M. F. Atiyah, R. Bott, and A. Shapiro :
“Clifford modules ,”
Topology
3 : Supplement 1
(July 1964 ),
pp. 3–38 .
MR
0167985
Zbl
0146.19001
article

People
BibTeX
@article {key0167985m,
AUTHOR = {Atiyah, M. F. and Bott, R. and Shapiro,
A.},
TITLE = {Clifford modules},
JOURNAL = {Topology},
FJOURNAL = {Topology},
VOLUME = {3},
NUMBER = {Supplement 1},
MONTH = {July},
YEAR = {1964},
PAGES = {3--38},
DOI = {10.1016/0040-9383(64)90003-5},
NOTE = {MR:0167985. Zbl:0146.19001.},
ISSN = {0040-9383},
}
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. Atiyah and R. Bott :
“On the periodicity theorem for complex vector bundles ,”
Acta Math.
112 : 1
(1964 ),
pp. 229–247 .
MR
0178470
Zbl
0131.38201
article

Abstract
People
BibTeX

The periodicity theorem for the infinite unitary group [Bott 1959] can be interpreted as a statement about complex vector bundles. As such it describes the relation between vector bundles over \( X \) and \( X\times S^2 \) , where \( X \) is a compact space and \( S^2 \) is the 2-sphere. This relation is most succinctly expressed by the formula
\[ K(X\times S^2) \simeq K(X)\otimes K(S^2), \]
where \( K(X) \) is the Grothendieck group of complex vector bundles over \( X \) . The general theory of these \( K \) -groups, as developed in [Atiyah and Hirzebruch 1961], has found many applications in topology and related fields. Since the periodicity theorem is the foundation stone of all this theory it seems desirable to have an elementary proof of it, and it is the purpose of this paper to present such a proof.

@article {key0178470m,
AUTHOR = {Atiyah, Michael and Bott, Raoul},
TITLE = {On the periodicity theorem for complex
vector bundles},
JOURNAL = {Acta Math.},
FJOURNAL = {Acta Mathematica},
VOLUME = {112},
NUMBER = {1},
YEAR = {1964},
PAGES = {229--247},
DOI = {10.1007/BF02391772},
NOTE = {MR:0178470. Zbl:0131.38201.},
ISSN = {0001-5962},
}
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, 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. 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) . A Russian translation was published in 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 and R. Bott :
“The Yang–Mills equations over Riemann surfaces ,”
Philos. Trans. R. Soc. Lond., A
308 : 1505
(1983 ),
pp. 523–615 .
MR
702806
Zbl
0509.14014
article

Abstract
People
BibTeX

The Yang–Mills functional over a Riemann surface is studied from the point of view of Morse theory. The main result is that this is a ‘perfect’ functional provided due account is taken of its gauge symmetry. This enables topological conclusions to be drawn about the critical sets and leads eventually to information about the moduli space of algebraic bundles over the Riemann surface. This in turn depends on the interplay between the holomorphic and unitary structures, which is analysed in detail.

@article {key702806m,
AUTHOR = {Atiyah, M. F. and Bott, R.},
TITLE = {The {Y}ang--{M}ills equations over {R}iemann
surfaces},
JOURNAL = {Philos. Trans. R. Soc. Lond., A},
FJOURNAL = {Philosophical Transactions of the Royal
Society of London. Series A. Mathematical
and Physical Sciences},
VOLUME = {308},
NUMBER = {1505},
YEAR = {1983},
PAGES = {523--615},
DOI = {10.1098/rsta.1983.0017},
NOTE = {MR:702806. Zbl:0509.14014.},
ISSN = {0080-4614},
CODEN = {PTRMAD},
}
M. F. Atiyah and R. Bott :
“The moment map and equivariant cohomology ,”
Topology
23 : 1
(1984 ),
pp. 1–28 .
MR
721448
Zbl
0521.58025
article

Abstract
People
BibTeX

The purpose of this note is to present a de Rham version of the localization theorems of equivariant cohomology, and to point out their relation to a recent result of Duistermaat and Heckman and also to a quite independent result of Witten. To a large extent all the material that we use has been around for some time, although equivariant cohomology is not perhaps familiar to analysts. Our contribution is therefore mainly an expository one linking together various points of view.

@article {key721448m,
AUTHOR = {Atiyah, M. F. and Bott, R.},
TITLE = {The moment map and equivariant cohomology},
JOURNAL = {Topology},
FJOURNAL = {Topology},
VOLUME = {23},
NUMBER = {1},
YEAR = {1984},
PAGES = {1--28},
DOI = {10.1016/0040-9383(84)90021-1},
NOTE = {MR:721448. Zbl:0521.58025.},
ISSN = {0040-9383},
CODEN = {TPLGAF},
}
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},
}