Michael F. Atiyah Mathematics’ Deep Reasons by Claudio Bartocci The age of unification How will the mathematics of the last fifty years appear to the eyes of future historians? As difficult as it is to make predictions that depend in large part on developments to come (Lakatos docet), we can hazard a guess that the second half of the twentieth century will probably be considered a period of extraordinary proliferation of new ideas, and at the same time, of the rediscovery of the fundamental unity of mathematics. In the first half of the century, marked by the triumphant of Hilbert’s program culminating in the great undertaking of the Bourbaki group, there was a widespread tendency – we might say – towards specialization and therefore towards the parcelling out of mathematical knowledge: doubtless, it was this tendency that made possible the rapid development of disciplines such as general topology, group theory, mathematical logic, differential geometry, functional analysis, algebraic and differential topology, commutative algebra and algebraic geometry (even if this last was destined to change its face in later years). On the contrary, the second fifty years are characterized above all as an “era of unification, where borders are crossed over, techniques have been moved from one field into the other, and things have become hybridised to an enormous extent”:1 cross-fertilization has prevailed as the dominant paradigm. There are many examples that lend support to this thesis: the admirable theoretical edifice constructed by Grothendieck; the development of research areas such as global analysis; the omnipresence of category theory methods; arithmetic geometry or Alain Connes’s non-commutative geometry; the formidable results obtained by mathematicians such as Vladimir Arnol’d, Yuri I. Manin, Shing Tung Yau, Simon K. Donaldson, Vaughan Jones, Edward Witten, Richard Borcherds, Robert Langlands and Andrew Wiles. But perhaps more than anyone else, the person who demonstrated how fruitful it can be to delve into different disciplinary contexts in order to arrive at fundamental discoveries, to dig down to the roots of problems that are apparently distinct in order to identify the profound reason that unites them, is Michael Francis Atiyah, without a doubt one of the most prolific and influential mathematicians of the past century. From algebraic geometry to K-theory Born in London on 22 April 1929 to a Lebanese father and Scottish mother, Atiyah spent the years of his boyhood in Khartoum, Sudan and Cairo.2 In 1945, at the end of World War II, his family moved to England; the young Michael was sent to Manchester Grammar School, beacuse it was reputed to be the best school for mathematics in England. After having spent two-years in the National Service, he enrolled at Trinity College in Cambridge: his first article, written in 1952 under the guidance of J. A. Todd, regarded a question of projective geometry. Having begun his doctoral studies, still at Cambridge, Atiyah chose as his supervisor one of the greatest English mathematicians, William V. D. Hodge, who in 1941 had published his famous treatise on harmonic integrals. Hodge directed him to the work of Chern on characteristic classes; Atiyah went to discover the new ideas – developed above all by the French school (André Weil, Henri Cartan, Jean- Pierre Serre) – which were emerging in those years in algebraic geometry.3 He got acquainted with vector bundles, coherent sheaves and their cohomology, and became a voracious reader of the Comptes rendus. In 1955-1956 Atiyah spent a long period Institute of Advanced Study in Princeton, where he came to know Raoul Bott, Friedrich Hirzebruch and Isadore M. Singer and Serre, and would broaden his horizons in an atmosphere that was very stimulating intellectually. The author warmly thanks Sir Michael Atiyah for reading a preliminary version of this article and for pointing out a few inaccuracies. 1 M. F. Atiyah, “Mathematics in the XXth Century”, Bulletin of the London Mathematical Society, 34 (2002), pp. 1- 15. 2 Atiyah himself provided detailed information about his life and his research activities in a series of talks, recorded in March of 1997, available (along with their transcripts) on the website http://peoplesarchives.com. See also M. F. Atiyah, Siamo tutti matematici, Di Renzo, Rome, 2007. 3 Cf. Jean Dieudonné, History of algebraic geometry, Wadsworth, Monterey 1985, chap. VIII. C. Bartocci –Michael F. Atiyah: Mathematics’ deep reasons– 1 Until 1959 the most part of Atiyah’s works were in algebraic geometry: of particular note, for example, is the 1957 article on the classification of vector bundles on an elliptic curve. In later years – after his return to Europe – his interests would turn above all to topology. This reorientation was principally due to the influence of the German mathematician Friedrich Hirzebruch, organizer of the famous Arbeitstagung in Bonn, in which Atiyah was a faithful participant. Hirzebruch had extended (and reinterpreted in a topological key) the classic Riemann-Roch theorem to algebraic manifolds of higher dimensions and, making use of Thom’s cobordism theory, had defined particular combinations of “characteristic numbers” that assume integer values not only for algebraic manifolds, but for all differentiable manifolds as well. Hirzebruch’s results, united to Grothendieck’s profound generalization of the theorem of Hirzebruch-Riemann-Roch and to Bott’s theorem of periodicity,4 are the ingredients at the base of topological K-theory, which Atiyah (in collaboration with Hirzebruch) set out between 1959 and 1962.5 Atiyah himself recalls: “… I saw that by mixing all these things together you ended up with some interesting topological consequences, and because of that we then thought it would be useful to introduce the topological K-group as a formal apparatus in which to carry this out”.6 K-theory – which can be thought of as a kind of generalized cohomology theory constructed beginning with the isomorphism classes of vector bundles – immediately showed itself to be a useful and versatile tool for tackling problems of various natures: one of the most remarkable example was the solution by Frank Adams in 1962 of the problem of the maximum number of nowhere vanishing and (pointwise) linearly independent tangent vector fields on an odd-dimensional sphere (by the famous “hairy ball theorem”, the 2n- sphere has no non-vanishing vector field for n ≥ 1). At the International Congress of Mathematicians in Stockholm in 1962, Atiyah presented the numerous, and in large part unexpected, applications of K-theory, and in conclusion, mentioned the new interpretation which admits the notion of “symbol of an elliptic operator” into this theory. In the unifying framework of “K-theory” converged methods and ideas coming from algebraic geometry, topology and functional analysis: the “index theorem” – already clearly stated in the summer of 1962 but still awaiting proof –, will be the culmination of a long series of researches. The index theorem Atiyah spent the years from 1957 to 1961 in Cambridge as a lecturer and tutorial fellow at Pembroke College, overburdened with many hours of teaching. In 1960 the topologist Henry Whitehead died at 55, and his chair in Oxford remained vacant: Atiyah applied for the chair unsuccessfully (the favorite was one of Whitehead’s students, Graham Higman) and as makeshift solution, which in any case freed him from excessive teaching duties, he accepted a position as a reader. Less than two years later, with the death of Titchmarsh, the prestigious chair of Savilian Professor would become free, and Atiyah was called to hold it.7 In the attempt to extend results valid in the case of algebraic manifolds to differentiable manifolds, Hirzebruch had proven that a certain combination of characteristic classes – the so-called Â-genus –, which is in general a rational number, turns out to be an integer in the case of spin manifolds (that is, manifolds whose second Stiefel-Whitney class is null). This result naturally falls in the context of K-theory, but its explanation appeared a real mystery at the time. 4 Bott’s theorem of periodicity regards the homotopy groups of the groups U(n) as n " ! . 5 For mathematical details not included here, see M. F. Atiyah, “K-theory past and present” in Sitzungsberichte der Berliner Mathematischen Gesellschaft, Berliner Mathematischen Gesellschaft, Berlin 2001, pp. 411-417 and M. F. Atiyah, “Papers on K-theory”, in Collected Works, vol. 2, Oxford University Press, New York, 1988, pp. 1-3. 6 M. F. Atiyah, http://www.peoplesarchives.com (Part 7, 36). 7 See N. Hitchin, “Geometria a Oxford: 1960-1990”, in La matematica. Tempi e luoghi, vol. 1, ed. by C. Bartocci and P. Odifreddi, Einaudi, Torino, 2007, pp. 711-734; much information about the genesis of the index theorem was drawn from this essay. C. Bartocci –Michael F. Atiyah: Mathematics’ deep reasons– 2 In January 1962 Isadore Singer decided to spend (at his own expense) a period of time in Oxford. Two days after his arrival, Atiyah and Singer had this exchange: ATIYAH: Why is the genus an integer for spin manifolds? SINGER: What’s up, Michael? You know the answer much better than I. 8 ATIYAH: There’s a deeper reason. Singer has a thorough knowledge of differential geometry and analysis, disciplines in which Atiyah was instead less well-versed. The key to the problem was found in a few months: the answer lay in the Dirac operator. As Atiyah tells it: I knew the formula, Hirzebruch’s work, I knew what the answer was; what I had to guess was the problem. We had to find out what was this object. We knew from algebraic geometry, what it should look like in algebraic cases. We knew that it had to do with spinors because of Hirzebruch’s formula. So the question was, there should be some differential equation which would play the role of the Cauchy-Riemann equations in the spinor case which ought to fit the left-hand side of the equation.9 The differential operator “rediscovered” by Atiyah and Singer, whose construction is based on an in- depth study of the geometric properties of spin manifolds, is closely related to the operator, well known to physicists for about thirty years, that appears in Dirac’s equations.