Celebratio Mathematica

M. F. Atiyah: “A note on the tangents of a twisted cubic,” Proc. Cambridge Philos. Soc. 48 (1952), pp. 204–205. MR 0048079 Zbl 0046.14604 article

M. F. Atiyah: “Complex fibre bundles and ruled surfaces,” Proc. London Math. Soc. (3) 5 (1955), pp. 407–434. MR 0076409 Zbl 0174.52804 article

M. F. Atiyah and I. M. Singer: “The index of elliptic operators, I,” Ann. Math. (2) 87 : 3 (May 1968), pp. 484–530. A Russian translation was published in Uspehi Mat. Nauk 23:5(143). MR 236950 Zbl 0164.24001 article

This is the first of a series of papers which will be devoted to a study of the index of elliptic operators on compact manifolds. The main result was announced in [Atiyah and Singer 1963] and, for manifolds with boundary, in [Atiyah 1964]. The long delay between these announcements and the present paper is due to several factors. On the one hand, a fairly detailed exposition has already appeared in [Palais 1965]. On the other hand, our original proof, reproduced with minor modifications in [Palais 1965], had a number of drawbacks. In the first place the use of cobordism, and the computational checking associated with this, were not very enlightening. More seriously, however, the method of proof did not lend itself to certain natural generalizations of the problem where appropriate cobordism groups were not known. The reader who is familiar with the Riemann–Roch theorem will realize that our original proof of the index theorem was modelled closely on Hirzebruch’s proof of the Riemann–Roch theorem. Naturally enough we were led to look for a proof modelled more on that of Grothendieck. While we have not completely succeeded in this aim, we have at least found a proof which is much more natural, does not use cobordism, and lends itself therefore to generalization. In spirit, at least, it has much in common with Grothendieck’s approach.

M. F. Atiyah and G. B. Segal: “The index of elliptic operators, II,” Ann. Math. (2) 87 : 3 (1968), pp. 531–545. Russian translation published in Uspehi Mat. Nauk 23:6(144) (1968). MR 0236951 Zbl 0164.24201 article

The purpose of this paper is to show how the index theorem of [Atiyah and Singer 1963] can be reformulated as a general “Lefschetz fixed-point theorem” on the lines of [Atiyah and Bott 1967]. In this way we shall obtain the main theorem of [Atiyah and Bott 1967], generalized to deal with arbitrary fixed-point sets, but only for transformations belonging to a compact group.

The content of this paper is essentially topological, and it should be viewed as a paper on the equivariant \( K \)-theory of manifolds. The analysis has all been done in [Atiyah and Singer 1968], and what we do here is simply to express the topological index in terms of fixed-point sets. This is quite independent of the main theorem of [Atiyah and Singer 1968] asserting the equality of the topological and analytical indices.

As in [Atiyah and Singer 1968], we avoid cohomology and use only \( K \)-theory. In paper III of this series, we shall pass over to cohomology obtaining explicit formulas in terms of characteristic classes.

M. F. Atiyah and I. M. Singer: “The index of elliptic operators, III,” Ann. Math. (2) 87 : 3 (May 1968), pp. 546–604. A Russian translation was published in Uspehi Mat. Nauk 24:1(145). MR 236952 Zbl 0164.24301 article

In [1968a], paper I of this series, the index of an elliptic operator was computed in terms of \( K \)-theory. In this paper, we carry out what is essentially a routine exercise by passing from \( K \)-theory to cohomology. In this way, we end up with the explicit cohomological formula for the index announced in [1963].

In [1968a] we also considered elliptic operators (or complexes) compatible with a compact group \( G \) of transformations. The index in this case is a character of \( G \), and the main theorem of [1968a] gave a construction for this in \( KG \)-theory. In [1968b], paper II of this series, the value of this index-character at an element \( g\in G \) was expressed as the index of a new “virtual operator” on the fixed point set of \( g \). This was referred to as a Lefschetz fixed-point formula. By combining this formula with the cohomological formula for the index, we obtain finally an explicit cohomological formula for the index-character. We shall describe this formula in detail for a number of important operators. In particular we draw attention to the “integrality theorems” obtained in this way for actions of finite groups on manifolds. Most of these do not depend on the analysis in [1968a], but are a consequence of combining the purely topological results of [1968b] and the present paper.

M. F. Atiyah and I. M. Singer: “The index of elliptic operators, IV,” Ann. Math. (2) 93 : 1 (January 1971), pp. 119–138. A Russian translation was published in Uspehi Mat. Nauk 27:4(166) (1972). MR 279833 Zbl 0212.28603 article

M. F. Atiyah and I. M. Singer: “The index of elliptic operators, V,” Ann. Math. (2) 93 : 1 (January 1971), pp. 139–149. A Russian translation was published in Uspehi Mat. Nauk 27:4(166) (1972). MR 279834 article

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

M. Atiyah: The geometry and physics of knots. Lezioni Lincee. Cambridge University Press, 1990. These notes arise from lectures presented in Florence under the auspices of the Accademia dei Lincei. Russian translation published as Geometriya i fizika uzlov (1995). See also Miniconference on geometry and physics (1989). MR 1078014 Zbl 0729.57002 book

N. Hitchin and M. Atiyah: Web of stories, 1990–1997. Collection of audio recordings: interviews of Sir Michael Atiyah by Nigel Hitchins during the latter’s tenure as Master of Trinity College. misc

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Michael F. Atiyah

Sir Michael Atityah OM
22 April 1929–11 January 2019

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Sir Michael Atityah OM22 April 1929–11 January 2019

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Sir Michael Atityah OM
22 April 1929–11 January 2019