. Witten :
“Michael Atiyah and the physics/geometry interface ,”
pp. 43–50
in
The founders of index theory: Reminiscences of and about Sir Michael Atiyah, Raoul Bott, Friedrich Hirzebruch, and I. M. Singer .
Edited by S.-T. Yau .
International Press (Somerville, MA ),
2003 .
Republished from Asian J. Math. 3 :1 (1999)2009 edition of the book.
incollection
People
BibTeX
@incollection {key91870136,
AUTHOR = {Witten, Edward},
TITLE = {Michael {A}tiyah and the physics/geometry
interface},
BOOKTITLE = {The founders of index theory: {R}eminiscences
of and about {S}ir {M}ichael {A}tiyah,
{R}aoul {B}ott, {F}riedrich {H}irzebruch,
and {I}.~{M}. {S}inger},
EDITOR = {Yau, Shing-Tung},
PUBLISHER = {International Press},
ADDRESS = {Somerville, MA},
YEAR = {2003},
PAGES = {43--50},
NOTE = {Republished from \textit{Asian J. Math.}
\textbf{3}:1 (1999). Also in the 2009
edition of the book.},
ISBN = {9781571461209},
}
N. Hitchin :
“Sir Michael Atiyah: A brief biography ,”
pp. 15–22
in
The founders of index theory: Reminiscences of and about Sir Michael Atiyah, Raoul Bott, Friedrich Hirzebruch, and I. M. Singer .
Edited by S.-T. Yau .
International Press (Somerville, MA ),
2003 .
Also in the 2009 edition of the book.
incollection
People
BibTeX
@incollection {key29362485,
AUTHOR = {Hitchin, Nigel},
TITLE = {Sir {M}ichael {A}tiyah: {A} brief biography},
BOOKTITLE = {The founders of index theory: {R}eminiscences
of and about {S}ir {M}ichael {A}tiyah,
{R}aoul {B}ott, {F}riedrich {H}irzebruch,
and {I}.~{M}. {S}inger},
EDITOR = {Yau, Shing-Tung},
PUBLISHER = {International Press},
ADDRESS = {Somerville, MA},
YEAR = {2003},
PAGES = {15--22},
NOTE = {Also in the 2009 edition of the book.},
ISBN = {9781571461209},
}
M. Atiyah :
“The conscience of science ,”
pp. 51–66
in
The founders of index theory: Reminiscences of and about Sir Michael Atiyah, Raoul Bott, Friedrich Hirzebruch, and I. M. Singer .
Edited by S.-T. Yau .
International Press (Somerville, MA ),
2003 .
Also in the 2009 edition of the book.
incollection
People
BibTeX
@incollection {key73905347,
AUTHOR = {Atiyah, Michael},
TITLE = {The conscience of science},
BOOKTITLE = {The founders of index theory: {R}eminiscences
of and about {S}ir {M}ichael {A}tiyah,
{R}aoul {B}ott, {F}riedrich {H}irzebruch,
and {I}.~{M}. {S}inger},
EDITOR = {Yau, Shing-Tung},
PUBLISHER = {International Press},
ADDRESS = {Somerville, MA},
YEAR = {2003},
PAGES = {51--66},
NOTE = {Also in the 2009 edition of the book.},
ISBN = {9781571461209},
}
G. Lusztig :
“Recollections about my teacher Michael Atiyah ,”
pp. 37–42
in
The founders of index theory: Reminiscences of and about Sir Michael Atiyah, Raoul Bott, Friedrich Hirzebruch, and I. M. Singer .
Edited by S.-T. Yau .
International Press (Somerville, MA ),
2003 .
Republished from Asian J. Math. 3 :1 (1999)2009 edition of the book.
incollection
People
BibTeX
@incollection {key99039880,
AUTHOR = {Lusztig, George},
TITLE = {Recollections about my teacher {M}ichael
{A}tiyah},
BOOKTITLE = {The founders of index theory: {R}eminiscences
of and about {S}ir {M}ichael {A}tiyah,
{R}aoul {B}ott, {F}riedrich {H}irzebruch,
and {I}.~{M}. {S}inger},
EDITOR = {Yau, Shing-Tung},
PUBLISHER = {International Press},
ADDRESS = {Somerville, MA},
YEAR = {2003},
PAGES = {37--42},
NOTE = {Republished from \textit{Asian J. Math.}
\textbf{3}:1 (1999). Also in the 2009
edition of the book.},
ISBN = {9781571461209},
}
P. Goddard :
“Sir Michael Atiyah and the early days of the Newton Institute ,”
pp. 23–28
in
The founders of index theory: Reminiscences of and about Sir Michael Atiyah, Raoul Bott, Friedrich Hirzebruch, and I. M. Singer .
Edited by S.-T. Yau .
International Press (Somerville, MA ),
2003 .
Also in the 2009 edition of the book.
incollection
People
BibTeX
@incollection {key56102705,
AUTHOR = {Goddard, P.},
TITLE = {Sir {M}ichael {A}tiyah and the early
days of the {N}ewton {I}nstitute},
BOOKTITLE = {The founders of index theory: {R}eminiscences
of and about {S}ir {M}ichael {A}tiyah,
{R}aoul {B}ott, {F}riedrich {H}irzebruch,
and {I}.~{M}. {S}inger},
EDITOR = {Yau, Shing-Tung},
PUBLISHER = {International Press},
ADDRESS = {Somerville, MA},
YEAR = {2003},
PAGES = {23--28},
NOTE = {Also in the 2009 edition of the book.},
ISBN = {9781571461209},
}
The founders of index theory: Reminiscences of Atiyah, Bott, Hirzebruch, and Singer .
Edited by S.-T. Yau .
International Press (Somerville, MA ),
2003 .
Republished in 2009 .
MR
2136846 Zbl
1072.01021 book
People
BibTeX
@book {key2136846m,
TITLE = {The founders of index theory: {R}eminiscences
of {A}tiyah, {B}ott, {H}irzebruch, and
{S}inger},
EDITOR = {Yau, S.-T.},
PUBLISHER = {International Press},
ADDRESS = {Somerville, MA},
YEAR = {2003},
PAGES = {liv+358},
NOTE = {Republished in 2009. MR:2136846. Zbl:1072.01021.},
ISBN = {9781571461209},
}
M. F. Atiyah :
“A personal history ,”
pp. 5–15
in
The founders of index theory: Reminiscences of and about Sir Michael Atiyah, Raoul Bott, Friedrich Hirzebruch, and I. M. Singer ,
2nd edition.
Edited by S.-T. Yau .
International Press (Somerville, MA ),
2009 .
Republished in Atiyah’s Collected works , vol. 6 .
incollection
People
BibTeX
@incollection {key72270854,
AUTHOR = {Atiyah, M. F.},
TITLE = {A personal history},
BOOKTITLE = {The founders of index theory: {R}eminiscences
of and about {S}ir {M}ichael {A}tiyah,
{R}aoul {B}ott, {F}riedrich {H}irzebruch,
and {I}.~{M}. {S}inger},
EDITOR = {Yau, Shing-Tung},
EDITION = {2nd},
PUBLISHER = {International Press},
ADDRESS = {Somerville, MA},
YEAR = {2009},
PAGES = {5--15},
NOTE = {Republished in Atiyah's \textit{Collected
works}, vol.~6.},
ISBN = {9781571461377},
}
S. Donaldson :
“Geometry in Oxford c.1980–85 ,”
pp. 35–42
in
The founders of index theory: Reminiscences of and about Sir Michael Atiyah, Raoul Bott, Friedrich Hirzebruch, and I. M. Singer ,
2nd edition.
Edited by S.-T. Yau .
International Press (Somerville, MA ),
2009 .
Republished from Asian J. Math. 3 :1 (1999)2003 edition of the book.
incollection
Abstract
People
BibTeX
The first part of this essay comprises some brief reminiscences from my time as a research student of Sir Michael Atiyah: these will be commonplace to my contemporaries, but perhaps younger mathematicians may be less familiar with the research interests of this period. In the second part of the essay I will discuss some current research questions.
@incollection {key12598427,
AUTHOR = {Donaldson, Simon},
TITLE = {Geometry in {O}xford c.1980--85},
BOOKTITLE = {The founders of index theory: {R}eminiscences
of and about {S}ir {M}ichael {A}tiyah,
{R}aoul {B}ott, {F}riedrich {H}irzebruch,
and {I}.~{M}. {S}inger},
EDITOR = {Yau, Shing-Tung},
EDITION = {2nd},
PUBLISHER = {International Press},
ADDRESS = {Somerville, MA},
YEAR = {2009},
PAGES = {35--42},
NOTE = {Republished from \textit{Asian J. Math.}
\textbf{3}:1 (1999). Also in the 2003
edition of the book.},
ISBN = {9781571461377},
}
L. Nirenberg :
“Memories of Michael Atiyah ,”
pp. 45–46
in
The founders of index theory: Reminiscences of and about Sir Michael Atiyah, Raoul Bott, Friedrich Hirzebruch, and I. M. Singer ,
2nd edition.
Edited by S.-T. Yau .
International Press (Somerville, MA ),
2009 .
incollection
People
BibTeX
@incollection {key31967768,
AUTHOR = {Nirenberg, Louis},
TITLE = {Memories of {M}ichael {A}tiyah},
BOOKTITLE = {The founders of index theory: {R}eminiscences
of and about {S}ir {M}ichael {A}tiyah,
{R}aoul {B}ott, {F}riedrich {H}irzebruch,
and {I}.~{M}. {S}inger},
EDITOR = {Yau, Shing-Tung},
EDITION = {2nd},
PUBLISHER = {International Press},
ADDRESS = {Somerville, MA},
YEAR = {2009},
PAGES = {45--46},
ISBN = {9781571461377},
}
M. Atiyah :
“Memories of Raoul Bott ,”
pp. 89–94
in
The founders of index theory: Reminiscences of Atiyah, Bott, Hirzebruch, and Singer .
Edited by S.-T. Yau .
International Press (Somerville, MA ),
2009 .
incollection
People
BibTeX
@incollection {key53064164,
AUTHOR = {Atiyah, Michael},
TITLE = {Memories of {R}aoul {B}ott},
BOOKTITLE = {The founders of index theory: {R}eminiscences
of {A}tiyah, {B}ott, {H}irzebruch, and
{S}inger},
EDITOR = {Yau, S.-T.},
PUBLISHER = {International Press},
ADDRESS = {Somerville, MA},
YEAR = {2009},
PAGES = {89--94},
ISBN = {9781571461377},
}
M. Atiyah :
“Letter to Raoul, Fritz, and Is ,”
pp. 323–324
in
The founders of index theory: Reminiscences of and about Sir Michael Atiyah, Raoul Bott, Friedrich Hirzebruch, and I. M. Singer ,
2nd edition.
Edited by S.-T. Yau .
International Press (Somerville, MA ),
2009 .
Also found in the 2003 edition of the book.
incollection
People
BibTeX
@incollection {key46159632,
AUTHOR = {Atiyah, Michael},
TITLE = {Letter to {R}aoul, {F}ritz, and {I}s},
BOOKTITLE = {The founders of index theory: {R}eminiscences
of and about {S}ir {M}ichael {A}tiyah,
{R}aoul {B}ott, {F}riedrich {H}irzebruch,
and {I}.~{M}. {S}inger},
EDITOR = {Yau, Shing-Tung},
EDITION = {2nd},
PUBLISHER = {International Press},
ADDRESS = {Somerville, MA},
YEAR = {2009},
PAGES = {323--324},
NOTE = {Also found in the 2003 edition of the
book.},
ISBN = {9781571461377},
}
R. Bott :
“A letter from Raoul Bott ,”
pp. 33–34
in
The founders of index theory: Reminiscences of and about Sir Michael Atiyah, Raoul Bott, Friedrich Hirzebruch, and I. M. Singer ,
2nd edition.
Edited by S.-T. Yau .
International Press (Somerville, MA ),
2009 .
Also in the 2003 edition of the book.
incollection
People
BibTeX
@incollection {key40178879,
AUTHOR = {Bott, Raoul},
TITLE = {A letter from {R}aoul {B}ott},
BOOKTITLE = {The founders of index theory: {R}eminiscences
of and about {S}ir {M}ichael {A}tiyah,
{R}aoul {B}ott, {F}riedrich {H}irzebruch,
and {I}.~{M}. {S}inger},
EDITOR = {Yau, Shing-Tung},
EDITION = {2nd},
PUBLISHER = {International Press},
ADDRESS = {Somerville, MA},
YEAR = {2009},
PAGES = {33--34},
NOTE = {Also in the 2003 edition of the book.},
ISBN = {9781571461377},
}
G. Segal :
“Being a graduate student of Michael Atiyah ,”
pp. 47–48
in
The founders of index theory: Reminiscences of and about Sir Michael Atiyah, Raoul Bott, Friedrich Hirzebruch, and I. M. Singer ,
2nd edition.
Edited by S.-T. Yau .
International Press (Somerville, MA ),
2009 .
Also in the 2003 edition of the book.
incollection
People
BibTeX
@incollection {key94049467,
AUTHOR = {Segal, Graeme},
TITLE = {Being a graduate student of {M}ichael
{A}tiyah},
BOOKTITLE = {The founders of index theory: {R}eminiscences
of and about {S}ir {M}ichael {A}tiyah,
{R}aoul {B}ott, {F}riedrich {H}irzebruch,
and {I}.~{M}. {S}inger},
EDITOR = {Yau, Shing-Tung},
EDITION = {2nd},
PUBLISHER = {International Press},
ADDRESS = {Somerville, MA},
YEAR = {2009},
PAGES = {47--48},
NOTE = {Also in the 2003 edition of the book.},
ISBN = {9781571461377},
}
M. Atiyah :
“Mathematics: Queen and servant of the sciences ,”
pp. 55–76
in
The founders of index theory: Reminiscences of and about Sir Michael Atiyah, Raoul Bott, Friedrich Hirzebruch, and I. M. Singer ,
2nd edition.
Edited by S.-T. Yau .
International Press (Somerville, MA ),
2009 .
Republished from Proc. Am. Phil. Soc. 137 :4 (1993)Asian J. Math. 3 :1 (1999)
incollection
People
BibTeX
@incollection {key74724142,
AUTHOR = {Atiyah, Michael},
TITLE = {Mathematics: {Q}ueen and servant of
the sciences},
BOOKTITLE = {The founders of index theory: {R}eminiscences
of and about {S}ir {M}ichael {A}tiyah,
{R}aoul {B}ott, {F}riedrich {H}irzebruch,
and {I}.~{M}. {S}inger},
EDITOR = {Yau, Shing-Tung},
EDITION = {2nd},
PUBLISHER = {International Press},
ADDRESS = {Somerville, MA},
YEAR = {2009},
PAGES = {55--76},
NOTE = {Republished from \textit{Proc. Am. Phil.
Soc.} \textbf{137}:4 (1993). See also
\textit{Asian J. Math.} \textbf{3}:1
(1999).},
ISBN = {9781571461377},
}
The founders of index theory: Reminiscences of and about Sir Michael Atiyah, Raoul Bott, Friedrich Hirzebruch, and I. M. Singer ,
2nd edition.
Edited by S.-T. Yau .
International Press (Somerville, MA ),
2009 .
Republication of 2003 original .
MR
2547480 Zbl
1195.01090 book
People
BibTeX
@book {key2547480m,
TITLE = {The founders of index theory: {R}eminiscences
of and about {S}ir {M}ichael {A}tiyah,
{R}aoul {B}ott, {F}riedrich {H}irzebruch,
and {I}.~{M}. Singer},
EDITOR = {Yau, S.-T.},
EDITION = {2nd},
PUBLISHER = {International Press},
ADDRESS = {Somerville, MA},
YEAR = {2009},
PAGES = {lii+393},
NOTE = {Republication of 2003 original. MR:2547480.
Zbl:1195.01090.},
ISBN = {9781571461377},
}
