M. F. Atiyah and I. M. Singer :
“The index of elliptic operators on compact manifolds Bull. Amer. Math. Soc.
69 : 3
(1963 ),
pp. 422–433 .
MR
157392 Zbl
0118.31203 article
Abstract
People
BibTeX
In [1960] Gel’fand posed the general problem of investigating the relationship between topological and analytical invariants of elliptic differential operators. In particular he suggested that it should be possible to express the index of an elliptic operator in topological terms. This problem has been taken up by Agranovič [1962a; 1962b], Dynin [1962b; 1961a; 1961b], Seeley [1961; 1963] and Vol’pert [1962] who have solved it in special cases. The purpose of this paper is to give a general formula for the index of an elliptic operator on any compact oriented differentiable manifold. As a special case of this formula we get the Hirzebruch–Riemann–Roch theorem for any compact complex manifold. This was previously known only for projective algebraic manifolds. Some other special cases, of interest in differential topology, are discussed.
@article {key157392m,
AUTHOR = {Atiyah, M. F. and Singer, I. M.},
TITLE = {The index of elliptic operators on compact
manifolds},
JOURNAL = {Bull. Amer. Math. Soc.},
FJOURNAL = {Bulletin of the American Mathematical
Society},
VOLUME = {69},
NUMBER = {3},
YEAR = {1963},
PAGES = {422--433},
DOI = {10.1090/S0002-9904-1963-10957-X},
NOTE = {MR:157392. Zbl:0118.31203.},
ISSN = {0002-9904},
}
M. F. Atiyah and R. Bott :
“The index problem for manifolds with boundary ,”
pp. 175–186
in
Differential Analysis: Papers presented at the international colloquium
(Bombay, 7–14 January 1964 ).
Tata Institute of Fundamental Research Studies in Mathematics 2 .
Oxford University Press (London ),
1964 .
MR
0185606 Zbl
0163.34603 incollection
Abstract
People
BibTeX
The aim of these lectures is to report on the progress of the index problem in the last year. We will describe an extension of the index formula for closed manifolds (see [Atiyah and Singer 1963]) to manifolds with boundary. The work of Section 4, i.e., the proof of the general index theorem from Theorem 1 was done in collaboration with Singer.
@incollection {key0185606m,
AUTHOR = {Atiyah, M. F. and Bott, R.},
TITLE = {The index problem for manifolds with
boundary},
BOOKTITLE = {Differential Analysis: {P}apers presented
at the international colloquium},
SERIES = {Tata Institute of Fundamental Research
Studies in Mathematics},
NUMBER = {2},
PUBLISHER = {Oxford University Press},
ADDRESS = {London},
YEAR = {1964},
PAGES = {175--186},
NOTE = {(Bombay, 7--14 January 1964). MR:0185606.
Zbl:0163.34603.},
ISSN = {0496-9480},
}
M. F. Atiyah :
“The index of elliptic operators on compact manifolds ”
in
Séminaire Bourbaki. 15e année: 1962/63 .
Séminaire Bourbaki .
Secrétariat Mathématique (Paris ),
1964 .
Exposé no. 253.
Reprint of article in Bull. Am. Math. Soc. 69 (1963)Séminaire Bourbaki 8 (1995)
Zbl
0124.31102 incollection
BibTeX
@incollection {key0124.31102z,
AUTHOR = {Atiyah, Michael F.},
TITLE = {The index of elliptic operators on compact
manifolds},
BOOKTITLE = {S\'eminaire {B}ourbaki. 15e ann\'ee:
1962/63},
ORGANIZATION = {S\'eminaire Bourbaki},
PUBLISHER = {Secr\'etariat Math\'ematique},
ADDRESS = {Paris},
YEAR = {1964},
NOTE = {Expos\'e no.~253. Reprint of article
in \textit{Bull. Am. Math. Soc.} \textbf{69}
(1963). See also \textit{S\'eminaire
Bourbaki} \textbf{8} (1995). Zbl:0124.31102.},
}
M. F. Atiyah :
“The index theorem for manifolds with boundary ,”
pp. 337–351
in
Seminar on the Atiyah–Singer index theorem .
Edited by R. S. Palais .
Annals of Mathematics Studies 57 .
Princeton University Press ,
1965 .
Appendix I.
Atiyah’s sole contribution to Seminar on the Atiyah–Singer index theorem (1965)Atiyah’s Collected works , vol. 3 . See also similarly-titled article in Differential analysis (1965)
incollection
People
BibTeX
@incollection {key42952549,
AUTHOR = {Atiyah, M. F.},
TITLE = {The index theorem for manifolds with
boundary},
BOOKTITLE = {Seminar on the {A}tiyah--{S}inger index
theorem},
EDITOR = {Palais, Richard S.},
SERIES = {Annals of Mathematics Studies},
NUMBER = {57},
PUBLISHER = {Princeton University Press},
YEAR = {1965},
PAGES = {337--351},
NOTE = {Appendix I. Atiyah's sole contribution
to \textit{Seminar on the Atiyah--Singer
index theorem} (1965). Republished in
Atiyah's \textit{Collected works}, vol.~3.
See also similarly-titled article in
\textit{Differential analysis} (1965).},
ISSN = {0066-2313},
}
M. F. Atiyah and R. Bott :
“A Lefschetz fixed point formula for elliptic differential operators Bull. Am. Math. Soc.
72 : 2
(1966 ),
pp. 245–250 .
MR
0190950 Zbl
0151.31801 article
Abstract
People
BibTeX
The classical Lefschetz fixed point formula expresses, under suitable circumstances, the number of fixed points of a continuous map \( f:X\to X \) in terms of the transformation induced by \( f \) on the cohomology of \( X \) . If \( X \) is not just a topological space but has some further structure, and if this structure is preserved by \( f \) , one would expect to be able to refine the Lefschetz formula and to say more about the nature of the fixed points. The purpose of this note is to present such a refinement (Theorem 1) when \( X \) is a compact differentiable manifold endowed with an elliptic differential operator (or more generally an elliptic complex). Taking essentially the classical operators of complex and Riemannian geometry we obtain a number of important special cases (Theorem 2, 3). The first of these was conjectured to us by Shimura and was proved by Eichler for dimension one.
@article {key0190950m,
AUTHOR = {Atiyah, M. F. and Bott, R.},
TITLE = {A {L}efschetz fixed point formula for
elliptic differential operators},
JOURNAL = {Bull. Am. Math. Soc.},
FJOURNAL = {Bulletin of the American Mathematical
Society},
VOLUME = {72},
NUMBER = {2},
YEAR = {1966},
PAGES = {245--250},
DOI = {10.1090/S0002-9904-1966-11483-0},
NOTE = {MR:0190950. Zbl:0151.31801.},
ISSN = {0002-9904},
}
M. F. Atiyah and R. Bott :
“A Lefschetz fixed point formula for elliptic complexes, I Ann. Math. (2)
86 : 2
(1967 ),
pp. 374–407 .
MR
0212836 Zbl
0161.43201 article
Abstract
People
BibTeX
@article {key0212836m,
AUTHOR = {Atiyah, M. F. and Bott, R.},
TITLE = {A {L}efschetz fixed point formula for
elliptic complexes, {I}},
JOURNAL = {Ann. Math. (2)},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {86},
NUMBER = {2},
YEAR = {1967},
PAGES = {374--407},
DOI = {10.2307/1970694},
NOTE = {MR:0212836. Zbl:0161.43201.},
ISSN = {0003-486X},
}
M. F. Atiyah :
“A Lefschetz fixed-point formula for elliptic differential operators ,”
pp. 38–39
in
Simposio internazionale di geometria algebrica
(Rome, 30 September–5 October 1965 ),
published as Rend. Mat. Appl.
V : 25 .
Issue edited by G. Castelnuovo .
Cremonese (Rome ),
1967 .
See also Bull. Amer. Math. Soc. 72 :2 (1966)
Zbl
0149.41201 incollection
People
BibTeX
@article {key0149.41201z,
AUTHOR = {Atiyah, Michael F.},
TITLE = {A {L}efschetz fixed-point formula for
elliptic differential operators},
JOURNAL = {Rend. Mat. Appl.},
FJOURNAL = {Rendiconti di Matematica e delle sue
Applicazioni},
VOLUME = {V},
NUMBER = {25},
YEAR = {1967},
PAGES = {38--39},
NOTE = {\textit{Simposio internazionale di geometria
algebrica} (Rome, 30 September--5 October
1965). Issue edited by G. Castelnuovo.
See also \textit{Bull. Amer. Math. Soc.}
\textbf{72}:2 (1966). Zbl:0149.41201.},
ISSN = {1120-7183},
}
M. F. Atiyah :
“Algebraic topology and elliptic operators Comm. Pure Appl. Math.
20
(1967 ),
pp. 237–249 .
MR
0211418 Zbl
0145.43804 article
BibTeX
@article {key0211418m,
AUTHOR = {Atiyah, M. F.},
TITLE = {Algebraic topology and elliptic operators},
JOURNAL = {Comm. Pure Appl. Math.},
FJOURNAL = {Communications on Pure and Applied Mathematics},
VOLUME = {20},
YEAR = {1967},
PAGES = {237--249},
DOI = {10.1002/cpa.3160200202},
NOTE = {MR:0211418. Zbl:0145.43804.},
ISSN = {0010-3640},
}
M. F. Atiyah and G. B. Segal :
“The index of elliptic operators, II ,”
Uspehi Mat. Nauk
23 : 6 (144)
(1968 ),
pp. 135–149 .
Russian translation of article in Ann. Math. 87 :3 (1968)
MR
0236953 article
People
BibTeX
@article {key0236953m,
AUTHOR = {Atiyah, M. F. and Segal, G. B.},
TITLE = {The index of elliptic operators, {II}},
JOURNAL = {Uspehi Mat. Nauk},
FJOURNAL = {Uspekhi Matematicheskikh Nauk},
VOLUME = {23},
NUMBER = {6 (144)},
YEAR = {1968},
PAGES = {135--149},
NOTE = {Russian translation of article in \textit{Ann.
Math.} \textbf{87}:3 (1968). MR:0236953.},
ISSN = {0042-1316},
}
M. F. Atiyah and G. B. Segal :
“The index of elliptic operators, II Ann. Math. (2)
87 : 3
(1968 ),
pp. 531–545 .
Russian translation published in Uspehi Mat. Nauk 23 :6(144) (1968)
MR
0236951 Zbl
0164.24201 article
Abstract
People
BibTeX
The purpose of this paper is to show how the index theorem of [Atiyah and Singer 1963] can be reformulated as a general “Lefschetz fixed-point theorem” on the lines of [Atiyah and Bott 1967]. In this way we shall obtain the main theorem of [Atiyah and Bott 1967], generalized to deal with arbitrary fixed-point sets, but only for transformations belonging to a compact group.
The content of this paper is essentially topological, and it should be viewed as a paper on the equivariant \( K \) -theory of manifolds. The analysis has all been done in [Atiyah and Singer 1968], and what we do here is simply to express the topological index in terms of fixed-point sets. This is quite independent of the main theorem of [Atiyah and Singer 1968] asserting the equality of the topological and analytical indices.
As in [Atiyah and Singer 1968], we avoid cohomology and use only \( K \) -theory. In paper III of this series, we shall pass over to cohomology obtaining explicit formulas in terms of characteristic classes.
@article {key0236951m,
AUTHOR = {Atiyah, M. F. and Segal, G. B.},
TITLE = {The index of elliptic operators, {II}},
JOURNAL = {Ann. Math. (2)},
FJOURNAL = {Annals of Mathematics, second series},
VOLUME = {87},
NUMBER = {3},
YEAR = {1968},
PAGES = {531--545},
DOI = {10.2307/1970716},
NOTE = {Russian translation published in \textit{Uspehi
Mat. Nauk} \textbf{23}:6(144) (1968).
MR:0236951. Zbl:0164.24201.},
ISSN = {0003-486X},
}
M. F. Atiyah and I. M. Singer :
“The index of elliptic operators, I Ann. Math. (2)
87 : 3
(May 1968 ),
pp. 484–530 .
A Russian translation was published in Uspehi Mat. Nauk 23 :5(143)
MR
236950 Zbl
0164.24001 article
Abstract
People
BibTeX
This is the first of a series of papers which will be devoted to a study of the index of elliptic operators on compact manifolds. The main result was announced in [Atiyah and Singer 1963] and, for manifolds with boundary, in [Atiyah 1964]. The long delay between these announcements and the present paper is due to several factors. On the one hand, a fairly detailed exposition has already appeared in [Palais 1965]. On the other hand, our original proof, reproduced with minor modifications in [Palais 1965], had a number of drawbacks. In the first place the use of cobordism, and the computational checking associated with this, were not very enlightening. More seriously, however, the method of proof did not lend itself to certain natural generalizations of the problem where appropriate cobordism groups were not known. The reader who is familiar with the Riemann–Roch theorem will realize that our original proof of the index theorem was modelled closely on Hirzebruch’s proof of the Riemann–Roch theorem. Naturally enough we were led to look for a proof modelled more on that of Grothendieck. While we have not completely succeeded in this aim, we have at least found a proof which is much more natural, does not use cobordism, and lends itself therefore to generalization. In spirit, at least, it has much in common with Grothendieck’s approach.
@article {key236950m,
AUTHOR = {Atiyah, M. F. and Singer, I. M.},
TITLE = {The index of elliptic operators, {I}},
JOURNAL = {Ann. Math. (2)},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {87},
NUMBER = {3},
MONTH = {May},
YEAR = {1968},
PAGES = {484--530},
DOI = {10.2307/1970715},
NOTE = {A Russian translation was published
in \textit{Uspehi Mat. Nauk} \textbf{23}:5(143).
MR:236950. Zbl:0164.24001.},
ISSN = {0003-486X},
}
M. F. Atiyah and I. M. Singer :
“The index of elliptic operators, III Ann. Math. (2)
87 : 3
(May 1968 ),
pp. 546–604 .
A Russian translation was published in Uspehi Mat. Nauk 24 :1(145)
MR
236952 Zbl
0164.24301 article
Abstract
People
BibTeX
In [1968a], paper I of this series, the index of an elliptic operator was computed in terms of \( K \) -theory. In this paper, we carry out what is essentially a routine exercise by passing from \( K \) -theory to cohomology. In this way, we end up with the explicit cohomological formula for the index announced in [1963].
In [1968a] we also considered elliptic operators (or complexes) compatible with a compact group \( G \) of transformations. The index in this case is a character of \( G \) , and the main theorem of [1968a] gave a construction for this in \( KG \) -theory. In [1968b], paper II of this series, the value of this index-character at an element \( g\in G \) was expressed as the index of a new “virtual operator” on the fixed point set of \( g \) . This was referred to as a Lefschetz fixed-point formula. By combining this formula with the cohomological formula for the index, we obtain finally an explicit cohomological formula for the index-character. We shall describe this formula in detail for a number of important operators. In particular we draw attention to the “integrality theorems” obtained in this way for actions of finite groups on manifolds. Most of these do not depend on the analysis in [1968a], but are a consequence of combining the purely topological results of [1968b] and the present paper.
@article {key236952m,
AUTHOR = {Atiyah, M. F. and Singer, I. M.},
TITLE = {The index of elliptic operators, {III}},
JOURNAL = {Ann. Math. (2)},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {87},
NUMBER = {3},
MONTH = {May},
YEAR = {1968},
PAGES = {546--604},
DOI = {10.2307/1970717},
NOTE = {A Russian translation was published
in \textit{Uspehi Mat. Nauk} \textbf{24}:1(145).
MR:236952. Zbl:0164.24301.},
ISSN = {0003-486X},
}
M. F. Atiyah and R. Bott :
“A Lefschetz fixed point formula for elliptic complexes, II: Applications Ann. Math. (2)
88 : 3
(November 1968 ),
pp. 451–491 .
MR
0232406 Zbl
0167.21703 article
Abstract
People
BibTeX
@article {key0232406m,
AUTHOR = {Atiyah, M. F. and Bott, R.},
TITLE = {A {L}efschetz fixed point formula for
elliptic complexes, {II}: {A}pplications},
JOURNAL = {Ann. Math. (2)},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {88},
NUMBER = {3},
MONTH = {November},
YEAR = {1968},
PAGES = {451--491},
DOI = {10.2307/1970721},
NOTE = {MR:0232406. Zbl:0167.21703.},
ISSN = {0003-486X},
}
M. F. At’ja and I. M. Zinger :
“The index of elliptic operators, I Uspehi Mat. Nauk
23 : 5(143)
(1968 ),
pp. 99–142 .
Russian translation of an article published in Ann. Math. (2) 87 :3 (1968)
MR
232402 article
People
BibTeX
@article {key232402m,
AUTHOR = {At\cprime ja, M. F. and Zinger, I. M.},
TITLE = {The index of elliptic operators, {I}},
JOURNAL = {Uspehi Mat. Nauk},
FJOURNAL = {Akademiya Nauk SSSR i Moskovskoe Matematicheskoe
Obshchestvo. Uspekhi Matematicheskikh
Nauk},
VOLUME = {23},
NUMBER = {5(143)},
YEAR = {1968},
PAGES = {99--142},
URL = {http://www.mathnet.ru/eng/rm5670},
NOTE = {Russian translation of an article published
in \textit{Ann. Math. (2)} \textbf{87}:3
(1968). MR:232402.},
ISSN = {0042-1316},
}
M. F. Atiyah :
“Global aspects of the theory of elliptic differential operators 57–64
in
Proceedings of the International Congress of Mathematics
(Moscow, 16–26 August 1966 ).
Mir (Moscow ),
1968 .
MR
0233378 Zbl
0204.41902 incollection
Abstract
BibTeX
The subject matter of this talk lies in the area between Analysis and Algebraic Topology. More specifically, I want to discuss the relations between the analysis of linear partial differential operators of elliptic type and the algebraic topology of linear groups of finite-dimensional vector spaces. I will try to show that these two topics are intimately related, and that the study of each is of great importance for the development of the other.
@incollection {key0233378m,
AUTHOR = {Atiyah, Michael F.},
TITLE = {Global aspects of the theory of elliptic
differential operators},
BOOKTITLE = {Proceedings of the {I}nternational {C}ongress
of {M}athematics},
PUBLISHER = {Mir},
ADDRESS = {Moscow},
YEAR = {1968},
PAGES = {57--64},
URL = {http://www.mathunion.org/ICM/ICM1966.1/Main/icm1966.1.0057.0064.ocr.pdf},
NOTE = {(Moscow, 16--26 August 1966). MR:0233378.
Zbl:0204.41902.},
}
M. F. Atiyah :
“Hyperbolic differential equations and algebraic geometry (after Petrowsky) 87–99
in
Séminaire Bourbaki 1966/1967 .
W. A. Benjamin (New York and Amsterdam ),
1968 .
Exposé no. 319.
Republished in 1995 .
Zbl
0201.12501 incollection
People
BibTeX
@incollection {key0201.12501z,
AUTHOR = {Atiyah, Michael F.},
TITLE = {Hyperbolic differential equations and
algebraic geometry (after {P}etrowsky)},
BOOKTITLE = {S\'eminaire {B}ourbaki 1966/1967},
PUBLISHER = {W. A. Benjamin},
ADDRESS = {New York and Amsterdam},
YEAR = {1968},
PAGES = {87--99},
URL = {http://www.numdam.org/item?id=SB_1966-1968__10__87_0},
NOTE = {Expos\'e no.~319. Republished in 1995.
Zbl:0201.12501.},
}
M. F. Atiyah and I. M. Singer :
“Index theory for skew-adjoint Fredholm operators Inst. Hautes Études Sci. Publ. Math.
37
(1969 ),
pp. 5–26 .
MR
285033 Zbl
0194.55503 article
People
BibTeX
@article {key285033m,
AUTHOR = {Atiyah, M. F. and Singer, I. M.},
TITLE = {Index theory for skew-adjoint {F}redholm
operators},
JOURNAL = {Inst. Hautes \'Etudes Sci. Publ. Math.},
FJOURNAL = {Institut des Hautes \'Etudes Scientifiques.
Publications Math\'ematiques},
VOLUME = {37},
YEAR = {1969},
PAGES = {5--26},
URL = {http://www.numdam.org/item?id=PMIHES_1969__37__5_0},
NOTE = {MR:285033. Zbl:0194.55503.},
ISSN = {0073-8301},
}
M. F. Atiyah :
“The signature of fibre-bundles ,”
pp. 73–84
in
Global analysis: Papers in honor of K. Kodaira .
Edited by S. Iyanaga and D. C. Spencer .
Princeton Mathematical Series 29 .
University of Tokyo Press ,
1969 .
MR
0254864 Zbl
0193.52302 incollection
Abstract
People
BibTeX
For a compact oriented differentiable manifold \( X \) of dimension \( 4k \) the signature (or index) of \( X \) is defined as the signature of the quadratic form in \( H^{2k}(X;\mathbb{R}) \) given by the cup product. Thus
\[\operatorname{Sign}(X) = p - q \]
where \( p \) is the number of \( + \) signs in a diagonalization of the given quadratic form and \( q \) is the number of \( - \) signs. If \( \operatorname{dim} X \) is not by divisible by 4 one defines \( \operatorname{Sign}(X) \) to be zero. Then one has the the multiplicative formula
\[\operatorname{Sign}(X \times Y) = \operatorname{Sign}(X)\cdot\operatorname{Sign}(Y)\]
In [Chern, Hirzebruch and Serre 1957] it was proved that this multiplicative formula continues to hold when \( X \times Y \) is replaced by a fibre bundle with base \( X \) and fibre \( Y \) provided that the fundamental group of \( X \) acts trivially us on the cohomology of \( Y \) .
In this paper we exhibit examples which show that this restriction on the action of \( \pi_1(X) \) is necessary, and that the signature is not multiplicative in general fibre-bundles .
@incollection {key0254864m,
AUTHOR = {Atiyah, M. F.},
TITLE = {The signature of fibre-bundles},
BOOKTITLE = {Global analysis: {P}apers in honor of
{K}.~{K}odaira},
EDITOR = {Sh\=okichi Iyanaga and Donald Clayton
Spencer},
SERIES = {Princeton Mathematical Series},
NUMBER = {29},
PUBLISHER = {University of Tokyo Press},
YEAR = {1969},
PAGES = {73--84},
NOTE = {MR:0254864. Zbl:0193.52302.},
ISSN = {0079-5194},
ISBN = {9780691080772},
}
M. F. At’ja and I. M. Zinger :
“The index of elliptic operators, III Uspehi Mat. Nauk
24 : 1(145)
(1969 ),
pp. 127–182 .
The English original was published in Ann. Math. (2) 87 :3 (1968)
MR
256417 article
People
BibTeX
@article {key256417m,
AUTHOR = {At\cprime ja, M. F. and Zinger, I. M.},
TITLE = {The index of elliptic operators, {III}},
JOURNAL = {Uspehi Mat. Nauk},
FJOURNAL = {Akademiya Nauk SSSR i Moskovskoe Matematicheskoe
Obshchestvo. Uspekhi Matematicheskikh
Nauk},
VOLUME = {24},
NUMBER = {1(145)},
YEAR = {1969},
PAGES = {127--182},
URL = {http://www.mathnet.ru/eng/rm5453},
NOTE = {The English original was published in
\textit{Ann. Math. (2)} \textbf{87}:3
(1968). MR:256417.},
ISSN = {0042-1316},
}
M. F. Atiyah :
“Topology of elliptic operators ,”
pp. 101–119
in
Global analysis
(Berkeley, CA, 1–26 July 1968 ).
Edited by S.-S. Chern and S. Smale .
Proceedings of Symposia in Pure Mathematics 16 .
American Mathematical Society (Providence, RI ),
1970 .
MR
0264700 Zbl
0207.22601 incollection
Abstract
People
BibTeX
@incollection {key0264700m,
AUTHOR = {Atiyah, Michael F.},
TITLE = {Topology of elliptic operators},
BOOKTITLE = {Global analysis},
EDITOR = {Shiing-Shen Chern and Stephen Smale},
SERIES = {Proceedings of Symposia in Pure Mathematics},
NUMBER = {16},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {1970},
PAGES = {101--119},
NOTE = {(Berkeley, CA, 1--26 July 1968). MR:0264700.
Zbl:0207.22601.},
ISSN = {0082-0717},
}
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. F. Atiyah :
“Global theory of elliptic operators ,”
pp. 21–30
in
Functional analysis and related topics
(Tokyo, 1969 ).
Edited by S. T. Kuroda .
University of Tokyo Press ,
1970 .
MR
0266247 Zbl
0193.43601 incollection
People
BibTeX
@incollection {key0266247m,
AUTHOR = {Atiyah, M. F.},
TITLE = {Global theory of elliptic operators},
BOOKTITLE = {Functional analysis and related topics},
EDITOR = {Shige Toshi Kuroda},
PUBLISHER = {University of Tokyo Press},
YEAR = {1970},
PAGES = {21--30},
NOTE = {(Tokyo, 1969). MR:0266247. Zbl:0193.43601.},
}
M. F. Atiyah :
“Resolution of singularities and division of distributions Comm. Pure Appl. Math.
23 : 2
(1970 ),
pp. 145–150 .
MR
0256156 Zbl
0188.19405 article
Abstract
BibTeX
In this note I shall show how Hironaka’s theorem [1964] on the resolution of singularities leads very quickly to a new proof of the Hörmander–Lojasiewicz theorem [Hörmander 1958; Lojasiewicz 1959] on the division of distributions and hence to the existence of temperate fundamental solutions for constant-coefficient differential operators. Since most of the difficulties in the general theory of partial differential operators arise from the singularities of the characteristic variety, it is quite natural to expect Hironaka’s theorem to be relevant. In fact, this note is primarily intended to draw the attention of analysts to the power of this theorem. It seems likely that its application in the field of partial differential equations may yield many results besides those described here.
@article {key0256156m,
AUTHOR = {Atiyah, M. F.},
TITLE = {Resolution of singularities and division
of distributions},
JOURNAL = {Comm. Pure Appl. Math.},
FJOURNAL = {Communications on Pure and Applied Matheamtics},
VOLUME = {23},
NUMBER = {2},
YEAR = {1970},
PAGES = {145--150},
DOI = {10.1002/cpa.3160230202},
NOTE = {MR:0256156. Zbl:0188.19405.},
ISSN = {0010-3640},
}
M. F. Atiyah, R. Bott, and L. Gårding :
“Lacunas for hyperbolic differential operators with constant coefficients, I Acta Math.
124 : 1
(July 1970 ),
pp. 109–189 .
A Russian translation was published in Usp. Mat. Nauk 26 :2(158)
MR
0470499 Zbl
0191.11203 article
Abstract
People
BibTeX
The theory of lacunas for hyperbolic differential operators was created by I. G. Petrovsky who published the basic paper of the subject in 1945. Although its results are very clear, the paper is difficult reading and has so far not lead to studies of the same scope. We shall clarify and generalize Petrovsky’s theory.
@article {key0470499m,
AUTHOR = {Atiyah, M. F. and Bott, R. and G\aa
rding, L.},
TITLE = {Lacunas for hyperbolic differential
operators with constant coefficients,
{I}},
JOURNAL = {Acta Math.},
FJOURNAL = {Acta Mathematica},
VOLUME = {124},
NUMBER = {1},
MONTH = {July},
YEAR = {1970},
PAGES = {109--189},
DOI = {10.1007/BF02394570},
NOTE = {A Russian translation was published
in \textit{Usp. Mat. Nauk} \textbf{26}:2(158).
MR:0470499. Zbl:0191.11203.},
ISSN = {0001-5962},
}
M. F. Atiyah :
“Riemann surfaces and spin structures Ann. Sci. École Norm. Sup. (4)
4 : 1
(1971 ),
pp. 47–62 .
MR
0286136 Zbl
0212.56402 article
Abstract
BibTeX
@article {key0286136m,
AUTHOR = {Atiyah, Michael F.},
TITLE = {Riemann surfaces and spin structures},
JOURNAL = {Ann. Sci. \'Ecole Norm. Sup. (4)},
VOLUME = {4},
NUMBER = {1},
YEAR = {1971},
PAGES = {47--62},
URL = {http://www.numdam.org/item?id=ASENS_1971_4_4_1_47_0},
NOTE = {MR:0286136. Zbl:0212.56402.},
ISSN = {0012-9593},
}
M. F. Atiyah and I. M. Singer :
“The index of elliptic operators, V Ann. Math. (2)
93 : 1
(January 1971 ),
pp. 139–149 .
A Russian translation was published in Uspehi Mat. Nauk 27 :4(166) (1972)
MR
279834 article
Abstract
People
BibTeX
The preceding papers of this series dealt with the index of elliptic pseudo-differential operators and families of such operators. In all this, our operators (and vector bundles) were over the complex numbers. In this paper we want to refine the preceding theory to deal with real operators, for example differential operators with real coefficients.
@article {key279834m,
AUTHOR = {Atiyah, M. F. and Singer, I. M.},
TITLE = {The index of elliptic operators, {V}},
JOURNAL = {Ann. Math. (2)},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {93},
NUMBER = {1},
MONTH = {January},
YEAR = {1971},
PAGES = {139--149},
DOI = {10.2307/1970757},
NOTE = {A Russian translation was published
in \textit{Uspehi Mat. Nauk} \textbf{27}:4(166)
(1972). MR:279834.},
ISSN = {0003-486X},
}
M. F. Atiyah and I. M. Singer :
“The index of elliptic operators, IV Ann. Math. (2)
93 : 1
(January 1971 ),
pp. 119–138 .
A Russian translation was published in Uspehi Mat. Nauk 27 :4(166) (1972)
MR
279833 Zbl
0212.28603 article
Abstract
People
BibTeX
@article {key279833m,
AUTHOR = {Atiyah, M. F. and Singer, I. M.},
TITLE = {The index of elliptic operators, {IV}},
JOURNAL = {Ann. Math. (2)},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {93},
NUMBER = {1},
MONTH = {January},
YEAR = {1971},
PAGES = {119--138},
DOI = {10.2307/1970756},
NOTE = {A Russian translation was published
in \textit{Uspehi Mat. Nauk} \textbf{27}:4(166)
(1972). MR:279833. Zbl:0212.28603.},
ISSN = {0003-486X},
}
M. F. At’ja, R. Bott, and L. Gording :
“Lacunas for hyperbolic differential operators with constant coefficients, I Usp. Mat. Nauk
26 : 2(158)
(1971 ),
pp. 25–100 .
Russian translation of an article in Acta Math. 124 :1 (1970)
MR
0606062 Zbl
0208.13201 article
People
BibTeX
@article {key0606062m,
AUTHOR = {At{\cprime}ja, M. F. and Bott, R. and
Gording, L.},
TITLE = {Lacunas for hyperbolic differential
operators with constant coefficients,
{I}},
JOURNAL = {Usp. Mat. Nauk},
FJOURNAL = {Uspekhi Matematicheskikh Nauk},
VOLUME = {26},
NUMBER = {2(158)},
YEAR = {1971},
PAGES = {25--100},
URL = {http://mi.mathnet.ru/eng/umn5186},
NOTE = {Russian translation of an article in
\textit{Acta Math.} \textbf{124}:1 (1970).
MR:0606062. Zbl:0208.13201.},
ISSN = {0042-1316},
}
M. F. At’ja and I. M. Zinger :
“The index of elliptic operators, IV Uspehi Mat. Nauk
27 : 4(166)
(1972 ),
pp. 161–178 .
Russian translation of an article published in Ann. Math. (2) 93 :1 (1971)
MR
385933 Zbl
0237.58017 article
People
BibTeX
@article {key385933m,
AUTHOR = {At\cprime ja, M. F. and Zinger, I. M.},
TITLE = {The index of elliptic operators, {IV}},
JOURNAL = {Uspehi Mat. Nauk},
FJOURNAL = {Akademiya Nauk SSSR i Moskovskoe Matematicheskoe
Obshchestvo. Uspekhi Matematicheskikh
Nauk},
VOLUME = {27},
NUMBER = {4(166)},
YEAR = {1972},
PAGES = {161--178},
URL = {http://mi.mathnet.ru/eng/umn5086},
NOTE = {Russian translation of an article published
in \textit{Ann. Math. (2)} \textbf{93}:1
(1971). MR:385933. Zbl:0237.58017.},
ISSN = {0042-1316},
}
M. F. Atiyah and I. M. Singer :
“The index of elliptic operators, V Usp. Mat. Nauk
27 : 4(166)
(1972 ).
Russian translation of an article published in Ann. Math. (2) 93 :1 (1971)
Zbl
0237.58018 article
People
BibTeX
@article {key0237.58018z,
AUTHOR = {Atiyah, Michael F. and Singer, I. M.},
TITLE = {The index of elliptic operators, {V}},
JOURNAL = {Usp. Mat. Nauk},
FJOURNAL = {Uspekhi Matematicheskikh Nauk (N.S.)},
VOLUME = {27},
NUMBER = {4(166)},
YEAR = {1972},
URL = {http://mi.mathnet.ru/eng/umn5087},
NOTE = {Russian translation of an article published
in \textit{Ann. Math. (2)} \textbf{93}:1
(1971). Zbl:0237.58018.},
ISSN = {0042-1316},
}
M. F. Atiyah, V. K. Patodi, and I. M. Singer :
“Spectral asymmetry and Riemannian geometry Bull. London Math. Soc.
5
(July 1973 ),
pp. 229–234 .
MR
331443 Zbl
0268.58010 article
Abstract
People
BibTeX
If \( A \) is a positive self-adjoint elliptic (linear) differential operator on a compact manifold then it has a discrete spectrum consisting of positive eigenvaues \( \{\lambda\} \) . In analogy with the classical Riemann zeta-function one can define, for \( \operatorname{Re}(s) \) large,
\[ \zeta_A(s) = \operatorname{Tr} A^{-s} = \sum\lambda^{-s}. \]
@article {key331443m,
AUTHOR = {Atiyah, M. F. and Patodi, V. K. and
Singer, I. M.},
TITLE = {Spectral asymmetry and {R}iemannian
geometry},
JOURNAL = {Bull. London Math. Soc.},
FJOURNAL = {The Bulletin of the London Mathematical
Society},
VOLUME = {5},
MONTH = {July},
YEAR = {1973},
PAGES = {229--234},
DOI = {10.1112/blms/5.2.229},
NOTE = {MR:331443. Zbl:0268.58010.},
ISSN = {0024-6093},
}
M. Atiyah, R. Bott, and V. K. Patodi :
“On the heat equation and the index theorem ,”
Matematika, Moskva
17 : 6
(1973 ),
pp. 3–48 .
Russian translation of an article in Invent. Math. 19 :4 (1973)
Zbl
0364.58016 article
People
BibTeX
@article {key0364.58016z,
AUTHOR = {Atiyah, Michael and Bott, Raoul and
Patodi, V. K.},
TITLE = {On the heat equation and the index theorem},
JOURNAL = {Matematika, Moskva},
VOLUME = {17},
NUMBER = {6},
YEAR = {1973},
PAGES = {3--48},
NOTE = {Russian translation of an article in
\textit{Invent. Math.} \textbf{19}:4
(1973). Zbl:0364.58016.},
}
M. Atiyah, R. Bott, and V. K. Patodi :
“On the heat equation and the index theorem Invent. Math.
19 : 4
(1973 ),
pp. 279–330 .
Dedicated to Sir William Hodge on his 70th birthday.
Errata were published in Invent. Math. 28 :3 (1975)Matematika 17 :6 (1973)
MR
0650828 Zbl
0257.58008 article
Abstract
People
BibTeX
@article {key0650828m,
AUTHOR = {Atiyah, M. and Bott, R. and Patodi,
V. K.},
TITLE = {On the heat equation and the index theorem},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {19},
NUMBER = {4},
YEAR = {1973},
PAGES = {279--330},
DOI = {10.1007/BF01425417},
NOTE = {Dedicated to Sir William Hodge on his
70th birthday. Errata were published
in \textit{Invent. Math.} \textbf{28}:3
(1975). A Russian translation was published
in \textit{Matematika} \textbf{17}:6
(1973). MR:0650828. Zbl:0257.58008.},
ISSN = {0020-9910},
}
M. F. Atiyah, R. Bott, and L. Gårding :
“Lacunas for hyperbolic differential operators with constant coefficients, II Acta Math.
131 : 1
(December 1973 ),
pp. 145–206 .
A Russian translation was published in Usp. Mat. Nauk 39 :3(237)
MR
0470500 Zbl
0266.35045 article
People
BibTeX
@article {key0470500m,
AUTHOR = {Atiyah, M. F. and Bott, R. and G\aa
rding, L.},
TITLE = {Lacunas for hyperbolic differential
operators with constant coefficients,
{II}},
JOURNAL = {Acta Math.},
FJOURNAL = {Acta Mathematica},
VOLUME = {131},
NUMBER = {1},
MONTH = {December},
YEAR = {1973},
PAGES = {145--206},
DOI = {10.1007/BF02392039},
NOTE = {A Russian translation was published
in \textit{Usp. Mat. Nauk} \textbf{39}:3(237).
MR:0470500. Zbl:0266.35045.},
ISSN = {0001-5962},
}
M. F. Atiyah :
The index of elliptic operators ,
1973 .
Notes distributed at seventy-ninth annual meeting of the American Mathematical Society, Dallas, TX, 25–28 January 1973.
Republished in Atiyah’s Collected works , vol. 3, pp. 475–498.
Republished in Fields Medallists’ lectures (1997)
Zbl
0263.58012 misc
BibTeX
@misc {key0263.58012z,
AUTHOR = {Atiyah, Michael F.},
TITLE = {The index of elliptic operators},
HOWPUBLISHED = {Notes distributed at seventy-ninth annual
meeting of the American Mathematical
Society, Dallas, TX, 25--28 January
1973},
YEAR = {1973},
PAGES = {19},
NOTE = {Republished in Atiyah's \textit{Collected
works}, vol.~3, pp. 475--498. Republished
in \textit{Fields Medallists' lectures}
(1997). Zbl:0263.58012.},
}
M. F. Atiyah :
Elliptic operators and compact groups .
Lecture Notes in Mathematics 401 .
Springer (Berlin ),
1974 .
MR
0482866 Zbl
0297.58009 book
BibTeX
@book {key0482866m,
AUTHOR = {Atiyah, Michael Francis},
TITLE = {Elliptic operators and compact groups},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {401},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1974},
PAGES = {93},
NOTE = {MR:0482866. Zbl:0297.58009.},
ISSN = {0075-8434},
ISBN = {9783540068556},
}
M. F. Atiyah, V. K. Patodi, and I. M. Singer :
“Spectral asymmetry and Riemannian geometry, II Math. Proc. Cambridge Philos. Soc.
78 : 3
(November 1975 ),
pp. 405–432 .
MR
397798 Zbl
0314.58016 article
Abstract
People
BibTeX
@article {key397798m,
AUTHOR = {Atiyah, M. F. and Patodi, V. K. and
Singer, I. M.},
TITLE = {Spectral asymmetry and {R}iemannian
geometry, {II}},
JOURNAL = {Math. Proc. Cambridge Philos. Soc.},
FJOURNAL = {Mathematical Proceedings of the Cambridge
Philosophical Society},
VOLUME = {78},
NUMBER = {3},
MONTH = {November},
YEAR = {1975},
PAGES = {405--432},
DOI = {10.1017/S0305004100051872},
NOTE = {MR:397798. Zbl:0314.58016.},
ISSN = {0305-0041},
}
M. F. Atiyah, V. K. Patodi, and I. M. Singer :
“Spectral asymmetry and Riemannian geometry, I Math. Proc. Cambridge Philos. Soc.
77
(1975 ),
pp. 43–69 .
MR
397797 Zbl
0297.58008 article
Abstract
People
BibTeX
The main purpose of this paper is to present a generalization of Hirzebruch’s signature theorem for the case of manifolds with boundary. Our result is in the framework of Riemannian geometry and can be viewed as analogous to the Gauss–Bonnet theorem for manifolds with boundary, although there is a very significant difference between the two cases which is, in a sense, the central topic of the paper.
@article {key397797m,
AUTHOR = {Atiyah, M. F. and Patodi, V. K. and
Singer, I. M.},
TITLE = {Spectral asymmetry and {R}iemannian
geometry, {I}},
JOURNAL = {Math. Proc. Cambridge Philos. Soc.},
FJOURNAL = {Mathematical Proceedings of the Cambridge
Philosophical Society},
VOLUME = {77},
YEAR = {1975},
PAGES = {43--69},
DOI = {10.1017/S0305004100049410},
NOTE = {MR:397797. Zbl:0297.58008.},
ISSN = {0305-0041},
}
M. F. Atiyah :
“Eigenvalues and Riemannian geometry ,”
pp. 5–9
in
Manifolds
(Tokyo, 1973 ).
Edited by A. Hattori .
University of Tokyo Press ,
1975 .
MR
0372928 Zbl
0319.53030 incollection
People
BibTeX
@incollection {key0372928m,
AUTHOR = {Atiyah, M. F.},
TITLE = {Eigenvalues and {R}iemannian geometry},
BOOKTITLE = {Manifolds},
EDITOR = {Akio Hattori},
PUBLISHER = {University of Tokyo Press},
YEAR = {1975},
PAGES = {5--9},
NOTE = {(Tokyo, 1973). MR:0372928. Zbl:0319.53030.},
ISBN = {9780860081098},
}
M. F. Atiyah :
“Classical groups and classical differential operators on manifolds ,”
pp. 5–48
in
Differential operators on manifolds
(Varenna, Italy, 24 August–2 September 1975 ).
Edited by E. Vesentini .
Cremonese (Rome ),
1975 .
MR
0650830 incollection
People
BibTeX
@incollection {key0650830m,
AUTHOR = {Atiyah, M. F.},
TITLE = {Classical groups and classical differential
operators on manifolds},
BOOKTITLE = {Differential operators on manifolds},
EDITOR = {E. Vesentini},
PUBLISHER = {Cremonese},
ADDRESS = {Rome},
YEAR = {1975},
PAGES = {5--48},
NOTE = {(Varenna, Italy, 24 August--2 September
1975). MR:0650830.},
}
M. Atiyah, R. Bott, and V. K. Patodi :
“Errata to: ‘On the heat equation and the index theorem’ Invent. Math.
28 : 3
(1975 ),
pp. 277–280 .
Errata for article in Invent. Math. 19 :4 (1973)
MR
0650829 Zbl
0301.58018 article
Abstract
People
BibTeX
The joint paper of the above title which appeared in Inventiones Math. 19 , 279–330 (1973), though correct in principle, contained some technical errors which we shall here explain and rectify. Our thanks are due to D. Epstein, Y. Colin de Verdiére and A. Vasquez whose computations and queries alerted us to our errors.
@article {key0650829m,
AUTHOR = {Atiyah, M. and Bott, R. and Patodi,
V. K.},
TITLE = {Errata to: ``{O}n the heat equation
and the index theorem''},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {28},
NUMBER = {3},
YEAR = {1975},
PAGES = {277--280},
DOI = {10.1007/BF01425562},
NOTE = {Errata for article in \textit{Invent.
Math.} \textbf{19}:4 (1973). MR:0650829.
Zbl:0301.58018.},
ISSN = {0020-9910},
}
M. F. Atiyah and E. Rees :
“Vector bundles on projective 3-space Invent. Math.
35 : 1
(1976 ),
pp. 131–153 .
MR
0419852 article
Abstract
People
BibTeX
On a compact complex manifold \( X \) it is an interesting problem to compare the continuous and holomorphic vector bundles. The case of line-bundles is classical and is well understood in the framework of sheaf theory. On the other hand for bundles \( E \) with \( \dim E \geq \dim X \) we are in the stable topological range and one can use \( K \) -theory. Much is known in this direction, for example the topological and holomorphic \( K \) -groups of all complex projective spaces are isomorphic.
This paper deals with what is perhaps the simplest case not covered by the methods indicated above. We shall consider 2-dimensional complex vector bundles over the 3-dimensional complex projective space \( P_3 \) . Our aim is to prove
Every continuous 2-dimensional vector bundle over \( P_3 \) admits a holomorphic structure.
@article {key0419852m,
AUTHOR = {Atiyah, M. F. and Rees, E.},
TITLE = {Vector bundles on projective 3-space},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {35},
NUMBER = {1},
YEAR = {1976},
PAGES = {131--153},
DOI = {10.1007/BF01390136},
NOTE = {MR:0419852.},
ISSN = {0020-9910},
}
M. F. Atiyah, V. K. Patodi, and I. M. Singer :
“Spectral asymmetry and Riemannian geometry, III Math. Proc. Cambridge Philos. Soc.
79 : 1
(1976 ),
pp. 71–99 .
MR
397799 Zbl
0325.58015 article
Abstract
People
BibTeX
In Parts I and II of this paper [1975a; 1975b] we studied the ‘spectral asymmetry’ of certain elliptic self-adjoint operators arising in Riemannian geometry. More precisely, for any elliptic self-adjoint operator \( A \) on a compact manifold we defined
\[ \eta_A(s) = \sum_{\lambda\neq 0}\operatorname{sign} \lambda \,|\lambda|^{-s}, \]
where \( \lambda \) runs over the eigenvalues of \( A \) . For the particular operators of interest in Riemannian geometry we showed that \( \eta_A(s) \) had an analytic continuation to the whole complex \( s \) -plane, with simple poles, and that \( s = 0 \) was not a pole. The real number \( \eta_A(0) \) , which is a measure of ‘spectral asymmetry’, was studied in detail particularly in relation to representations of the fundamental group.
@article {key397799m,
AUTHOR = {Atiyah, M. F. and Patodi, V. K. and
Singer, I. M.},
TITLE = {Spectral asymmetry and {R}iemannian
geometry, {III}},
JOURNAL = {Math. Proc. Cambridge Philos. Soc.},
FJOURNAL = {Mathematical Proceedings of the Cambridge
Philosophical Society},
VOLUME = {79},
NUMBER = {1},
YEAR = {1976},
PAGES = {71--99},
DOI = {10.1017/S0305004100052105},
NOTE = {MR:397799. Zbl:0325.58015.},
ISSN = {0305-0041},
}
M. F. Atiyah :
“Elliptic operators, discrete groups and von Neumann algebras ,”
pp. 43–72
in
Colloque “Analyse et Topologie” en l’honneur de Henri Cartan
(Orsay, 1974 ).
Astérisque 32–33 .
Société Mathématique de France (Paris ),
1976 .
MR
0420729 Zbl
0323.58015 incollection
People
BibTeX
@incollection {key0420729m,
AUTHOR = {Atiyah, Michael F.},
TITLE = {Elliptic operators, discrete groups
and von {N}eumann algebras},
BOOKTITLE = {Colloque ``{A}nalyse et {T}opologie''
en l'honneur de {H}enri {C}artan},
SERIES = {Ast\'erisque},
NUMBER = {32--33},
PUBLISHER = {Soci\'et\'e Math\'ematique de France},
ADDRESS = {Paris},
YEAR = {1976},
PAGES = {43--72},
NOTE = {(Orsay, 1974). MR:0420729. Zbl:0323.58015.},
ISSN = {0303-1179},
}
M. Atiyah and W. Schmid :
“A geometric construction of the discrete series for semisimple Lie groups Invent. Math.
42 : 1
(1977 ),
pp. 1–62 .
Republished in Harmonic analysis and representations of semisimple Lie groups (1980)Invent. Math. 54 :2 (1979)
MR
0463358 Zbl
0373.22001 article
Abstract
People
BibTeX
The purpose of this paper is to give a new and, to a large extent, self-contained account of the principal results concerning the discrete series. The main novelty in our presentation is that we use (a weak form of) the geometric realization to construct the discrete series representations and to obtain information about their characters.
@article {key0463358m,
AUTHOR = {Atiyah, Michael and Schmid, Wilfried},
TITLE = {A geometric construction of the discrete
series for semisimple {L}ie groups},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {42},
NUMBER = {1},
YEAR = {1977},
PAGES = {1--62},
DOI = {10.1007/BF01389783},
NOTE = {Republished in \textit{Harmonic analysis
and representations of semisimple Lie
groups} (1980). Errata published in
\textit{Invent. Math.} \textbf{54}:2
(1979). MR:0463358. Zbl:0373.22001.},
ISSN = {0020-9910},
}
M. Atiyah and W. Schmid :
“Erratum: ‘A geometric construction of the discrete series for semisimple Lie groups’ Invent. Math.
54 : 2
(1979 ),
pp. 189–192 .
Erratum for article in Invent. Math. 42 :1 (1977)Harmonic analysis and representations of semisimple Lie groups (1980)
MR
550183 article
Abstract
People
BibTeX
In the above paper [Atiyah and Schmid 1977] a key role is played by a result of Borel [1963], concerning discrete subgroups \( \Gamma \) of semisimple Lie groups \( G \) . He proves that if \( G \) is linear, one can find a torsion-free \( \Gamma \) with \( \Gamma\backslash G \) compact. Unfortunately we applied this result in [Atiyah and Schmid 1977] even for non-linear \( G \) , in which case the existence of such \( \Gamma \) is seriously in doubt, as pointed out to us by P. Deligne and J.-P. Serre. The difficulty is that a torsion-free subgroup of the adjoint group lifts to a cocompact subgroup \( \Gamma \subset G \) which contains the (finite) center \( Z \) of \( G \) , and there may be an obstruction to removing this torsion subgroup. As it stands, [Atiyah and Schmid 1977] is correct only for linear \( G \) , and we shall now indicate how to extend the proof to cover all \( G \) .
@article {key550183m,
AUTHOR = {Atiyah, Michael and Schmid, Wilfried},
TITLE = {Erratum: ``{A} geometric construction
of the discrete series for semisimple
{L}ie groups''},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {54},
NUMBER = {2},
YEAR = {1979},
PAGES = {189--192},
DOI = {10.1007/BF01408936},
NOTE = {Erratum for article in \textit{Invent.
Math.} \textbf{42}:1 (1977). See also
\textit{Harmonic analysis and representations
of semisimple Lie groups} (1980). MR:550183.},
ISSN = {0020-9910},
}
M. Atiyah and W. Schmid :
“A geometric construction of the discrete series for semisimple Lie groups ,”
pp. 317–378
in
Harmonic analysis and representations of semisimple Lie groups
(Liège, 5–17 September 1977 ).
Edited by J. A. Wolf, M. Cahen, and M. de Wilde .
Mathematical Physics and Applied Mathematics 5 .
D. Reidel (Dordrecht ),
1980 .
Republished from Invent. Math. 42 :1 (1977)
Zbl
0466.22012 incollection
People
BibTeX
@incollection {key0466.22012z,
AUTHOR = {Atiyah, Michael and Schmid, Wilfried},
TITLE = {A geometric construction of the discrete
series for semisimple {L}ie groups},
BOOKTITLE = {Harmonic analysis and representations
of semisimple {L}ie groups},
EDITOR = {Joseph Albert Wolf and Michel Cahen
and M. de Wilde},
SERIES = {Mathematical Physics and Applied Mathematics},
NUMBER = {5},
PUBLISHER = {D. Reidel},
ADDRESS = {Dordrecht},
YEAR = {1980},
PAGES = {317--378},
NOTE = {(Li\`ege, 5--17 September 1977). Republished
from \textit{Invent. Math.} \textbf{42}:1
(1977). Zbl:0466.22012.},
ISSN = {0165-2419},
ISBN = {9789027710420},
}
M. Atiyah and W. Schmid :
“Erratum to the article ‘A geometric construction of the discrete series for semisimple Lie groups’ ”
in
Harmonic analysis and representations of semisimple Lie groups
(Liège, 5–17 September 1977 ).
Edited by J. A. Wolf, M. Cahen, and M. de Wilde .
Mathematical Physics and Applied Mathematics 5 .
D. Reidel (Dordrecht ),
1980 .
Erratum for article in same volume . Republished from Invent. Math. 54 :2 (1979)
Zbl
0466.22013 incollection
Abstract
People
BibTeX
In the above paper [Atiyah and Schmid 1977] a key role is played by a result of Borel [1963], concerning discrete subgroups \( \Gamma \) of semisimple Lie groups \( G \) . He proves that if \( G \) is linear, one can find a torsion-free \( \Gamma \) with \( \Gamma\backslash G \) compact. Unfortunately we applied this result in [Atiyah and Schmid 1977] even for non-linear \( G \) , in which case the existence of such \( \Gamma \) is seriously in doubt, as pointed out to us by P. Deligne and J.-P. Serre. The difficulty is that a torsion-free subgroup of the adjoint group lifts to a cocompact subgroup \( \Gamma \subset G \) which contains the (finite) center \( Z \) of \( G \) , and there may be an obstruction to removing this torsion subgroup. As it stands, [Atiyah and Schmid 1977] is correct only for linear \( G \) , and we shall now indicate how to extend the proof to cover all \( G \) .
@incollection {key0466.22013z,
AUTHOR = {Atiyah, Michael and Schmid, Wilfried},
TITLE = {Erratum to the article ``{A} geometric
construction of the discrete series
for semisimple {L}ie groups''},
BOOKTITLE = {Harmonic analysis and representations
of semisimple {L}ie groups},
EDITOR = {Joseph Albert Wolf and Michel Cahen
and M. de Wilde},
SERIES = {Mathematical Physics and Applied Mathematics},
NUMBER = {5},
PUBLISHER = {D. Reidel},
ADDRESS = {Dordrecht},
YEAR = {1980},
NOTE = {(Li\`ege, 5--17 September 1977). Erratum
for article in same volume. Republished
from \textit{Invent. Math.} \textbf{54}:2
(1979). Zbl:0466.22013.},
ISSN = {0165-2419},
ISBN = {9789027710420},
}
M. F. Atiyah, H. Donnelly, and I. M. Singer :
“Geometry and analysis of Shimizu \( L \) -functions Proc. Nat. Acad. Sci. U.S.A.
79 : 18
(September 1982 ),
pp. 5751 .
MR
674920 Zbl
0503.12007 article
Abstract
People
BibTeX
The values of 0 of Shimizu \( L \) -functions are realized as the signature defects of framed manifolds. This settles a conjecture of Hirzebruch [Hirzebruch, F. (1973) Enseign. Math. 19, 183–281] affirmatively. The proof employs the spectral theory of elliptic operators.
@article {key674920m,
AUTHOR = {Atiyah, M. F. and Donnelly, H. and Singer,
I. M.},
TITLE = {Geometry and analysis of {S}himizu \$L\$-functions},
JOURNAL = {Proc. Nat. Acad. Sci. U.S.A.},
FJOURNAL = {Proceedings of the National Academy
of Sciences of the United States of
America},
VOLUME = {79},
NUMBER = {18},
MONTH = {September},
YEAR = {1982},
PAGES = {5751},
URL = {http://www.pnas.org/content/79/18/5751.short},
NOTE = {MR:674920. Zbl:0503.12007.},
ISSN = {0027-8424},
}
M. F. Atiyah, H. Donnelly, and I. M. Singer :
“Eta invariants, signature defects of cusps, and values of \( L \) -functions Ann. Math. (2)
118 : 1
(July 1983 ),
pp. 131–177 .
An addendum to this article was published in Ann. Math. (2) 119 :3 (1984)
MR
707164 Zbl
0531.58048 article
Abstract
People
BibTeX
The purpose of this paper is to prove a conjecture of Hirzebruch [1973] which gives a topological meaning to certain values of \( L \) -functions arising in totally real number fields. This conjecture was based on the very detailed investigation made by Hirzebruch for the case of real quadratic fields, and hinged on the fine structure of the cusp singularities of the Hilbert modular surfaces. It may therefore be helpful to recall the motivation and background of the conjecture.
@article {key707164m,
AUTHOR = {Atiyah, M. F. and Donnelly, H. and Singer,
I. M.},
TITLE = {Eta invariants, signature defects of
cusps, and values of \$L\$-functions},
JOURNAL = {Ann. Math. (2)},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {118},
NUMBER = {1},
MONTH = {July},
YEAR = {1983},
PAGES = {131--177},
DOI = {10.2307/2006957},
NOTE = {An addendum to this article was published
in \textit{Ann. Math. (2)} \textbf{119}:3
(1984). MR:707164. Zbl:0531.58048.},
ISSN = {0003-486X},
}
M. F. Atiyah, H. Donnelly, and I. M. Singer :
“Signature defects of cusps and values of \( L \) -functions: The nonsplit case Ann. Math. (2)
119 : 3
(1984 ),
pp. 635–637 .
Addendum to an article published in Ann. Math. (2) 118 :1 (1983)
MR
744866 Zbl
0577.58030 article
Abstract
People
BibTeX
This note is a supplement to our paper [Atiyah, Donnelly and Singer 1983]. In [1973] Hirzebruch conjectured that the values at zero of the Shimizu \( L \) -functions are realized as the signature defects of cusps associated to Hilbert modular varieties. In [Atiyah, Donnelly and Singer 1983] we claimed to have established the Hirzebruch conjecture but, as was pointed out to us by W. Müller, we only dealt with the “split” case. In fact our method of proof extends with essentially no change to the non-split case as we shall now explain.
@article {key744866m,
AUTHOR = {Atiyah, M. F. and Donnelly, H. and Singer,
I. M.},
TITLE = {Signature defects of cusps and values
of \$L\$-functions: {T}he nonsplit case},
JOURNAL = {Ann. Math. (2)},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {119},
NUMBER = {3},
YEAR = {1984},
PAGES = {635--637},
DOI = {10.2307/2007088},
NOTE = {Addendum to an article published in
\textit{Ann. Math. (2)} \textbf{118}:1
(1983). MR:744866. Zbl:0577.58030.},
ISSN = {0003-486X},
}
M. F. At’ya, R. Bott, and L. Gårding :
“Lacunas for hyperbolic differential operators with constant coefficients, II Usp. Mat. Nauk
39 : 3(237)
(1984 ),
pp. 171–224 .
Russian translation of an article in Acta Math. 131 :1 (1973)
MR
747794 Zbl
0568.35058 article
People
BibTeX
@article {key747794m,
AUTHOR = {At{\cprime}ya, M. F. and Bott, R. and
G\aa rding, L.},
TITLE = {Lacunas for hyperbolic differential
operators with constant coefficients,
{II}},
JOURNAL = {Usp. Mat. Nauk},
FJOURNAL = {Uspekhi Matematicheskikh Nauk},
VOLUME = {39},
NUMBER = {3(237)},
YEAR = {1984},
PAGES = {171--224},
URL = {http://mi.mathnet.ru/eng/umn2374},
NOTE = {Russian translation of an article in
\textit{Acta Math.} \textbf{131}:1 (1973).
MR:747794. Zbl:0568.35058.},
ISSN = {0042-1316},
}
M. F. Atiyah :
“Characters of semi-simple Lie groups ,”
pp. 489–558
in
Collected works ,
vol. 4: Index theory 2 .
Oxford Science Publications .
Clarendon Press and Oxford University Press (Oxford, New York ),
1988 .
Lectures given at Mathematical Institute, Oxford, 1976.
incollection
BibTeX
@incollection {key48537300,
AUTHOR = {Atiyah, M. F.},
TITLE = {Characters of semi-simple {L}ie groups},
BOOKTITLE = {Collected works},
VOLUME = {4: Index theory~2},
SERIES = {Oxford Science Publications},
PUBLISHER = {Clarendon Press and Oxford University
Press},
ADDRESS = {Oxford, New York},
YEAR = {1988},
PAGES = {489--558},
NOTE = {Lectures given at Mathematical Institute,
Oxford, 1976.},
ISBN = {9780198532767},
}
M. Atiyah :
Collected works ,
vol. 4: Index theory 2 .
Oxford Science Publications .
The Clarendon Press and Oxford University Press (Oxford and New York ),
1988 .
MR
951895 book
BibTeX
@book {key951895m,
AUTHOR = {Atiyah, Michael},
TITLE = {Collected works},
VOLUME = {4: Index theory~2},
SERIES = {Oxford Science Publications},
PUBLISHER = {The Clarendon Press and Oxford University
Press},
ADDRESS = {Oxford and New York},
YEAR = {1988},
PAGES = {xxiii+617},
NOTE = {MR:951895.},
ISBN = {9780198532781},
}
M. Atiyah :
Collected works ,
vol. 3: Index theory 1 .
Oxford Science Publications .
The Clarendon Press and Oxford University Press (Oxford and New York ),
1988 .
MR
951894 Zbl
0724.53001 book
BibTeX
@book {key951894m,
AUTHOR = {Atiyah, Michael},
TITLE = {Collected works},
VOLUME = {3: Index theory~1},
SERIES = {Oxford Science Publications},
PUBLISHER = {The Clarendon Press and Oxford University
Press},
ADDRESS = {Oxford and New York},
YEAR = {1988},
PAGES = {xxii+593},
NOTE = {MR:951894. Zbl:0724.53001.},
ISBN = {9780198532774},
}
M. F. Atiyah :
“The index of elliptic operators ,”
pp. 475–498
in
Collected works ,
vol. 3: Index theory 1 .
Oxford Science Publications .
Oxford University Press ,
1988 .
Colloquium Lectures (Dallas, 1973), American Mathematical Society.
BibTeX
@incollection {key45696042,
AUTHOR = {Atiyah, M. F.},
TITLE = {The index of elliptic operators},
BOOKTITLE = {Collected works},
VOLUME = {3: Index theory 1},
SERIES = {Oxford Science Publications},
PUBLISHER = {Oxford University Press},
YEAR = {1988},
PAGES = {475--498},
NOTE = {Colloquium Lectures (Dallas, 1973),
American Mathematical Society.},
}
V. Vassiliev :
“The mathematical legacy of ‘Lacunas for hyperbolic differential equations’ ,”
pp. xxiii–xxviii
in
Raoul Bott: Collected papers ,
vol. 2: Differential operators .
Edited by R. D. MacPherson .
Contemporary Mathematicians .
Birkhäuser (Boston, MA ),
1994 .
Commentary on two-part article published in Acta Math. 124 (1970)Acta Math. 131 (1973)
MR
1290364 incollection
People
BibTeX
@incollection {key1290364m,
AUTHOR = {Vassiliev, Victor},
TITLE = {The mathematical legacy of ``{L}acunas
for hyperbolic differential equations''},
BOOKTITLE = {Raoul {B}ott: {C}ollected papers},
EDITOR = {Robert D. MacPherson},
VOLUME = {2: Differential operators},
SERIES = {Contemporary Mathematicians},
PUBLISHER = {Birkh\"auser},
ADDRESS = {Boston, MA},
YEAR = {1994},
PAGES = {xxiii--xxviii},
NOTE = {Commentary on two-part article published
in \textit{Acta Math.} \textbf{124}
(1970) and \textit{Acta Math.} \textbf{131}
(1973). MR:1290364.},
ISSN = {0884-7037},
ISBN = {9780817636463},
}
M. F. Atiyah :
“Hyperbolic differential equations and algebraic geometry (after Petrowsky) 87–99
in
Séminaire Bourbaki 10: Années 1966/67–1967/68 .
Société Mathématique de France (Paris ),
1995 .
Exposé no. 319.
Republication of 1968 original .
MR
1610456 incollection
People
BibTeX
@incollection {key1610456m,
AUTHOR = {Atiyah, Michael F.},
TITLE = {Hyperbolic differential equations and
algebraic geometry (after {P}etrowsky)},
BOOKTITLE = {S\'eminaire {B}ourbaki 10: {A}nn\'ees
1966/67--1967/68},
PUBLISHER = {Soci\'et\'e Math\'ematique de France},
ADDRESS = {Paris},
YEAR = {1995},
PAGES = {87--99},
URL = {http://www.numdam.org/item?id=SB_1966-1968__10__87_0},
NOTE = {Expos\'e no.~319. Republication of 1968
original. MR:1610456.},
ISBN = {9782856290439},
}
M. F. Atiyah :
“The index of elliptic operators on compact manifolds ,”
pp. 159–169
in
Séminaire Bourbaki ,
vol. 8, Années 1962–63, 1963–64 .
Astérisque .
Séminaire Bourbaki .
Société Mathématique de France (Paris ),
1995 .
Exposé no. 253.
Reprint of article in Bull. Am. Math. Soc. 69 (1963)Séminaire Bourbaki. 15e année: 1962/63 (1964)
MR
1611539 incollection
BibTeX
@incollection {key1611539m,
AUTHOR = {Atiyah, Michael F.},
TITLE = {The index of elliptic operators on compact
manifolds},
BOOKTITLE = {S\'eminaire Bourbaki},
VOLUME = {8, Ann\'ees 1962--63, 1963--64},
SERIES = {Ast\'erisque},
ORGANIZATION = {S\'eminaire Bourbaki},
PUBLISHER = {Soci\'et\'e Math\'ematique de France},
ADDRESS = {Paris},
YEAR = {1995},
PAGES = {159--169},
NOTE = {Expos\'e no.~253. Reprint of article
in \textit{Bull. Am. Math. Soc.} \textbf{69}
(1963). See also \textit{S\'eminaire
Bourbaki. 15e ann\'ee: 1962/63} (1964).
MR:1611539.},
ISSN = {0303-1179},
ISBN = {9782856290415},
}
M. F. Atiyah :
“The index of elliptic operators ,”
pp. 115–127
in
Fields Medallists’ lectures .
Edited by M. F. Atiyah and D. Iagolnitzer .
World Scientific Series in 20th Century Mathematics 5 .
World Scientific (River Edge, NJ ),
1997 .
Republication of notes distributed at AMS conference (1973) .
MR
1622942 incollection
Abstract
People
BibTeX
The index theorem is an outgrowth of the Riemann–Roch theorem in algebraic geometry and in these lectures I shall follow its historical development, starting from the theory of algebraic curves and gradually leading up to the modern developments. Since the Riemann–Roch theorem has been a central theorem in algebraic geometry the history of the theorem is to a great extent a history of algebraic geometry. My purpose therefore is really to use the theorem as a focus for a general historical survey.
@incollection {key1622942m,
AUTHOR = {Atiyah, Michael F.},
TITLE = {The index of elliptic operators},
BOOKTITLE = {Fields {M}edallists' lectures},
EDITOR = {Michael Francis Atiyah and Daniel Iagolnitzer},
SERIES = {World Scientific Series in 20th Century
Mathematics},
NUMBER = {5},
PUBLISHER = {World Scientific},
ADDRESS = {River Edge, NJ},
YEAR = {1997},
PAGES = {115--127},
NOTE = {Republication of notes distributed at
AMS conference (1973). MR:1622942.},
ISBN = {9789810231026},
}
