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
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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. 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 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. 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},
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Obshchestvo. Uspekhi Matematicheskikh
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YEAR = {1969},
PAGES = {127--182},
URL = {http://www.mathnet.ru/eng/rm5453},
NOTE = {The English original was published in
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ISSN = {0042-1316},
}
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 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. 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. 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. 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, 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, 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 :
“A survey of \( K \) -theory ,”
pp. 1–9
in
\( K \) -theory and operator algebrasUniversity of Georgia, Athens, GA, 21–25 April 1975 ).
Edited by B. B. Morrel and I. M. Singer .
Springer (Berlin ),
1977 .
MR
0474299 Zbl
0345.55005 incollection
People
BibTeX
@incollection {key0474299m,
AUTHOR = {Atiyah, M. F.},
TITLE = {A survey of \$K\$-theory},
BOOKTITLE = {\$K\$-theory and operator algebras},
EDITOR = {Bernard B. Morrel and I. M. Singer},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1977},
PAGES = {1--9},
NOTE = {(University of Georgia, Athens, GA,
21--25 April 1975). MR:0474299. Zbl:0345.55005.},
ISBN = {9783540081333},
}
M. F. Atiyah, N. J. Hitchin, and I. M. Singer :
“Deformations of instantons Proc. Nat. Acad. Sci. U.S.A.
74 : 7
(July 1977 ),
pp. 2662–2663 .
MR
458424 Zbl
0356.58011 article
Abstract
People
BibTeX
@article {key458424m,
AUTHOR = {Atiyah, M. F. and Hitchin, N. J. and
Singer, I. M.},
TITLE = {Deformations of instantons},
JOURNAL = {Proc. Nat. Acad. Sci. U.S.A.},
FJOURNAL = {Proceedings of the National Academy
of Sciences of the United States of
America},
VOLUME = {74},
NUMBER = {7},
MONTH = {July},
YEAR = {1977},
PAGES = {2662--2663},
URL = {http://www.pnas.org/content/74/7/2662.short},
NOTE = {MR:458424. Zbl:0356.58011.},
ISSN = {0027-8424},
}
M. F. Atiyah, N. J. Hitchin, and I. M. Singer :
“Self-duality in four-dimensional Riemannian geometry Proc. Roy. Soc. London Ser. A
362 : 1711
(September 1978 ),
pp. 425–461 .
MR
506229 Zbl
0389.53011 article
Abstract
People
BibTeX
@article {key506229m,
AUTHOR = {Atiyah, M. F. and Hitchin, N. J. and
Singer, I. M.},
TITLE = {Self-duality in four-dimensional {R}iemannian
geometry},
JOURNAL = {Proc. Roy. Soc. London Ser. A},
FJOURNAL = {Proceedings of the Royal Society. London.
Series A. Mathematical and Physical
Sciences},
VOLUME = {362},
NUMBER = {1711},
MONTH = {September},
YEAR = {1978},
PAGES = {425--461},
DOI = {10.1098/rspa.1978.0143},
NOTE = {MR:506229. Zbl:0389.53011.},
ISSN = {0080-4630},
}
M. F. Atiyah :
“Remarks on Morse theory 1–5
in
Recent developments in gauge theories .
Edited by G. ’t Hooft, C. Itzykson, A. Jaffe, H. Lehmann, P. K. Mitter, I. M. Singer, and R. Stora .
NATO Advanced Study Institutes Series B. Physics. 59 .
Plenum Press (New York and London ),
1980 .
Republished in Atiyah’s Collected works , vol. 5 .
incollection
Abstract
People
BibTeX
Morse theory is a topological approach to the Calculus of Variations. It aims to relate the critical points of a functional to the topology of the function space on which the functional is defined. It is only directly applicable in special rather restrictive conditions, notably for problems involving one independent variable. However I will discuss a number of special examples, in some of which the Morse theory really works, and others in which it clearly fails but where nevertheless some aspects appear still to survive. These examples include those of physical interest and it would be interesting to investigate these further. One can make a number of speculations in this direction.
@incollection {key12001912,
AUTHOR = {Atiyah, M. F.},
TITLE = {Remarks on {M}orse theory},
BOOKTITLE = {Recent developments in gauge theories},
EDITOR = {'t Hooft, G. and Itzykson, C. and Jaffe,
A. and Lehmann, H. and Mitter, P. K.
and Singer, I. M. and Stora, R.},
SERIES = {NATO Advanced Study Institutes Series
B. Physics.},
NUMBER = {59},
PUBLISHER = {Plenum Press},
ADDRESS = {New York and London},
YEAR = {1980},
PAGES = {1--5},
DOI = {10.1007/978-1-4684-7571-5},
NOTE = {Republished in Atiyah's \textit{Collected
works}, vol.~5.},
ISSN = {0258-1221},
ISBN = {9780306404795},
}
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,
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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 and I. M. Singer :
“Dirac operators coupled to vector potentials Proc. Nat. Acad. Sci. U.S.A.
81 : 8
(April 1984 ),
pp. 2597–2600 .
MR
742394 Zbl
0547.58033 article
Abstract
People
BibTeX
@article {key742394m,
AUTHOR = {Atiyah, M. F. and Singer, I. M.},
TITLE = {Dirac operators coupled to vector potentials},
JOURNAL = {Proc. Nat. Acad. Sci. U.S.A.},
FJOURNAL = {Proceedings of the National Academy
of Sciences of the United States of
America},
VOLUME = {81},
NUMBER = {8},
MONTH = {April},
YEAR = {1984},
PAGES = {2597--2600},
URL = {http://www.pnas.org/content/81/8/2597.short},
NOTE = {MR:742394. Zbl:0547.58033.},
ISSN = {0027-8424},
}
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},
}
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},
}
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},
}
Interview with Michael Atiyah and Isadore Singer ,”
Eur. Math. Soc. Newsl.
(September 2004 ),
pp. 24–30 .
Republished in Notices Amer. Math. Soc. 52 :2 (2005)Mitt. Dtsch. Math.-Ver. 12 :4 (2004)
article
People
BibTeX
@article {key75663535,
TITLE = {Interview with {M}ichael {A}tiyah and
{I}sadore {S}inger},
JOURNAL = {Eur. Math. Soc. Newsl.},
FJOURNAL = {European Mathematical Society Newsletter},
MONTH = {September},
YEAR = {2004},
PAGES = {24--30},
NOTE = {Republished in \textit{Notices Amer.
Math. Soc.} \textbf{52}:2 (2005) and
\textit{Mitt. Dtsch. Math.-Ver.} \textbf{12}:4
(2004).},
ISSN = {1027-488X},
}
Atiyah and Singer receive 2004 Abel Prize Notices Amer. Math. Soc.
51 : 6
(2004 ),
pp. 649–650 .
MR
2064152 Zbl
1168.01308 article
People
BibTeX
@article {key2064152m,
TITLE = {Atiyah and {S}inger receive 2004 {A}bel
{P}rize},
JOURNAL = {Notices Amer. Math. Soc.},
FJOURNAL = {Notices of the American Mathematical
Society},
VOLUME = {51},
NUMBER = {6},
YEAR = {2004},
PAGES = {649--650},
URL = {http://www.ams.org/notices/200406/comm-abel.pdf},
NOTE = {MR:2064152. Zbl:1168.01308.},
ISSN = {0002-9920},
}
M. Raussen and C. Skau :
“Interview with Michael Atiyah and Isadore Singer Mitt. Dtsch. Math.-Ver.
12 : 4
(2004 ),
pp. 272–281 .
Originally published in the European Mathematical Society Newsletter , September 2004Notices Amer. Math. Soc. 52 :2 (2005)
MR
2110686 Zbl
1087.01013 article
People
BibTeX
@article {key2110686m,
AUTHOR = {Raussen, Martin and Skau, Christian},
TITLE = {Interview with {M}ichael {A}tiyah and
{I}sadore {S}inger},
JOURNAL = {Mitt. Dtsch. Math.-Ver.},
FJOURNAL = {Deutsche Mathematiker-Vereinigung},
VOLUME = {12},
NUMBER = {4},
YEAR = {2004},
PAGES = {272--281},
DOI = {10.1515/dmvm-2004-0089},
NOTE = {Originally published in the \textit{European
Mathematical Society Newsletter}, September
2004. Also published in \textit{Notices
Amer. Math. Soc.} \textbf{52}:2 (2005).
MR:2110686. Zbl:1087.01013.},
ISSN = {0947-4471},
}
M. Raussen and C. Skau :
“Interview with Michael Atiyah and Isadore Singer Notices Amer. Math. Soc.
52 : 2
(February 2005 ),
pp. 225–233 .
Originally published in the European Mathematical Society Newsletter , September 2004Mitt. Dtsch. Math.-Ver. 12 :4 (2004)
MR
2110633 Zbl
1142.01347 article
People
BibTeX
@article {key2110633m,
AUTHOR = {Raussen, Martin and Skau, Christian},
TITLE = {Interview with {M}ichael {A}tiyah and
{I}sadore {S}inger},
JOURNAL = {Notices Amer. Math. Soc.},
FJOURNAL = {Notices of the American Mathematical
Society},
VOLUME = {52},
NUMBER = {2},
MONTH = {February},
YEAR = {2005},
PAGES = {225--233},
URL = {http://www.ams.org/notices/200502/comm-interview.pdf},
NOTE = {Originally published in the \textit{European
Mathematical Society Newsletter}, September
2004. Also published in \textit{Mitt.
Dtsch. Math.-Ver.} \textbf{12}:4 (2004).
MR:2110633. Zbl:1142.01347.},
ISSN = {0002-9920},
}
M. Atiyah :
“The interaction between geometry and physics ,”
pp. 1–15
in
The unity of mathematics
(Cambridge, MA, 31 August–4 September 2003 ).
Edited by P. I. Ètingof, V. Retakh, and I. M. Singer .
Progress in Mathematics 244 .
Birkhäuser (Boston, MA ),
2006 .
MR
2181802 Zbl
1118.58002 incollection
People
BibTeX
@incollection {key2181802m,
AUTHOR = {Atiyah, Michael},
TITLE = {The interaction between geometry and
physics},
BOOKTITLE = {The unity of mathematics},
EDITOR = {Pavel I. \`Etingof and Vladimir Retakh
and Isadore Manuel Singer},
SERIES = {Progress in Mathematics},
NUMBER = {244},
PUBLISHER = {Birkh\"auser},
ADDRESS = {Boston, MA},
YEAR = {2006},
PAGES = {1--15},
NOTE = {(Cambridge, MA, 31 August--4 September
2003). MR:2181802. Zbl:1118.58002.},
ISSN = {0743-1643},
ISBN = {9780817640767},
}
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},
}
D. Burns, L. Ji, K. Liu, A. Ranicki, and S.-T. Yau :
“Preface ,”
pp. 1–3
in
Special issue: In honor of Michael Atiyah and Isadore Singer ,
published as Pure Appl. Math. Q.
6 : 2
(2010 ).
MR
2761847 incollection
People
BibTeX
@article {key2761847m,
AUTHOR = {Burns, Daniel and Ji, Lizhen and Liu,
Kefeng and Ranicki, Andrew and Yau,
Shing-Tung},
TITLE = {Preface},
JOURNAL = {Pure Appl. Math. Q.},
VOLUME = {6},
NUMBER = {2},
YEAR = {2010},
PAGES = {1--3},
NOTE = {\textit{Special issue: {I}n honor of
{M}ichael {A}tiyah and {I}sadore {S}inger}.
MR:2761847.},
ISSN = {1558-8599},
}
Special issue: In honor of Michael Atiyah and Isadore Singer ,
published as Pure Appl. Math. Q.
6 : 2
(2010 ).
book
People
BibTeX
@book {key18429134,
TITLE = {Special issue: {I}n honor of {M}ichael
{A}tiyah and {I}sadore {S}inger},
YEAR = {2010},
PAGES = {645},
NOTE = {Published as \textit{Pure Appl. Math.
Q.} \textbf{6}:2.},
ISSN = {1558-8599},
}
