Introduction to the Atiyah issue

By Simon Donaldson

Sir Michael Atiyah (1929–2019) was a giant figure in twentieth century mathematics. His publications, beginning in 1952, span nearly 70 years and he made fundamental contributions to the development of many fields, both directly through his own work and through his wider influence, inspiration, and leadership.

Atiyah began his career in Cambridge, England, but spent a formative year (1955–56) at the Institute for Advanced Study, Princeton. There he met, among others, his long-time collaborators Bott, Hirzebruch, and Singer (making up “the gang of four” as Yau has affectionately referred to them). A few years later Atiyah began his monumental work developing the sister subjects of -theory (with Hirzebruch) and index theory (with Singer). While the focus of this work was in algebraic topology and global analysis, many of the ideas were rooted in algebraic geometry, the area in which Atiyah had begun his research and to which he made important contributions. During the bulk of this period Atiyah held the Savilian chair in Oxford before another spell 1971–73 in the Institute for Advanced Study, this time as a permanent member. A few years after his return to Oxford, Atiyah’s research took a new direction in which, partly with Singer, he was a pioneer and leader in the development of the interaction between modern geometry and physics, a development which has had a profound influence, reaching into nearly all areas of mathematics, over the half century since.

Many tributes and personal recollections of Atiyah have been published since his death. These include the article of Connes and Kouneiher Reference 7 and a collection of recollections Reference 10 both of which appear in the Notices of the AMS. An article of Hitchin Reference 11 gives an inside account of the interaction between geometry and physics in Atiyah’s work. All these convey the exceptional quality of Atiyah’s character and talents, both as a mathematician and more broadly. As a mathematician, Atiyah was driven by a search for beauty, depth of understanding and connections between ideas—not particularly interested in “problem solving”. One example is his multiple treatments of the Bott periodicity theorem (see the article of Segal in this issue) including one of his most beautiful papers Reference 1. The theorem drops out from a general construction (the index of families) and a wonderfully simple but subtle observation, somewhat reminiscent of the way one evaluates a one-dimensional Gaussian integral by going to two dimensions. (In brief, the question involves a relation between two spaces and and the trick is to take the product with an extra factor of to get and then exploit a symmetry between the last two factors.)

Atiyah filled many prominent positions and was an outstanding advocate and spokesman for mathematics and science. In 1990 he curtailed his research activity for a while to take up three simultaneous positions: President of the Royal Society in London, Master of Trinity College, and founding Director of the Isaac Newton Institute in Cambridge. Before that he had served as President of the London Mathematical Society and later, after his nominal retirement to Edinburgh, he was President of the Royal Society of Edinburgh.

Perhaps Atiyah’s most exceptional quality was as a communicator. He liked to develop his mathematical ideas in conversations, for which he had enormous stamina, and—talking to everybody—he was able to make seminal connections between previously unrelated fields. His lectures were an inspiration; frequently developing some simple idea in a beautiful and surprising direction in which, as if by magic, all difficulties disappeared—under his guidance the audience would see the whole picture. His writing, both in mathematical papers and for more general readership, had the same qualities and is gathered together in the seven volumes of Collected works published by Oxford University Press. (Within these Atiyah explains the depth of preparation that would go into his magical lectures.) Last, but far from least, Atiyah was an extremely kind man, thoughtful for others, as many of the recollections in Reference 10 attest.

This issue of the Bulletin contains three articles on Atiyah’s work. Graeme Segal focuses on his work in algebraic topology Reference 12, Dan Freed on index theory Reference 9, and my own contribution on his early work in algebraic geometry and later work influenced by the gauge theories coming from physics Reference 8. As the reader will see, we have taken slightly different approaches to the task and there are some overlaps between the accounts which we have not tried to remove—all of these topics fitted together within Atiyah’s overall vision. On the other hand we have not covered everything: there are very substantial contributions we do not discuss. These include work on representation theory Reference 4, contributions to analysis of distributions Reference 2, hyperbolic partial differential equations Reference 3, and, later, to knot and manifold invariants such as Reference 5 and configuration spaces Reference 6.

Table of Contents

  1. Untitled Section

References

[1]
M. F. Atiyah, Bott periodicity and the index of elliptic operators, Quart. J. Math. Oxford Ser. (2) 19 (1968), 113–140, DOI 10.1093/qmath/19.1.113. MR228000, Show rawAMSref\bib{kn:QJM}{article}{ author={Atiyah, M. F.}, title={Bott periodicity and the index of elliptic operators}, journal={Quart. J. Math. Oxford Ser. (2)}, volume={19}, date={1968}, pages={113--140}, issn={0033-5606}, review={\MR {228000}}, doi={10.1093/qmath/19.1.113}, } Close amsref.
[2]
M. F. Atiyah, Resolution of singularities and division of distributions, Comm. Pure Appl. Math. 23 (1970), 145–150, DOI 10.1002/cpa.3160230202. MR256156, Show rawAMSref\bib{kn:A1}{article}{ author={Atiyah, M. F.}, title={Resolution of singularities and division of distributions}, journal={Comm. Pure Appl. Math.}, volume={23}, date={1970}, pages={145--150}, issn={0010-3640}, review={\MR {256156}}, doi={10.1002/cpa.3160230202}, } Close amsref.
[3]
M. F. Atiyah, R. Bott, and L. Gårding, Lacunas for hyperbolic differential operators with constant coefficients. I, Acta Math. 124 (1970), 109–189, DOI 10.1007/BF02394570. MR470499, Show rawAMSref\bib{kn:A2}{article}{ author={Atiyah, M. F.}, author={Bott, R.}, author={G{\aa }rding, L.}, title={Lacunas for hyperbolic differential operators with constant coefficients. I}, journal={Acta Math.}, volume={124}, date={1970}, pages={109--189}, issn={0001-5962}, review={\MR {470499}}, doi={10.1007/BF02394570}, } Close amsref.
[4]
M. Atiyah and W. Schmid, A geometric construction of the discrete series for semisimple Lie groups, Invent. Math. 42 (1977), 1–62, DOI 10.1007/BF01389783. MR463358, Show rawAMSref\bib{kn:A3}{article}{ author={Atiyah, Michael}, author={Schmid, Wilfried}, title={A geometric construction of the discrete series for semisimple Lie groups}, journal={Invent. Math.}, volume={42}, date={1977}, pages={1--62}, issn={0020-9910}, review={\MR {463358}}, doi={10.1007/BF01389783}, } Close amsref.
[5]
M. Atiyah, New invariants of - and -dimensional manifolds, The mathematical heritage of Hermann Weyl (Durham, NC, 1987), Proc. Sympos. Pure Math., vol. 48, Amer. Math. Soc., Providence, RI, 1988, pp. 285–299, DOI 10.1090/pspum/048/974342. MR974342, Show rawAMSref\bib{kn:A4}{article}{ author={Atiyah, Michael}, title={New invariants of $3$- and $4$-dimensional manifolds}, conference={ title={The mathematical heritage of Hermann Weyl}, address={Durham, NC}, date={1987}, }, book={ series={Proc. Sympos. Pure Math.}, volume={48}, publisher={Amer. Math. Soc., Providence, RI}, }, date={1988}, pages={285--299}, review={\MR {974342}}, doi={10.1090/pspum/048/974342}, } Close amsref.
[6]
M. Atiyah and P. Sutcliffe, The geometry of point particles, Royal Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 458 (2002), no. 2021, 1089–1115, DOI 10.1098/rspa.2001.0913. MR1902577, Show rawAMSref\bib{kn:A5}{article}{ author={Atiyah, Michael}, author={Sutcliffe, Paul}, title={The geometry of point particles}, journal={Royal Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci.}, volume={458}, date={2002}, number={2021}, pages={1089--1115}, issn={1364-5021}, review={\MR {1902577}}, doi={10.1098/rspa.2001.0913}, } Close amsref.
[7]
A. Connes and J. Kouneiher, Sir Michael Atiyah, a knight mathematician: a tribute to Michael Atiyah, an inspiration and a friend, Notices Amer. Math. Soc. 66 (2019), no. 10, 1660–1671. MR3969866, Show rawAMSref\bib{kn:CK}{article}{ author={Connes, Alain}, author={Kouneiher, Joseph}, title={Sir Michael Atiyah, a knight mathematician: a tribute to Michael Atiyah, an inspiration and a friend}, journal={Notices Amer. Math. Soc.}, volume={66}, date={2019}, number={10}, pages={1660--1671}, issn={0002-9920}, review={\MR {3969866}}, } Close amsref.
[8]
S. Donaldson, Atiyah’s work on holomorphic vector bundles and gauge theories, Bull. Amer. Math. Soc. 58 (2021), no. 4, 567–610.
[9]
D. Freed, The Atiyah–Singer index theorem, Bull. Amer. Math. Soc. 58 (2021), no. 4, 517–566.
[10]
N. J. Hitchin (ed.), Memories of Sir Michael Atiyah, Notices Amer. Math. Soc. 66 (2019), no. 11, 1834–1852. With remembrances. MR3971088, Show rawAMSref\bib{kn:NJH0}{article}{ title={Memories of Sir Michael Atiyah}, author={Hitchin~(ed.), Nigel J.}, note={With remembrances}, journal={Notices Amer. Math. Soc.}, volume={66}, date={2019}, number={11}, pages={1834--1852}, issn={0002-9920}, review={\MR {3971088}}, } Close amsref.
[11]
N. J. Hitchin, Michael Atiyah: Geometry and Physics To appear in: Literature and History of Mathematical Science, CMSA Series in Mathematics, International Press, Boston
[12]
Graeme Segal, Michael Atiyah’s work in algebraic topology, Bull. Amer. Math. Soc. 58 (2021), no. 4, 481–510.

Article Information

Author Information
Simon Donaldson
Simons Center for Geometry and Physics, Stony Brook University, New York; and Department of Mathematics, Imperial College, London, United Kingdom
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Journal Information
Bulletin of the American Mathematical Society, Volume 58, Issue 4, ISSN 0273-0979, published by the American Mathematical Society, Providence, Rhode Island.
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