Atiyah–Singer index theorem[*][[Atiyah–Singer index theorem (theorem that the difference between the dimensions of the kernel and cokernel of a differential operator on a manifold is the integral of a characteristic class)|]] Atiyah algebroid[*][[Atiyah algebroid (Lie algebroid associated to a principal bundle)|]] Atiyah–Hirzebruch spectral sequence[*][[Atiyah–Hirzebruch spectral sequence (computational tool from homological algebra)|]] Atiyah–Segal completion theorem[*][[Atiyah–Segal completion theorem (Mathematical result about equivariant K-theory in homotopy theory)|]] Atiyah–Hitchin–Singer theorem[*][[Atiyah–Hitchin–Singer theorem (the theorem that the space of SU(2) anti self dual Yang–Mills fields on a 4-sphere with index k > 0 has dimension 8k – 3)|]] Atiyah–Bott fixed-point theorem[*][[Atiyah–Bott fixed-point theorem (theorem)|]] Atiyah–Bott formula[*][[Atiyah–Bott formula |]] Atiyah–Hitchin space[*][[Atiyah–Hitchin space (4-dimensional, asymptotically locally flat (ALF) hyperkähler manifold, which is the moduli space of two monopoles in the SU(2) Bogomolny equation)|]] Atiyah-Jänich theorem[*][[Atiyah-Jänich theorem (mathematical theorem in functional analysis)|]] Atiyah conjecture on configurations[*][[Atiyah conjecture on configurations (Mathematical conjecture)|]] Atiyah–Jones conjecture[*][[Atiyah–Jones conjecture (theorem about the homology of the moduli space of instantons over a sphere, conjectured by M. F. Atiyah and J. D. S. Jones (1978), proved by C. P. Boyer, J. C. Hurtubise, and B. M. Mann et al. (1998, 1999))|]] Atiyah conjecture[*][[Atiyah conjecture |]]
A colaborat cu mai mulți matematicieni, printre care Raoul Bott, Isadore Singer și Friedrich Hirzebruch.
Cu acesta din urmă a pus bazele Teoriei K din domeniul spațiilor topologice.
Dar cea mai valoroasă contribuție a sa o constituie Teorema indicelui Atiyah-Singer, care indică numărul de soluții independente ale anumitor tipuri de ecuații diferențiale.
Cărți
Această subsecțiune enumeră toate cărțile scrise de Atiyah; omite câteva cărți pe care le-a editat.
Atiyah, Michael F.; Macdonald, Ian G. (), Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., MR0242802. A classic textbook covering standard commutative algebra.
Atiyah, Michael F. (), Vector fields on manifolds, Arbeitsgemeinschaft für Forschung des Landes Nordrhein-Westfalen, Heft 200, Cologne: Westdeutscher Verlag, MR0263102. Reprinted as (Atiyah 1988b, item 50).
Atiyah, Michael F. (), Elliptic operators and compact groups, Lecture Notes in Mathematics, Vol. 401, Berlin, New York: Springer-Verlag, MR0482866. Reprinted as (Atiyah 1988c, item 78).
Atiyah, Michael F. (), Geometry of Yang–Mills fields, Scuola Normale Superiore Pisa, Pisa, MR0554924. Reprinted as (Atiyah 1988e, item 99).
Atiyah, Michael F. (), Collected works. Vol. 1 Early papers: general papers, Oxford Science Publications, The Clarendon Press Oxford University Press, ISBN978-0-19-853275-0, MR0951892.
Atiyah, Michael F. (), Collected works. Vol. 2 K-theory, Oxford Science Publications, The Clarendon Press Oxford University Press, ISBN978-0-19-853276-7, MR0951892.
Atiyah, Michael F. (), Collected works. Vol. 3 Index theory: 1, Oxford Science Publications, The Clarendon Press Oxford University Press, ISBN978-0-19-853277-4, MR0951892.
Atiyah, Michael F. (), Collected works. Vol. 4 Index theory:2, Oxford Science Publications, The Clarendon Press Oxford University Press, ISBN978-0-19-853278-1, MR0951892.
Atiyah, Michael F. (), Collected works. Vol. 5 Gauge theories, Oxford Science Publications, The Clarendon Press Oxford University Press, ISBN978-0-19-853279-8, MR0951892.
Atiyah, Michael F. (), K-theory, Advanced Book Classics (ed. 2nd), Addison-Wesley, ISBN978-0-201-09394-0, MR1043170. First edition (1967) reprinted as (Atiyah 1988b, item 45).
Atiyah, Michael F. (), Collected works. Vol. 6, Oxford Science Publications, The Clarendon Press Oxford University Press, ISBN978-0-19-853099-2, MR2160826.
Atiyah, Michael F.; Hirzebruch, Friedrich (), „Vector bundles and homogeneous spaces”, Proc. Sympos. Pure Math. AMS, 3: 7–38. Reprinted in (Atiyah 1988b, paper 28).
Atiyah, Michael F.; Segal, Graeme B. (), „Equivariant K-Theory and Completion”, Journal of Differential Geometry, 3: 1–18. Reprinted in (Atiyah 1988b, paper 49).
Atiyah, Michael F. (), „Elliptic operators, discrete groups and von Neumann algebras”, Colloque "Analyse et Topologie" en l'Honneur de Henri Cartan (Orsay, 1974), Asterisque, 32–33, Soc. Math. France, Paris, pp.43–72, MR0420729. Reprinted in (Atiyah 1988d, paper 89). Formulation of the Atiyah "Conjecture" on the rationality of the L2-Betti numbers.
Atiyah, Michael F.; Singer, Isadore M. (), „The Index of Elliptic Operators I”, Annals of Mathematics, The Annals of Mathematics, Vol. 87, No. 3, 87 (3): 484–530, doi:10.2307/1970715, JSTOR1970715. This gives a proof using K theory instead of cohomology. Reprinted in (Atiyah 1988c, paper 64).
Atiyah, Michael F.; Segal, Graeme B. (), „The Index of Elliptic Operators: II”, Annals of Mathematics, Second Series, The Annals of Mathematics, Vol. 87, No. 3, 87 (3): 531–545, doi:10.2307/1970716, JSTOR1970716. This reformulates the result as a sort of Lefschetz fixed point theorem, using equivariant K theory. Reprinted in (Atiyah 1988c, paper 65).
Atiyah, Michael F.; Singer, Isadore M. (), „The Index of Elliptic Operators III”, Annals of Mathematics, Second Series, 87 (3): 546–604, doi:10.2307/1970717, JSTOR1970717. This paper shows how to convert from the K-theory version to a version using cohomology. Reprinted in (Atiyah 1988c, paper 66).
Atiyah, Michael F.; Singer, Isadore M. (), „The Index of Elliptic Operators IV”, Annals of Mathematics, Second Series, The Annals of Mathematics, Vol. 93, No. 1, 93 (1): 119–138, doi:10.2307/1970756, JSTOR1970756 This paper studies families of elliptic operators, where the index is now an element of the K-theory of the space parametrizing the family. Reprinted in (Atiyah 1988c, paper 67).
Atiyah, Michael F.; Singer, Isadore M. (), „The Index of Elliptic Operators V”, Annals of Mathematics, Second Series, The Annals of Mathematics, Vol. 93, No. 1, 93 (1): 139–149, doi:10.2307/1970757, JSTOR1970757. This studies families of real (rather than complex) elliptic operators, when one can sometimes squeeze out a little extra information. Reprinted in (Atiyah 1988c, paper 68).
Atiyah, Michael F.; Bott, Raoul (), „A Lefschetz Fixed Point Formula for Elliptic Complexes: I”, Annals of Mathematics, Second Series, The Annals of Mathematics, Vol. 86, No. 2, 86 (2): 374–407, doi:10.2307/1970694, JSTOR1970694 (reprinted in (Atiyah 1988c, paper 61))and Atiyah, Michael F.; Bott, Raoul (), „A Lefschetz Fixed Point Formula for Elliptic Complexes: II. Applications”, Annals of Mathematics, Second Series, 88 (3): 451–491, doi:10.2307/1970721, JSTOR1970721. Reprinted in (Atiyah 1988c, paper 62). These give the proofs and some applications of the results announced in the previous paper.
Boyer, Charles P.; Hurtubise, J. C.; Mann, B. M.; Milgram, R. J. (), „The topology of instanton moduli spaces. I. The Atiyah–Jones conjecture”, Annals of Mathematics, Second Series, The Annals of Mathematics, Vol. 137, No. 3, 137 (3): 561–609, doi:10.2307/2946532, ISSN0003-486X, JSTOR2946532, MR1217348
Barth, Wolf P.; Hulek, Klaus; Peters, Chris A.M.; Van de Ven, Antonius (), Compact Complex Surfaces, Berlin: Springer, p.334, ISBN978-3-540-00832-3
Gel'fand, Israel M. (), „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.
Palais, Richard S. (), Seminar on the Atiyah–Singer Index Theorem, Annals of Mathematics Studies, 57, S.l.: Princeton Univ Press, ISBN0-691-08031-3. This describes the original proof of the index theorem. (Atiyah and Singer never published their original proof themselves, but only improved versions of it.)