قضیه اندیس عطیه-سینگر

قضیه اندیس عطیه-سینگر

گرایش	هندسه دیفرانسیل
نخستین اثبات توسط	مایکل عطیه و ایسادور سینگر
نخستین اثبات در تاریخ	1963
نتایج	قضیه چرن-گاو-بونت قضیه گروتندیک-ریمان-رخ قضیه علامت هیزربروخ قضیه روخلین

در هندسه دیفرانسیل، قضیه اندیس عطیه-سینگر (به انگلیسی: Atiyah-Singer Index Theorem)، توسط مایکل عطیه و ایسادور سینگر اثبات شد، بیان می‌دارد که برای یک عملگر دیفرانسیل بیضوی روی یک منیفلد فشرده، اندیس تحلیلی (مربوط به تعد فضای جواب‌ها) برابر با اندیس توپولوژیکی (که برحسب داده‌های توپولوژیکی تعریف می‌شود) است. این قضیه، بسیاری از قضایای دیگری چون قضیه چرن-گاوس-بونت و قضیه ریمان-رخ را به عنوان حالت‌های خاص در بر گرفته و کاربردهایی در فیزیک نظری دارد.^{صفحه 11}.

ارجاعات

• مشارکت‌کنندگان ویکی‌پدیا. «Atiyah-Singer Index Theorem». در دانشنامهٔ ویکی‌پدیای انگلیسی، بازبینی‌شده در ۹ ژوئن ۲۰۲۱.

منابع

مقالات عطیه در جلدهای 3 و 4 از مجموعه آثارش گردآوری شده است،(Atiyah 1988a, 1988b)

• 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
• Atiyah, M. F. (1976), "Elliptic operators, discrete groups and von Neumann algebras", Colloque "Analyse et Topologie" en l'Honneur de Henri Cartan (Orsay, 1974), Asterisque, vol. 32–33, Soc. Math. France, Paris, pp. 43–72, MR 0420729
• Atiyah, M. F.; Segal, G. B. (1968), "The Index of Elliptic Operators: II", Annals of Mathematics, Second Series, 87 (3): 531–545, doi:10.2307/1970716, JSTOR 1970716 This reformulates the result as a sort of Lefschetz fixed point theorem, using equivariant K-theory.
• 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.
• 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.
• Atiyah, M. F.; Bott, R. (1966), "A Lefschetz Fixed Point Formula for Elliptic Differential Operators", Bull. Am. Math. Soc., 72 (2): 245–50, doi:10.1090/S0002-9904-1966-11483-0. This states a theorem calculating the Lefschetz number of an endomorphism of an elliptic complex.
• Atiyah, M. F.; Bott, R. (1967), "A Lefschetz Fixed Point Formula for Elliptic Complexes: I", Annals of Mathematics, Second series, 86 (2): 374–407, doi:10.2307/1970694, JSTOR 1970694 and Atiyah, M. F.; Bott, R. (1968), "A Lefschetz Fixed Point Formula for Elliptic Complexes: II. Applications", Annals of Mathematics, Second Series, 88 (3): 451–491, doi:10.2307/1970721, JSTOR 1970721 These give the proofs and some applications of the results announced in the previous paper.
• 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
• Atiyah, Michael; Schmid, Wilfried (1977), "A geometric construction of the discrete series for semisimple Lie groups", Invent. Math., 42: 1–62, Bibcode:1977InMat..42....1A, doi:10.1007/BF01389783, MR 0463358, S2CID 189831012, Atiyah, Michael; Schmid, Wilfried (1979), "Erratum", Invent. Math., 54 (2): 189–192, Bibcode:1979InMat..54..189A, doi:10.1007/BF01408936, MR 0550183
• Atiyah, Michael (1988a), Collected works. Vol. 3. Index theory: 1, Oxford Science Publications, New York: The Clarendon Press, Oxford University Press, ISBN 978-0-19-853277-4, MR 0951894
• Atiyah, Michael (1988b), Collected works. Vol. 4. Index theory: 2, Oxford Science Publications, New York: The Clarendon Press, Oxford University Press, ISBN 978-0-19-853278-1, MR 0951895
• Baum, P.; Fulton, W.; Macpherson, R. (1979), "Riemann-Roch for singular varieties", Acta Mathematica, 143: 155–191, doi:10.1007/BF02684299, S2CID 83458307, Zbl 0332.14003
• Berline, Nicole; Getzler, Ezra; Vergne, Michèle (1992), Heat Kernels and Dirac Operators, Berlin: Springer, ISBN 978-3-540-53340-5 This gives an elementary proof of the index theorem for the Dirac operator, using the heat equation and supersymmetry.
• Bismut, Jean-Michel (1984), "The Atiyah–Singer Theorems: A Probabilistic Approach. I. The index theorem", J. Funct. Analysis, 57: 56–99, doi:10.1016/0022-1236(84)90101-0 Bismut proves the theorem for elliptic complexes using probabilistic methods, rather than heat equation methods.
• 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
• Connes, A. (1986), "Non-commutative differential geometry", Publications Mathématiques, 62: 257–360, doi:10.1007/BF02698807, S2CID 122740195, Zbl 0592.46056
• Connes, A. (1994), Noncommutative Geometry, San Diego: Academic Press, ISBN 978-0-12-185860-5, Zbl 0818.46076
• 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, 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
• Donaldson, S.K.; Sullivan, D. (1989), "Quasiconformal 4-manifolds", Acta Mathematica, 163: 181–252, doi:10.1007/BF02392736, Zbl 0704.57008
• 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, شابک ‎۰−۳۸۷−۱۳۶۱۹−۳. On page 120 Gel'fand suggests that the index of an elliptic operator should be expressible in terms of topological data.
• 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
• Getzler, E. (1988), "A short proof of the local Atiyah–Singer index theorem", Topology, 25: 111–117, doi:10.1016/0040-9383(86)90008-X
• Gilkey, Peter B. (1994), Invariance Theory, the Heat Equation, and the Atiyah–Singer Theorem, ISBN 978-0-8493-7874-4 Free online textbook that proves the Atiyah–Singer theorem with a heat equation approach
• Higson, Nigel; Roe, John (2000), Analytic K-homology, Oxford University Press, ISBN 9780191589201
• Hilsum, M. (1999), "Structures riemaniennes L^p et K-homologie", Annals of Mathematics, 149 (3): 1007–1022, arXiv:math/9905210, doi:10.2307/121079, JSTOR 121079, S2CID 119708566
• 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
• 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
• Kirby, R.; Siebenmann, L.C. (1977), Foundational Essays on Topological Manifolds, Smoothings and Triangulations, Annals of Mathematics Studies in Mathematics, vol. 88, Princeton: Princeton University Press and Tokio University Press
• Melrose, Richard B. (1993), The Atiyah–Patodi–Singer Index Theorem, Wellesley, Mass.: Peters, ISBN 978-1-56881-002-7 Free online textbook.
• Novikov, S.P. (1965), "Topological invariance of the rational Pontrjagin classes" (PDF), Doklady Akademii Nauk SSSR, 163: 298–300
• 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.)
• Shanahan, P. (1978), The Atiyah–Singer index theorem: an introduction, Lecture Notes in Mathematics, vol. 638, Springer, CiteSeerX 10.1.1.193.9222, doi:10.1007/BFb0068264, ISBN 978-0-387-08660-6
• Singer, I.M. (1971), "Future extensions of index theory and elliptic operators", Prospects in Mathematics, Annals of Mathematics Studies in Mathematics, vol. 70, pp. 171–185
• 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
• Sullivan, D.; Teleman, N. (1983), "An analytic proof of Novikov's theorem on rational Pontrjagin classes", Publications Mathématiques, Paris, 58: 291–293, doi:10.1007/BF02953773, S2CID 8348213, Zbl 0531.58045
• Teleman, N. (1980), "Combinatorial Hodge theory and signature operator", Inventiones Mathematicae, 61 (3): 227–249, Bibcode:1980InMat..61..227T, doi:10.1007/BF01390066, S2CID 122247909
• Teleman, N. (1983), "The index of signature operators on Lipschitz manifolds", Publications Mathématiques, 58: 251–290, doi:10.1007/BF02953772, S2CID 121497293, Zbl 0531.58044
• Teleman, N. (1984), "The index theorem on topological manifolds", Acta Mathematica, 153: 117–152, doi:10.1007/BF02392376, Zbl 0547.58036
• Teleman, N. (1985), "Transversality and the index theorem", Integral Equations and Operator Theory, 8 (5): 693–719, doi:10.1007/BF01201710, S2CID 121137053
• Thom, R. (1956), "Les classes caractéristiques de Pontrjagin de variétés triangulées", Symp. Int. Top. Alg. Mexico, pp. 54–67
• Witten, Edward (1982), "Supersymmetry and Morse theory", J. Diff. Geom., 17 (4): 661–692, doi:10.4310/jdg/1214437492, MR 0683171
• Shing-Tung Yau, ed. (2009) [First published in 2005], The Founders of Index Theory (2nd ed.), Somerville, Mass.: International Press of Boston, ISBN 978-1571461377 - Personal accounts on Atiyah, Bott, Hirzebruch and Singer.