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 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 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. 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. 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. 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) . 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, 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. 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. 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},
}
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) and 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},
}