M. F. Atiyah and R. S. Ward :
“Instantons and algebraic geometry Comm. Math. Phys.
55 : 2
(1977 ),
pp. 117–124 .
MR
0494098 Zbl
0362.14004 article
Abstract
People
BibTeX
Minimum action solutions for \( \mathit{SU}(2) \) Yang–Mills fields in Euclidean 4-space correspond, via the Penrose twistor transform, to algebraic bundles on the complex projective 3-space. These bundles in turn correspond to algebraic curves. The implication of these results for the Yang–Mills fields is described. In particular all solutions are rational and can be constructed from a series of Ansätze \( A_l \) for \( l \geq 1 \) .
@article {key0494098m,
AUTHOR = {Atiyah, M. F. and Ward, R. S.},
TITLE = {Instantons and algebraic geometry},
JOURNAL = {Comm. Math. Phys.},
FJOURNAL = {Communications in Mathematical Physics},
VOLUME = {55},
NUMBER = {2},
YEAR = {1977},
PAGES = {117--124},
DOI = {10.1007/BF01626514},
NOTE = {MR:0494098. Zbl:0362.14004.},
ISSN = {0010-3616},
}
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 :
“Geometry of Yang–Mills fields ,”
pp. 216–221
in
Mathematical problems in theoretical physics
(Rome, 6–15 June 1977 ).
Edited by G. Dell’Antonio .
Lecture Notes in Physics 80 .
Springer (Berlin ),
1978 .
See also Geometry of Yang–Mills fields (1979)
MR
518436 incollection
Abstract
People
BibTeX
In this talk I shall explain how information about classical solutions of Yang–Mills equations can be obtained, rather surprisingly, from algebraic geometry. Although direct physical interest is restricted to the case of four dimensions I shall begin by discussing the two-dimensional case. Besides preparing the ground for the four-dimensional problem this has independent mathematical (and possibly physical) interest, and very complete results can be obtained.
@incollection {key518436m,
AUTHOR = {Atiyah, M. F.},
TITLE = {Geometry of {Y}ang--{M}ills fields},
BOOKTITLE = {Mathematical problems in theoretical
physics},
EDITOR = {Dell'Antonio, Gianfausto},
SERIES = {Lecture Notes in Physics},
NUMBER = {80},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1978},
PAGES = {216--221},
NOTE = {(Rome, 6--15 June 1977). See also \textit{Geometry
of Yang--Mills fields} (1979). MR:518436.},
ISSN = {0075-8450},
ISBN = {9783540088530},
}
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, N. J. Hitchin, V. G. Drinfel’d, and Yu. I. Manin :
“Construction of instantons Phys. Lett. A
65 : 3
(1978 ),
pp. 185–187 .
MR
598562 article
Abstract
People
BibTeX
@article {key598562m,
AUTHOR = {Atiyah, M. F. and Hitchin, N. J. and
Drinfel{\cprime}d, V. G. and Manin,
Yu. I.},
TITLE = {Construction of instantons},
JOURNAL = {Phys. Lett. A},
FJOURNAL = {Physics Letters A},
VOLUME = {65},
NUMBER = {3},
YEAR = {1978},
PAGES = {185--187},
DOI = {10.1016/0375-9601(78)90141-X},
NOTE = {MR:598562.},
ISSN = {0031-9163},
}
M. F. Atiyah and J. D. S. Jones :
“Topological aspects of Yang–Mills theory Comm. Math. Phys.
61 : 2
(1978 ),
pp. 97–118 .
MR
503187 Zbl
0387.55009 article
Abstract
People
BibTeX
@article {key503187m,
AUTHOR = {Atiyah, M. F. and Jones, J. D. S.},
TITLE = {Topological aspects of {Y}ang--{M}ills
theory},
JOURNAL = {Comm. Math. Phys.},
FJOURNAL = {Communications in Mathematical Physics},
VOLUME = {61},
NUMBER = {2},
YEAR = {1978},
PAGES = {97--118},
DOI = {10.1007/BF01609489},
NOTE = {MR:503187. Zbl:0387.55009.},
ISSN = {0010-3616},
}
M. F. Atiyah :
Geometry of Yang–Mills fields .
Lezioni Fermiane .
Accademia Nazionale dei Lincei & Scuola Normale Superiore Pisa (Pisa ),
1979 .
Reprinted in Atiyah’s Collected works , vol. 5 (1988)Mathematical problems in theoretical physics (1978)
MR
554924 Zbl
0435.58001 book
BibTeX
@book {key554924m,
AUTHOR = {Atiyah, M. F.},
TITLE = {Geometry of {Y}ang--{M}ills fields},
SERIES = {Lezioni Fermiane},
PUBLISHER = {Accademia Nazionale dei Lincei \& Scuola
Normale Superiore Pisa},
ADDRESS = {Pisa},
YEAR = {1979},
PAGES = {98},
NOTE = {Reprinted in Atiyah's \textit{Collected
works}, vol.~5 (1988). See also article
in \textit{Mathematical problems in
theoretical physics} (1978). MR:554924.
Zbl:0435.58001.},
ISBN = {9788876423031},
}
M. F. Atiyah :
“Real and complex geometry in four dimensions ,”
pp. 1–10
in
The Chern Symposium 1979
(Berkeley, CA, June 1979 ).
Edited by W. Y. Hsiang .
Springer (New York ),
1980 .
MR
609554 Zbl
0454.53043 incollection
Abstract
People
BibTeX
Some fifty years ago Einstein’s theory of general relativity provided a great stimulus for different geometry, but after a period of fruitful interaction the interests of geometers and physicists diverged. Within the past few years there has been a resurgence of geometrical ideas in physics, arising partly from the popularity of gauge theories in elementary-particle physics and partly from the work of Hawking and Penrose on black holes. A characteristic feature in both cases has been the significance of global or topological properties, and it is this feature which has particularly attracted mathematicians.
The ideas and problems arising in this way have had a considerable impact on geometry, especially the differential geometry of four dimensions. Through the remarkable twistor theory of Penrose [1977] there is also an intimate connection with complex analytic geometry. In this lecture I shall attempt to survey these mathematical developments with especial emphasis on Penrose’s ideas and their application.
@incollection {key609554m,
AUTHOR = {Atiyah, M. F.},
TITLE = {Real and complex geometry in four dimensions},
BOOKTITLE = {The {C}hern {S}ymposium 1979},
EDITOR = {Wu Yi Hsiang},
PUBLISHER = {Springer},
ADDRESS = {New York},
YEAR = {1980},
PAGES = {1--10},
NOTE = {(Berkeley, CA, June 1979). MR:609554.
Zbl:0454.53043.},
ISBN = {9780387905372},
}
M. F. Atiyah and R. Bott :
“Yang–Mills and bundles over algebraic curves ,”
pp. 11–20
in
Geometry and analysis: Papers dedicated to the memory of V. K. Patodi .
Indian Academy of Sciences (Bangalore ),
1980 .
Republished in Proc. Indian Acad. Sci. Math. Sci. 90 :1 (1981)
MR
592249 Zbl
0482.14007 incollection
People
BibTeX
@incollection {key592249m,
AUTHOR = {Atiyah, M. F. and Bott, R.},
TITLE = {Yang--{M}ills and bundles over algebraic
curves},
BOOKTITLE = {Geometry and analysis: {P}apers dedicated
to the memory of {V}.~{K}. {P}atodi},
PUBLISHER = {Indian Academy of Sciences},
ADDRESS = {Bangalore},
YEAR = {1980},
PAGES = {11--20},
NOTE = {Republished in \textit{Proc. Indian
Acad. Sci. Math. Sci.} \textbf{90}:1
(1981). MR:592249. Zbl:0482.14007.},
ISBN = {9780387102702},
}
M. F. Atiyah :
“Geometrical aspects of gauge theories 881–885
in
Proceedings of the International Congress of Mathematicians
(Helsinki, 15–23 August 1978 ),
vol. 2 .
Edited by O. Lehto .
Academia Scientiarum Fennica (Helsinki ),
1980 .
MR
562703 Zbl
0428.58018 incollection
People
BibTeX
@incollection {key562703m,
AUTHOR = {Atiyah, M. F.},
TITLE = {Geometrical aspects of gauge theories},
BOOKTITLE = {Proceedings of the {I}nternational {C}ongress
of {M}athematicians},
EDITOR = {Lehto, Olli},
VOLUME = {2},
PUBLISHER = {Academia Scientiarum Fennica},
ADDRESS = {Helsinki},
YEAR = {1980},
PAGES = {881--885},
URL = {http://www.mathunion.org/ICM/ICM1978.2/Main/icm1978.2.0881.0886.ocr.pdf},
NOTE = {(Helsinki, 15--23 August 1978). MR:562703.
Zbl:0428.58018.},
ISBN = {9789514103520},
}
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 :
“Green’s functions for self-dual four-manifolds ,”
pp. 129–158
in
Mathematical analysis and applications, Part A ,
published as Adv. in Math. Suppl. Stud.
7A .
Issue edited by L. Nachbin .
Academic Press (New York ),
1981 .
MR
634238 Zbl
0484.53036 incollection
Abstract
People
BibTeX
The Penrose twistor theory converts problems of four-dimensional Riemannian geometry into complex analytic geometry of three dimensions. It applies whenever the Riemannian curvature is self-dual, in the sense that the Weyl tensor (the conformally invariant part of the Riemannian curvature) is self-dual. For a general exposition of this theory (in the positive definite case) see [Atiyah, Hitchin and Singer 1978]. Now on a Riemannian four-manifold there is a conformally invariant version of the Laplace operator (obtained by adding one-sixth of the scalar curvature of the usual Laplacian) and this has therefore a conformally invariant Green’s function (under appropriate global hypotheses). The purpose of this chapter is to show what corresponds to this Green’s function in the twistor theory. It turns out that we get a very well-known object in complex analytic geometry associated to the diagonal in \( Z \times Z \) , where \( Z \) is the twistor space.
@article {key634238m,
AUTHOR = {Atiyah, M. F.},
TITLE = {Green's functions for self-dual four-manifolds},
JOURNAL = {Adv. in Math. Suppl. Stud.},
FJOURNAL = {Advances in Mathematics. Supplementary
Studies},
VOLUME = {7A},
YEAR = {1981},
PAGES = {129--158},
NOTE = {\textit{Mathematical analysis and applications,
{P}art {A}}. Issue edited by L. Nachbin.
MR:634238. Zbl:0484.53036.},
ISBN = {9780125128018},
}
M. F. Atiyah and R. Bott :
“Yang–Mills and bundles over algebraic curves 11–20
in
Geometry and analysis: Papers dedicated to the memory of V. K. Patodi ,
published as Proc. Indian Acad. Sci., Math. Sci.
90 : 1 .
Indian Academy of Sciences (Bangalore ),
1981 .
MR
653942 Zbl
0499.14005 incollection
People
BibTeX
@article {key653942m,
AUTHOR = {Atiyah, M. F. and Bott, R.},
TITLE = {Yang--{M}ills and bundles over algebraic
curves},
JOURNAL = {Proc. Indian Acad. Sci., Math. Sci.},
FJOURNAL = {Proceedings of the Indian Academy of
Sciences. Mathematical Sciences},
VOLUME = {90},
NUMBER = {1},
YEAR = {1981},
PAGES = {11--20},
DOI = {10.1007/BF02867013},
NOTE = {\textit{Geometry and analysis: {P}apers
dedicated to the memory of {V}.~{K}.
{P}atodi}. MR:653942. Zbl:0499.14005.},
ISSN = {0253-4142},
ISBN = {9780387102702},
}
M. F. Atiyah :
“Convexity and commuting Hamiltonians Bull. London Math. Soc.
14 : 1
(1982 ),
pp. 1–15 .
MR
642416 Zbl
0482.58013 article
BibTeX
@article {key642416m,
AUTHOR = {Atiyah, M. F.},
TITLE = {Convexity and commuting {H}amiltonians},
JOURNAL = {Bull. London Math. Soc.},
FJOURNAL = {Bulletin of the London Mathematical
Society},
VOLUME = {14},
NUMBER = {1},
YEAR = {1982},
PAGES = {1--15},
DOI = {10.1112/blms/14.1.1},
NOTE = {MR:642416. Zbl:0482.58013.},
ISSN = {0024-6093},
}
M. F. Atiyah :
“Gauge theory and algebraic geometry ,”
pp. 1–20
in
Differential geometry and differential equations
(Beijing, 1980 ),
vol. 1 .
Edited by S.-S. Chern and W.-t. Wu .
Science Press, Gordon and Breach (Beijing, New York ),
1982 .
Zbl
0544.14012 incollection
People
BibTeX
@incollection {key0544.14012z,
AUTHOR = {Atiyah, Michael F.},
TITLE = {Gauge theory and algebraic geometry},
BOOKTITLE = {Differential geometry and differential
equations},
EDITOR = {Shiing-Shen Chern and Wen-ts\"un Wu},
VOLUME = {1},
PUBLISHER = {Science Press, Gordon and Breach},
ADDRESS = {Beijing, New York},
YEAR = {1982},
PAGES = {1--20},
NOTE = {(Beijing, 1980). Zbl:0544.14012.},
ISBN = {9780677310008},
}
M. F. Atiyah :
“Solutions of classical equations ,”
pp. 207–219
in
Gauge theories: Fundamental interactions and rigorous results
(Poiana Braşov, Romania, summer 1981 ).
Edited by P. Dita, V. Georgescu, and R. Purice .
Progress in Physics 5 .
Birkhä user (Boston ),
1982 .
Zbl
0539.70028 incollection
People
BibTeX
@incollection {key0539.70028z,
AUTHOR = {Atiyah, Michael F.},
TITLE = {Solutions of classical equations},
BOOKTITLE = {Gauge theories: {F}undamental interactions
and rigorous results},
EDITOR = {Dita, Petre and Georgescu, Vladimir
and Purice, Radu},
SERIES = {Progress in Physics},
NUMBER = {5},
PUBLISHER = {Birkh\"a user},
ADDRESS = {Boston},
YEAR = {1982},
PAGES = {207--219},
NOTE = {(Poiana Bra\c{s}ov, Romania, summer
1981). Zbl:0539.70028.},
ISSN = {0736-7422},
ISBN = {9783764330958},
}
M. F. Atiyah :
“Geometry of monopoles ,”
pp. 3–20
in
Monopoles in quantum field theory
(Trieste, December 1981 ).
Edited by N. S. Craigie, P. Goddard, and W. Nahm .
World Scientific (Singapore ),
1982 .
MR
766750 incollection
People
BibTeX
@incollection {key766750m,
AUTHOR = {Atiyah, M. F.},
TITLE = {Geometry of monopoles},
BOOKTITLE = {Monopoles in quantum field theory},
EDITOR = {N. S. Craigie and P. Goddard and W.
Nahm},
PUBLISHER = {World Scientific},
ADDRESS = {Singapore},
YEAR = {1982},
PAGES = {3--20},
NOTE = {(Trieste, December 1981). MR:766750.},
ISBN = {9789971950293},
}
M. F. Atiyah and A. N. Pressley :
“Convexity and loop groups ,”
pp. 33–63
in
Arithmetic and geometry: Papers dedicated to I. R. Shafarevich ,
vol. 2 .
Edited by M. Artin and I. R. Šafarevič .
Progress in Mathematics 36 .
Birkhäuser (Boston ),
1983 .
MR
717605 Zbl
0529.22013 incollection
People
BibTeX
@incollection {key717605m,
AUTHOR = {Atiyah, M. F. and Pressley, A. N.},
TITLE = {Convexity and loop groups},
BOOKTITLE = {Arithmetic and geometry: {P}apers dedicated
to {I}.~{R}. {S}hafarevich},
EDITOR = {Michael Artin and Igor R. \v{S}afarevi\v{c}},
VOLUME = {2},
SERIES = {Progress in Mathematics},
NUMBER = {36},
PUBLISHER = {Birkh\"auser},
ADDRESS = {Boston},
YEAR = {1983},
PAGES = {33--63},
NOTE = {MR:717605. Zbl:0529.22013.},
ISSN = {0743-1643},
ISBN = {9783764331320},
}
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 :
“Angular momentum, convex polyhedra and algebraic geometry Proc. Edinburgh Math. Soc. (2)
26 : 2
(1983 ),
pp. 121–133 .
MR
705256 Zbl
0521.58026 article
Abstract
BibTeX
The three families of classical groups of linear transformations (complex, orthogonal, symplectic) give rise to the three great branches of differential geometry (complex analytic, Riemannian and symplectic). Complex analytic geometry derives most of its interest from complex algebraic geometry, while symplectic geometry provides the general framework for Hamiltonian mechanics.
These three classical groups “intersect” in the unitary group and the three branches of differential geometry correspondingly “intersect” in Kähler geometry, which includes the study of algebraic varieties in projective space. This is the basic reason why Hodge was successful in applying Riemannian methods to algebraic geometry in his theory of harmonic forms.
@article {key705256m,
AUTHOR = {Atiyah, M. F.},
TITLE = {Angular momentum, convex polyhedra and
algebraic geometry},
JOURNAL = {Proc. Edinburgh Math. Soc. (2)},
FJOURNAL = {Proceedings of the Edinburgh Mathematical
Society, second series},
VOLUME = {26},
NUMBER = {2},
YEAR = {1983},
PAGES = {121--133},
DOI = {10.1017/S0013091500016837},
NOTE = {MR:705256. Zbl:0521.58026.},
ISSN = {0013-0915},
}
M. Atiyah :
“Anomalies and index theory 313–322
in
Supersymmetry and supergravity/nonperturbative QCD
(Mahabaleshwar, India, 5–19 January 1984 ).
Edited by P. Roy and V. Singh .
Lecture Notes in Mathematics 208 .
Springer (Berlin ),
1984 .
MR
774599 incollection
Abstract
People
BibTeX
There is at present considerable interest amongst physicists in various anomalies that occur in gauge theories and their relation to the index theory of the Dirac operator. Numerous papers have appeared on the subject [Atiyah and Singer 1984; Alvarez, Singer and Zumino 1984; Alverez-Gaumé and Witten 1983; Alverez-Gaumé and Ginsparg 1983; Baulieu 1984a; 1984b; Sumitani 1984] and it is the purpose of this lecture to provide a general introduction and guide to the subject. I will concentrate on the main ideas since the detailed calculations are fully available in the literature.
@incollection {key774599m,
AUTHOR = {Atiyah, Michael},
TITLE = {Anomalies and index theory},
BOOKTITLE = {Supersymmetry and supergravity/nonperturbative
{QCD}},
EDITOR = {Probir Roy and Virendra Singh},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {208},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1984},
PAGES = {313--322},
DOI = {10.1007/3-540-13390-9_10},
NOTE = {(Mahabaleshwar, India, 5--19 January
1984). MR:774599.},
ISSN = {0075-8434},
ISBN = {9780387133904},
}
M. F. Atiyah :
“The moment map in symplectic geometry ,”
pp. 43–51
in
Global Riemannian geometry
(Durham University, UK, July 1983 ).
Edited by T. Willmore and N. J. Hitchin .
Mathematics and its Applications .
Ellis Horwood (Chichester, UK ),
1984 .
MR
757204 Zbl
0615.53023 incollection
People
BibTeX
@incollection {key757204m,
AUTHOR = {Atiyah, M. F.},
TITLE = {The moment map in symplectic geometry},
BOOKTITLE = {Global {R}iemannian geometry},
EDITOR = {Thomas Willmore and Nigel J. Hitchin},
SERIES = {Mathematics and its Applications},
PUBLISHER = {Ellis Horwood},
ADDRESS = {Chichester, UK},
YEAR = {1984},
PAGES = {43--51},
NOTE = {(Durham University, UK, July 1983).
MR:757204. Zbl:0615.53023.},
ISBN = {9780470200179},
}
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},
}
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 :
“Instantons in two and four dimensions Comm. Math. Phys.
93 : 4
(1984 ),
pp. 437–451 .
MR
763752 Zbl
0564.58040 article
Abstract
BibTeX
@article {key763752m,
AUTHOR = {Atiyah, M. F.},
TITLE = {Instantons in two and four dimensions},
JOURNAL = {Comm. Math. Phys.},
FJOURNAL = {Communications in Mathematical Physics},
VOLUME = {93},
NUMBER = {4},
YEAR = {1984},
PAGES = {437--451},
DOI = {10.1007/BF01212288},
URL = {http://projecteuclid.org/getRecord?id=euclid.cmp/1103941176},
NOTE = {MR:763752. Zbl:0564.58040.},
ISSN = {0010-3616},
}
M. F. Atiyah :
“The Yang–Mills equations and the structure of 4-manifolds ,”
pp. 11–17
in
Global Riemannian geometry
(Durham University, UK, July 1983 ).
Edited by T. Willmore and N. J. Hitchin .
Series in Mathematics and its Applications .
Ellis Horwood (Chichester, UK ),
1984 .
MR
757200 Zbl
0614.57007 incollection
People
BibTeX
@incollection {key757200m,
AUTHOR = {Atiyah, M. F.},
TITLE = {The {Y}ang--{M}ills equations and the
structure of 4-manifolds},
BOOKTITLE = {Global {R}iemannian geometry},
EDITOR = {Thomas Willmore and Nigel J. Hitchin},
SERIES = {Series in Mathematics and its Applications},
PUBLISHER = {Ellis Horwood},
ADDRESS = {Chichester, UK},
YEAR = {1984},
PAGES = {11--17},
NOTE = {(Durham University, UK, July 1983).
MR:757200. Zbl:0614.57007.},
ISBN = {9780470200179},
}
M. F. Atiyah and N. J. Hitchin :
“Low energy scattering of nonabelian monopoles Phys. Lett. A
107 : 1
(1985 ),
pp. 21–25 .
MR
778313 Zbl
1177.53069 article
Abstract
People
BibTeX
@article {key778313m,
AUTHOR = {Atiyah, M. F. and Hitchin, N. J.},
TITLE = {Low energy scattering of nonabelian
monopoles},
JOURNAL = {Phys. Lett. A},
FJOURNAL = {Physics Letters A},
VOLUME = {107},
NUMBER = {1},
YEAR = {1985},
PAGES = {21--25},
DOI = {10.1016/0375-9601(85)90238-5},
NOTE = {MR:778313. Zbl:1177.53069.},
ISSN = {0375-9601},
}
M. Atiyah :
“Eigenvalues of the Dirac operator 251–260
in
Arbeitstagung Bonn 1984
(Max-Planck-Institut für Mathematik, Bonn, 15–22 June 1984 ).
Edited by F. Hirzebruch, J. Schwermer, and S. Suter .
Lecture Notes in Mathematics 1111 .
Springer (Berlin ),
1985 .
MR
797424 Zbl
0568.53022 incollection
Abstract
People
BibTeX
In recent years mathematicians have learnt a great deal from physicists and in particular from the work of Edward Witten. In a recent preprint [1984], Vafa and Witten have proved some striking results about the eigenvalues of the Dirac operator, and this talk will present their results. I shall concentrate entirely on the mathematical parts of their preprint leaving aside the physical interpretation which is their main motivation.
@incollection {key797424m,
AUTHOR = {Atiyah, Michael},
TITLE = {Eigenvalues of the {D}irac operator},
BOOKTITLE = {Arbeitstagung {B}onn 1984},
EDITOR = {Friedrich Hirzebruch and Joachim Schwermer
and Silke Suter},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {1111},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1985},
PAGES = {251--260},
DOI = {10.1007/BFb0084593},
NOTE = {(Max-Planck-Institut f\"ur Mathematik,
Bonn, 15--22 June 1984). MR:797424.
Zbl:0568.53022.},
ISSN = {0075-8434},
ISBN = {9780387151953},
}
M. F. Atiyah :
“Circular symmetry and stationary-phase approximation ,”
pp. 43–59
in
Colloque en l’honneur de Laurent Schwartz
(École Polytechnique, Palaiseau, 30 May–3 June 1983 ).
Astérisque 131 .
Société mathématique de France (Paris ),
1985 .
MR
816738 Zbl
0578.58039 incollection
People
BibTeX
@incollection {key816738m,
AUTHOR = {Atiyah, M. F.},
TITLE = {Circular symmetry and stationary-phase
approximation},
BOOKTITLE = {Colloque en l'honneur de {L}aurent {S}chwartz},
SERIES = {Ast\'erisque},
NUMBER = {131},
PUBLISHER = {Soci\'et\'e math\'ematique de France},
ADDRESS = {Paris},
YEAR = {1985},
PAGES = {43--59},
NOTE = {(\'Ecole Polytechnique, Palaiseau, 30
May--3 June 1983). MR:816738. Zbl:0578.58039.},
ISSN = {0303-1179},
}
M. Atiyah :
“Topological aspects of anomalies ,”
pp. 22–32
in
Symposium on anomalies, geometry, topology
(University of Chicago, 28–30 March 1985 ).
Edited by W. A. Bardeen and A. R. White .
World Scientific (Singapore ),
1985 .
MR
850843 Zbl
0651.58035 incollection
People
BibTeX
@incollection {key850843m,
AUTHOR = {Atiyah, Michael},
TITLE = {Topological aspects of anomalies},
BOOKTITLE = {Symposium on anomalies, geometry, topology},
EDITOR = {William A. Bardeen and Alan R. White},
PUBLISHER = {World Scientific},
ADDRESS = {Singapore},
YEAR = {1985},
PAGES = {22--32},
NOTE = {(University of Chicago, 28--30 March
1985). MR:850843. Zbl:0651.58035.},
ISBN = {9789971978693},
}
M. Atiyah :
“Commentary on the article of Manin 103–109
in
Arbeitstagung Bonn 1984
(Max-Planck-Institut für Mathematik, Bonn, 15–22 June 1984 ).
Edited by F. Hirzebruch, J. Schwermer, and S. Suter .
Lecture Notes in Mathematics 1111 .
Springer (Berlin ),
1985 .
The article is Yu. I. Manin, “New dimensions in geometry,” from the same volume.
MR
797417 Zbl
0595.53071 incollection
Abstract
People
BibTeX
Manin’s stimulating contribution to the 25th Arbeitstagung which, in his absence, I attempted to present, provided me with an opportunity of adding some further reflections of my own. This commentary, which is therefore a very personal response to Manin’s article, consists of very general and speculative remarks about large areas of contemporary mathematics. Such speculations are, for good reason, rarely put down on paper but the record of the 25th Arbeitstagung provides a rather singular occasion where ideas of this type may not be out of place.
@incollection {key797417m,
AUTHOR = {Atiyah, Michael},
TITLE = {Commentary on the article of {M}anin},
BOOKTITLE = {Arbeitstagung {B}onn 1984},
EDITOR = {Friedrich Hirzebruch and Joachim Schwermer
and Silke Suter},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {1111},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1985},
PAGES = {103--109},
DOI = {10.1007/BFb0084586},
NOTE = {(Max-Planck-Institut f\"ur Mathematik,
Bonn, 15--22 June 1984). The article
is {Y}u.\ {I}. {M}anin, ``{N}ew dimensions
in geometry,'' from the same volume.
MR:797417. Zbl:0595.53071.},
ISSN = {0075-8434},
ISBN = {9780387151953},
}
M. Atiyah, N. J. Hitchin, J. T. Stuart, and M. Tabor :
“Low-energy scattering of nonabelian magnetic monopoles [and discussion] Philos. Trans. Roy. Soc. London Ser. A
315 : 1533
(1985 ),
pp. 459–469 .
MR
836746 article
Abstract
People
BibTeX
The Bogomolny equations represent static configurations of magnetic monopoles, for some non-Abelian group such as \( \mathit{SU}(2) \) . Geodesic motion on this configuration space represents slowly moving interacting monopoles. Geometric information on this space can then be used to investigate the scattering of monopoles. The results show surprising features, including a 90 degrees scattering and the conversion of angular momentum into electric charge.
@article {key836746m,
AUTHOR = {Atiyah, Michael and Hitchin, N. J. and
Stuart, J. T. and Tabor, M.},
TITLE = {Low-energy scattering of nonabelian
magnetic monopoles [and discussion]},
JOURNAL = {Philos. Trans. Roy. Soc. London Ser.
A},
FJOURNAL = {Philosophical Transactions of the Royal
Society A: Mathematical, Physical \&
Engineering Sciences},
VOLUME = {315},
NUMBER = {1533},
YEAR = {1985},
PAGES = {459--469},
DOI = {10.1098/rsta.1985.0052},
NOTE = {MR:836746.},
ISSN = {0080-4614},
}
M. F. Atiyah :
“Magnetic monopoles in hyperbolic spaces ,”
pp. 1–33
in
Vector bundles on algebraic varieties
(Bombay, 9–16 January 1984 ).
Studies in Mathematics 11 .
Tata Institute for Fundamental Research (Bombay ),
1987 .
MR
893593 Zbl
0722.53063 incollection
BibTeX
@incollection {key893593m,
AUTHOR = {Atiyah, M. F.},
TITLE = {Magnetic monopoles in hyperbolic spaces},
BOOKTITLE = {Vector bundles on algebraic varieties},
SERIES = {Studies in Mathematics},
NUMBER = {11},
PUBLISHER = {Tata Institute for Fundamental Research},
ADDRESS = {Bombay},
YEAR = {1987},
PAGES = {1--33},
NOTE = {(Bombay, 9--16 January 1984). MR:893593.
Zbl:0722.53063.},
ISBN = {9780195620146},
}
M. Atiyah :
Collected works ,
vol. 5: Gauge theories .
Oxford Science Publications .
The Clarendon Press and Oxford University Press (Oxford, New York ),
1988 .
MR
951896 Zbl
0691.53003 book
BibTeX
@book {key951896m,
AUTHOR = {Atiyah, Michael},
TITLE = {Collected works},
VOLUME = {5: Gauge theories},
SERIES = {Oxford Science Publications},
PUBLISHER = {The Clarendon Press and Oxford University
Press},
ADDRESS = {Oxford, New York},
YEAR = {1988},
PAGES = {xxiii+685},
NOTE = {MR:951896. Zbl:0691.53003.},
ISBN = {9780198532798},
}
