Atiyah–Singer index theorem (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Atiyah–Singer index theorem" in English language version.

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alainconnes.org

Connes & Moscovici 1990. Connes, A.; Moscovici, H. (1990), "Cyclic cohomology, the Novikov conjecture and hyperbolic groups" (PDF), Topology, 29 (3): 345–388, doi:10.1016/0040-9383(90)90003-3, Zbl 0759.58047

ams.org

mathscinet.ams.org

Atiyah, Bott & Patodi 1973. Atiyah, M.; Bott, R.; Patodi, V. K. (1973), "On the heat equation and the index theorem", Invent. Math., 19 (4): 279–330, Bibcode:1973InMat..19..279A, doi:10.1007/BF01425417, MR 0650828, S2CID 115700319. Atiyah, M.; Bott, R.; Patodi, V. K. (1975), "Errata", Invent. Math., 28 (3): 277–280, Bibcode:1975InMat..28..277A, doi:10.1007/BF01425562, MR 0650829
Witten 1982. Witten, Edward (1982), "Supersymmetry and Morse theory", J. Diff. Geom., 17 (4): 661–692, doi:10.4310/jdg/1214437492, MR 0683171

arxiv.org

Kayani 2020. Kayani, U. (2020). "Dynamical supersymmetry enhancement of black hole horizons". arXiv:1910.01080 [hep-th].
Hamilton 2020, p. 11. Hamilton, M. J. D. (2020). "The Higgs boson for mathematicians. Lecture notes on gauge theory and symmetry breaking". arXiv:1512.02632 [math.DG].

books.google.com

Palais 1965. Palais, Richard S. (1965), Seminar on the Atiyah–Singer Index Theorem, Annals of Mathematics Studies, vol. 57, S.l.: Princeton Univ Press, ISBN 978-0-691-08031-4 This describes the original proof of the theorem (Atiyah and Singer never published their original proof themselves, but only improved versions of it.)

doi.org

Atiyah & Singer 1963. Atiyah, Michael F.; Singer, Isadore M. (1963), "The Index of Elliptic Operators on Compact Manifolds", Bull. Amer. Math. Soc., 69 (3): 422–433, doi:10.1090/S0002-9904-1963-10957-X An announcement of the index theorem.
Gel'fand 1960. Gel'fand, I. M. (1960), "On elliptic equations", Russ. Math. Surv., 15 (3): 113–123, Bibcode:1960RuMaS..15..113G, doi:10.1070/rm1960v015n03ABEH004094 reprinted in volume 1 of his collected works, p. 65–75, ISBN 0-387-13619-3. On page 120 Gel'fand suggests that the index of an elliptic operator should be expressible in terms of topological data.
Atiyah & Singer 1968a. Atiyah, Michael F.; Singer, Isadore M. (1968a), "The Index of Elliptic Operators I", Annals of Mathematics, 87 (3): 484–530, doi:10.2307/1970715, JSTOR 1970715 This gives a proof using K-theory instead of cohomology.
Atiyah & Singer (1968a); Atiyah & Singer (1968b); Atiyah & Singer (1971a); Atiyah & Singer (1971b). Atiyah, Michael F.; Singer, Isadore M. (1968a), "The Index of Elliptic Operators I", Annals of Mathematics, 87 (3): 484–530, doi:10.2307/1970715, JSTOR 1970715 This gives a proof using K-theory instead of cohomology. Atiyah, Michael F.; Singer, Isadore M. (1968b), "The Index of Elliptic Operators III", Annals of Mathematics, Second Series, 87 (3): 546–604, doi:10.2307/1970717, JSTOR 1970717 This paper shows how to convert from the K-theory version to a version using cohomology. Atiyah, Michael F.; Singer, Isadore M. (1971a), "The Index of Elliptic Operators IV", Annals of Mathematics, Second Series, 93 (1): 119–138, doi:10.2307/1970756, JSTOR 1970756 This paper studies families of elliptic operators, where the index is now an element of the K-theory of the space parametrizing the family. Atiyah, Michael F.; Singer, Isadore M. (1971b), "The Index of Elliptic Operators V", Annals of Mathematics, Second Series, 93 (1): 139–149, doi:10.2307/1970757, JSTOR 1970757. This studies families of real (rather than complex) elliptic operators, when one can sometimes squeeze out a little extra information.
Kirby & Siebenmann 1969. Kirby, R.; Siebenmann, L.C. (1969), "On the triangulation of manifolds and the Hauptvermutung", Bull. Amer. Math. Soc., 75 (4): 742–749, doi:10.1090/S0002-9904-1969-12271-8
Kasparov 1972. Kasparov, G.G. (1972), "Topological invariance of elliptic operators, I: K-homology", Math. USSR Izvestija (Engl. Transl.), 9 (4): 751–792, Bibcode:1975IzMat...9..751K, doi:10.1070/IM1975v009n04ABEH001497
Atiyah, Bott & Patodi 1973. Atiyah, M.; Bott, R.; Patodi, V. K. (1973), "On the heat equation and the index theorem", Invent. Math., 19 (4): 279–330, Bibcode:1973InMat..19..279A, doi:10.1007/BF01425417, MR 0650828, S2CID 115700319. Atiyah, M.; Bott, R.; Patodi, V. K. (1975), "Errata", Invent. Math., 28 (3): 277–280, Bibcode:1975InMat..28..277A, doi:10.1007/BF01425562, MR 0650829
Getzler 1983. Getzler, E. (1983), "Pseudodifferential operators on supermanifolds and the Atiyah–Singer index theorem", Commun. Math. Phys., 92 (2): 163–178, Bibcode:1983CMaPh..92..163G, doi:10.1007/BF01210843, S2CID 55438589
Witten 1982. Witten, Edward (1982), "Supersymmetry and Morse theory", J. Diff. Geom., 17 (4): 661–692, doi:10.4310/jdg/1214437492, MR 0683171
Teleman 1983. Teleman, N. (1983), "The index of signature operators on Lipschitz manifolds", Publications Mathématiques de l'Institut des Hautes Études Scientifiques, 58: 251–290, doi:10.1007/BF02953772, S2CID 121497293, Zbl 0531.58044
Teleman 1984. Teleman, N. (1984), "The index theorem on topological manifolds", Acta Mathematica, 153: 117–152, doi:10.1007/BF02392376, Zbl 0547.58036
Connes 1986. Connes, A. (1986), "Non-commutative differential geometry", Publications Mathématiques de l'Institut des Hautes Études Scientifiques, 62: 257–360, doi:10.1007/BF02698807, S2CID 122740195, Zbl 0592.46056
Donaldson & Sullivan 1989. Donaldson, S.K.; Sullivan, D. (1989), "Quasiconformal 4-manifolds", Acta Mathematica, 163: 181–252, doi:10.1007/BF02392736, Zbl 0704.57008
Connes & Moscovici 1990. Connes, A.; Moscovici, H. (1990), "Cyclic cohomology, the Novikov conjecture and hyperbolic groups" (PDF), Topology, 29 (3): 345–388, doi:10.1016/0040-9383(90)90003-3, Zbl 0759.58047
Connes, Sullivan & Teleman 1994. Connes, A.; Sullivan, D.; Teleman, N. (1994), "Quasiconformal mappings, operators on Hilbert space and local formulae for characteristic classes", Topology, 33 (4): 663–681, doi:10.1016/0040-9383(94)90003-5, Zbl 0840.57013
Shanahan, P. (1978), The Atiyah-Singer Index Theorem, Lecture Notes in Mathematics, vol. 638, Springer, CiteSeerX 10.1.1.193.9222, doi:10.1007/BFb0068264, ISBN 978-0-387-08660-6

harvard.edu

ui.adsabs.harvard.edu

Gel'fand 1960. Gel'fand, I. M. (1960), "On elliptic equations", Russ. Math. Surv., 15 (3): 113–123, Bibcode:1960RuMaS..15..113G, doi:10.1070/rm1960v015n03ABEH004094 reprinted in volume 1 of his collected works, p. 65–75, ISBN 0-387-13619-3. On page 120 Gel'fand suggests that the index of an elliptic operator should be expressible in terms of topological data.
Kasparov 1972. Kasparov, G.G. (1972), "Topological invariance of elliptic operators, I: K-homology", Math. USSR Izvestija (Engl. Transl.), 9 (4): 751–792, Bibcode:1975IzMat...9..751K, doi:10.1070/IM1975v009n04ABEH001497
Atiyah, Bott & Patodi 1973. Atiyah, M.; Bott, R.; Patodi, V. K. (1973), "On the heat equation and the index theorem", Invent. Math., 19 (4): 279–330, Bibcode:1973InMat..19..279A, doi:10.1007/BF01425417, MR 0650828, S2CID 115700319. Atiyah, M.; Bott, R.; Patodi, V. K. (1975), "Errata", Invent. Math., 28 (3): 277–280, Bibcode:1975InMat..28..277A, doi:10.1007/BF01425562, MR 0650829
Getzler 1983. Getzler, E. (1983), "Pseudodifferential operators on supermanifolds and the Atiyah–Singer index theorem", Commun. Math. Phys., 92 (2): 163–178, Bibcode:1983CMaPh..92..163G, doi:10.1007/BF01210843, S2CID 55438589

jstor.org

Atiyah & Singer 1968a. Atiyah, Michael F.; Singer, Isadore M. (1968a), "The Index of Elliptic Operators I", Annals of Mathematics, 87 (3): 484–530, doi:10.2307/1970715, JSTOR 1970715 This gives a proof using K-theory instead of cohomology.
Atiyah & Singer (1968a); Atiyah & Singer (1968b); Atiyah & Singer (1971a); Atiyah & Singer (1971b). Atiyah, Michael F.; Singer, Isadore M. (1968a), "The Index of Elliptic Operators I", Annals of Mathematics, 87 (3): 484–530, doi:10.2307/1970715, JSTOR 1970715 This gives a proof using K-theory instead of cohomology. Atiyah, Michael F.; Singer, Isadore M. (1968b), "The Index of Elliptic Operators III", Annals of Mathematics, Second Series, 87 (3): 546–604, doi:10.2307/1970717, JSTOR 1970717 This paper shows how to convert from the K-theory version to a version using cohomology. Atiyah, Michael F.; Singer, Isadore M. (1971a), "The Index of Elliptic Operators IV", Annals of Mathematics, Second Series, 93 (1): 119–138, doi:10.2307/1970756, JSTOR 1970756 This paper studies families of elliptic operators, where the index is now an element of the K-theory of the space parametrizing the family. Atiyah, Michael F.; Singer, Isadore M. (1971b), "The Index of Elliptic Operators V", Annals of Mathematics, Second Series, 93 (1): 139–149, doi:10.2307/1970757, JSTOR 1970757. This studies families of real (rather than complex) elliptic operators, when one can sometimes squeeze out a little extra information.

mi.ras.ru

Novikov 1965. Novikov, S.P. (1965), "Topological invariance of the rational Pontrjagin classes" (PDF), Doklady Akademii Nauk SSSR, 163: 298–300

mit.edu

www-math.mit.edu

Melrose 1993. Melrose, Richard B. (1993), The Atiyah–Patodi–Singer Index Theorem, Wellesley, Mass.: Peters, ISBN 978-1-56881-002-7 Free online textbook.

numdam.org

Teleman 1983. Teleman, N. (1983), "The index of signature operators on Lipschitz manifolds", Publications Mathématiques de l'Institut des Hautes Études Scientifiques, 58: 251–290, doi:10.1007/BF02953772, S2CID 121497293, Zbl 0531.58044
Connes 1986. Connes, A. (1986), "Non-commutative differential geometry", Publications Mathématiques de l'Institut des Hautes Études Scientifiques, 62: 257–360, doi:10.1007/BF02698807, S2CID 122740195, Zbl 0592.46056

projecteuclid.org

Getzler 1983. Getzler, E. (1983), "Pseudodifferential operators on supermanifolds and the Atiyah–Singer index theorem", Commun. Math. Phys., 92 (2): 163–178, Bibcode:1983CMaPh..92..163G, doi:10.1007/BF01210843, S2CID 55438589
Index Theorems on Open Spaces
Some Remarks on the Paper of Callias

psu.edu

citeseerx.ist.psu.edu

Shanahan, P. (1978), The Atiyah-Singer Index Theorem, Lecture Notes in Mathematics, vol. 638, Springer, CiteSeerX 10.1.1.193.9222, doi:10.1007/BFb0068264, ISBN 978-0-387-08660-6

semanticscholar.org

api.semanticscholar.org

Atiyah, Bott & Patodi 1973. Atiyah, M.; Bott, R.; Patodi, V. K. (1973), "On the heat equation and the index theorem", Invent. Math., 19 (4): 279–330, Bibcode:1973InMat..19..279A, doi:10.1007/BF01425417, MR 0650828, S2CID 115700319. Atiyah, M.; Bott, R.; Patodi, V. K. (1975), "Errata", Invent. Math., 28 (3): 277–280, Bibcode:1975InMat..28..277A, doi:10.1007/BF01425562, MR 0650829
Getzler 1983. Getzler, E. (1983), "Pseudodifferential operators on supermanifolds and the Atiyah–Singer index theorem", Commun. Math. Phys., 92 (2): 163–178, Bibcode:1983CMaPh..92..163G, doi:10.1007/BF01210843, S2CID 55438589
Teleman 1983. Teleman, N. (1983), "The index of signature operators on Lipschitz manifolds", Publications Mathématiques de l'Institut des Hautes Études Scientifiques, 58: 251–290, doi:10.1007/BF02953772, S2CID 121497293, Zbl 0531.58044
Connes 1986. Connes, A. (1986), "Non-commutative differential geometry", Publications Mathématiques de l'Institut des Hautes Études Scientifiques, 62: 257–360, doi:10.1007/BF02698807, S2CID 122740195, Zbl 0592.46056

stackexchange.com

math.stackexchange.com

"algebraic topology - How to understand the Todd class?". Mathematics Stack Exchange. Retrieved 2021-02-05.

zbmath.org

Cartan-Schwartz 1965. Cartan-Schwartz (1965), Séminaire Henri Cartan. Théoreme d'Atiyah-Singer sur l'indice d'un opérateur différentiel elliptique. 16 annee: 1963/64 dirigee par Henri Cartan et Laurent Schwartz. Fasc. 1; Fasc. 2. (French), École Normale Supérieure, Secrétariat mathématique, Paris, Zbl 0149.41102
Atiyah 1970. Atiyah, M. F. (1970), "Global Theory of Elliptic Operators", Proc. Int. Conf. on Functional Analysis and Related Topics (Tokyo, 1969), University of Tokio, Zbl 0193.43601
Sullivan 1979. Sullivan, D. (1979), "Hyperbolic geometry and homeomorphisms", J.C. Candrell, "Geometric Topology", Proc. Georgia Topology Conf. Athens, Georgia, 1977, New York: Academic Press, pp. 543–595, ISBN 978-0-12-158860-1, Zbl 0478.57007
Teleman 1983. Teleman, N. (1983), "The index of signature operators on Lipschitz manifolds", Publications Mathématiques de l'Institut des Hautes Études Scientifiques, 58: 251–290, doi:10.1007/BF02953772, S2CID 121497293, Zbl 0531.58044
Teleman 1984. Teleman, N. (1984), "The index theorem on topological manifolds", Acta Mathematica, 153: 117–152, doi:10.1007/BF02392376, Zbl 0547.58036
Connes 1986. Connes, A. (1986), "Non-commutative differential geometry", Publications Mathématiques de l'Institut des Hautes Études Scientifiques, 62: 257–360, doi:10.1007/BF02698807, S2CID 122740195, Zbl 0592.46056
Donaldson & Sullivan 1989. Donaldson, S.K.; Sullivan, D. (1989), "Quasiconformal 4-manifolds", Acta Mathematica, 163: 181–252, doi:10.1007/BF02392736, Zbl 0704.57008
Connes & Moscovici 1990. Connes, A.; Moscovici, H. (1990), "Cyclic cohomology, the Novikov conjecture and hyperbolic groups" (PDF), Topology, 29 (3): 345–388, doi:10.1016/0040-9383(90)90003-3, Zbl 0759.58047
Connes, Sullivan & Teleman 1994. Connes, A.; Sullivan, D.; Teleman, N. (1994), "Quasiconformal mappings, operators on Hilbert space and local formulae for characteristic classes", Topology, 33 (4): 663–681, doi:10.1016/0040-9383(94)90003-5, Zbl 0840.57013