Michael Atiyah

How to use this course

Michael Atiyah worked across more of mathematics than almost anyone of his generation. He began as an algebraic geometer, built a new cohomology theory from vector bundles, and with Isadore Singer proved that the number of solutions of an elliptic differential equation is fixed by topology. In mid-career he turned to the equations of gauge theory and became one of the main bridges between geometry and theoretical physics. He also ran the Royal Society, Trinity College and a new national institute for mathematics.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. The lessons follow his work in rough chronological order: algebraic geometry (1952–1958), K-theory (1959–1966), index theory (1962–1976), symplectic geometry and gauge theory (1977–1988), and the interaction with quantum field theory (from 1988).

Timeline

Khartoum, Cairo, Manchester, 1929–1949

Michael Francis Atiyah was born in Hampstead, London, on 22 April 1929. His father, Edward Atiyah, was Lebanese, educated at Oxford, and worked in the Sudan as a civil servant; he was also an author. His mother, Jean Levens, came from a Scottish family. Atiyah grew up in Khartoum and attended the Diocesan school there from 1934 to 1941.

During the war he went to Victoria College, an English-language school in Egypt, first in Cairo and then in Alexandria. In 1945 the family moved to England, and he spent two years at Manchester Grammar School, whose strong mathematics teaching prepared him for Cambridge.

From 1947 to 1949 he did national service with the Royal Electrical and Mechanical Engineers. He then went up to Cambridge.

Cambridge and Hodge, 1949–1955

Atiyah entered Trinity College, Cambridge, in 1949. He was an undergraduate during a period when Cambridge pure mathematics was strong in analysis and number theory, while geometry meant mostly the projective geometry of the Italian school. He was supervised by A. S. Besicovitch and J. A. Todd, graduated in 1952, and chose W. V. D. Hodge as his research supervisor.

Hodge had shown how harmonic forms connect the topology of a complex manifold with its algebraic geometry. Through him Atiyah learned the new tools arriving from France: sheaf cohomology, fibre bundles and characteristic classes in the work of Henri Cartan, Jean-Pierre Serre, Shiing-Shen Chern and André Weil. His first independent work began with a flaw he found in a published paper on holomorphic vector bundles. Correcting it led to results on ruled surfaces and projective bundles that won the Smith’s Prize in 1954. He was elected a Fellow of Trinity the same year.

His thesis, Some applications of topological methods in algebraic geometry, was submitted in 1955. It already showed the habit that would define his career: take a classical geometric question and answer it with the most powerful modern machinery available. On 30 July 1955 he married Lily Brown, born in Edinburgh in 1928.

Princeton, Bonn, Oxford, 1955–1963

Atiyah spent 1955–56 at the Institute for Advanced Study in Princeton as a Commonwealth Fellow. The year was decisive. There he met Friedrich Hirzebruch, who had just proved his Riemann–Roch theorem, and Raoul Bott and Isadore Singer, who would become his two closest collaborators. He worked on vector bundles and characteristic classes, including the classification of bundles on elliptic curves (Lesson 04).

Back in Cambridge he became a lecturer in 1957 and a Fellow of Pembroke College in 1958. From 1957 he attended Hirzebruch’s annual Arbeitstagung in Bonn, where in that first year Alexander Grothendieck’s version of the Riemann–Roch theorem was a central topic. Grothendieck’s group of coherent sheaves, Hirzebruch’s integrality theorems and Bott’s new periodicity theorem were the ingredients from which Atiyah and Hirzebruch built topological K-theory (Lesson 05).

In 1961, after the death of J. H. C. Whitehead, Atiyah moved to Oxford as a Reader and Fellow of St Catherine’s College. He was elected a Fellow of the Royal Society in 1962. That spring Singer visited Oxford, and the two began the work that led to the index theorem. In 1963 Atiyah became Savilian Professor of Geometry, a chair once held by Edmond Halley.

Vector bundles on curves

For the general reader. A vector bundle attaches a vector space to each point of a space, varying continuously or algebraically. On a curve shaped like a doughnut (an elliptic curve), Atiyah found every possible bundle and arranged them all in a single picture. The answer was so clean that it became a model for decades of later work.

The setting

Let \(E\) be an elliptic curve over an algebraically closed field \(k\), that is, a smooth projective curve of genus 1 with a base point. Every vector bundle is a direct sum of indecomposable bundles (Atiyah proved a Krull–Schmidt theorem for bundles in 1956), so it suffices to classify indecomposables. For a bundle \(V\) write \(r\) for its rank and \(d\) for its degree. Riemann–Roch on a curve of genus 1 reads

Ideas of the proof. Degree-0 line bundles correspond to points of \(E\) via \(p\mapsto\mathcal O(p-p_0)\). Extensions of \(F_{r-1}\) by \(\mathcal O\) are classified by \(\mathrm{Ext}^1(F_{r-1},\mathcal O)\cong H^1(F_{r-1}^\vee)\), which turns out to be one-dimensional, so the non-split extension is unique up to isomorphism. For nonzero degree Atiyah gave an inductive construction relating \(\mathcal E(r,d)\) to \(\mathcal E(r,d-r)\) (tensoring with a degree-1 line bundle) and to bundles of smaller rank through the Euclidean algorithm on \((r,d)\). When \(\gcd(r,d)=1\) the indecomposable bundles are exactly the stable ones, a fact later central to the work of Mumford, Narasimhan and Seshadri on moduli of bundles.

The Atiyah class

In the same year Atiyah asked when a holomorphic bundle \(V\) on a complex manifold \(X\) carries a holomorphic connection. He defined a class \(a(V)\in H^1(X,\Omega^1_X\otimes\mathrm{End}\,V)\), the extension class of the bundle of first-order jets of \(V\), and proved that \(V\) admits a holomorphic connection if and only if \(a(V)=0\). On a compact Kähler manifold the trace of \(a(V)\) is \(c_1(V)\) up to a normalizing constant, and more generally the traces of its powers recover the Chern classes. So a bundle with a holomorphic connection on a compact Kähler manifold has vanishing rational Chern classes.

Topological K-theory

For the general reader. Count the vector bundles on a space, allowing yourself to add and subtract them formally, and you get a group that behaves like a cohomology theory. Atiyah and Hirzebruch built this K-theory around 1959–61. It turned out to be the right language for index theory and for problems in homotopy theory that ordinary cohomology could not reach.

Definitions

For a compact Hausdorff space \(X\), isomorphism classes of complex vector bundles form a commutative monoid under \(\oplus\), and a semiring with \(\otimes\). Its Grothendieck group is \(K^0(X)\), whose elements are formal differences \([V]-[W]\). For a based space, the reduced group \(\tilde K^0(X)\) is the kernel of restriction to the base point. Atiyah and Hirzebruch set \(\tilde K^{-n}(X)=\tilde K^0(\Sigma^nX)\) and obtained a cohomology theory, satisfying all the Eilenberg–Steenrod axioms except the dimension axiom.

Bott proved periodicity in 1957–59 for the homotopy groups of the unitary group using Morse theory. Atiyah and Bott later gave an elementary proof using clutching functions, which is the one in most textbooks. Periodicity lets one extend K-theory to a \(\mathbb Z/2\)-graded theory \(K^0\oplus K^1\).

So rationally K-theory sees nothing new. Its power lies in torsion and in integrality: \(\mathrm{ch}\) of a genuine bundle has very special denominators. The Atiyah–Hirzebruch spectral sequence \(H^p(X;K^q(\mathrm{pt}))\Rightarrow K^{p+q}(X)\) organizes the comparison.

Adams operations and the Hopf invariant

A map \(f:S^{4n-1}\to S^{2n}\) has a Hopf invariant \(h(f)\in\mathbb Z\): in the cohomology of the mapping cone \(X=S^{2n}\cup_f e^{4n}\) with generators \(x\in H^{2n}\) and \(y\in H^{4n}\), one has \(x^2=h(f)\,y\). Frank Adams proved in 1960, with a long argument using secondary cohomology operations, that \(h(f)\) can be odd only for \(n=1,2,4\). This settles which spheres are H-spaces and shows that real division algebras exist only in dimensions 1, 2, 4 and 8. In 1966 Adams and Atiyah gave a proof of a few lines in K-theory.

The Adams operations \(\psi^k\) are ring homomorphisms of \(K^0(X)\) with \(\psi^k(L)=L^k\) for line bundles and \(\psi^k(x)\equiv x^p \bmod p\) when \(k=p\) is prime. On \(\tilde K^0(S^{2m})\cong\mathbb Z\) the operation \(\psi^k\) is multiplication by \(k^m\). For the mapping cone, \(\tilde K^0(X)\) is free on classes \(a,b\) restricting to generators on \(S^{2n}\) and \(S^{4n}\), with \(a^2=h\,b\), and

Since \(\psi^2(a)\equiv a^2=hb \pmod 2\), \(\mu_2\equiv h\pmod2\). Now use \(\psi^2\psi^3=\psi^6=\psi^3\psi^2\). Comparing the coefficients of \(b\) gives

If \(h\) is odd then \(\mu_2\) is odd, so \(2^n\) divides \(3^n-1\). Exercise 5.1 shows that this forces \(n\in\{1,2,4\}\).

The index theorem

For the general reader. A system of linear differential equations on a closed manifold has some number of independent solutions, and its adjoint has some other number. The difference, called the index, does not change when the equations are deformed. Atiyah and Singer showed that it can be computed from topology alone. The theorem contains as special cases the Gauss–Bonnet theorem, the Riemann–Roch theorem and Hirzebruch’s signature theorem.

Elliptic operators

Let \(M\) be a closed manifold of dimension \(n\) and \(P:\Gamma(E)\to\Gamma(F)\) a linear differential operator of order \(m\) between vector bundles. In local coordinates \(P=\sum_{|\alpha|\le m}a_\alpha(x)\partial^\alpha\); its principal symbol at a covector \(\xi\) is \(\sigma_P(x,\xi)=\sum_{|\alpha|=m}a_\alpha(x)(i\xi)^\alpha\in\mathrm{Hom}(E_x,F_x)\). The operator is elliptic if \(\sigma_P(x,\xi)\) is invertible for all \(\xi\ne0\). Elliptic operators on closed manifolds are Fredholm, so

is finite, and it depends only on the homotopy class of the symbol through invertible symbols. Israel Gelfand had asked in 1960 for a topological formula for the index.

The theorem

The symbol of an elliptic operator defines a class \([\sigma_P]\in K^0_c(T^*M)\), the K-theory with compact supports of the cotangent bundle. Atiyah and Singer defined a topological index \(t\text{-}\mathrm{ind}:K^0_c(T^*M)\to\mathbb Z\) by embedding \(M\) in \(\mathbb R^N\), pushing the symbol class forward by a Thom isomorphism to \(K^0_c(\mathbb R^{2N})\cong\mathbb Z\).

The first proof, announced in 1963, followed Hirzebruch’s proof of Riemann–Roch through cobordism theory. The published proof (Atiyah–Singer I, 1968) uses K-theory: both indices are homomorphisms on \(K^0_c(T^*M)\) satisfying the same axioms (normalization on a point and compatibility with embeddings), and these axioms determine them. Later proofs use the heat equation (Atiyah–Bott–Patodi, 1973, building on Vijay Kumar Patodi’s work) and supersymmetric quantum mechanics.

The four classical complexes

For an operator built from geometry, the integrand becomes a characteristic class of \(M\):

The last line was new. It explained why the \(\hat A\)-genus of a spin manifold is an integer, and showed it to be an obstruction to positive scalar curvature (Lichnerowicz’s argument: positive scalar curvature forces \(\ker D=0\)). Atiyah and Singer extended the theorem to families of operators, to operators invariant under a group, and to real operators, where the index can live in \(KO\)-groups and take values in \(\mathbb Z/2\).

Fixed points and spectral asymmetry

For the general reader. Lefschetz showed how to count the fixed points of a map of a space using algebra. Atiyah and Bott found a far-reaching version for maps that respect a differential equation. A decade later, with Patodi and Singer, Atiyah extended the index theorem to manifolds with boundary, where a new invariant measures the imbalance between positive and negative eigenvalues.

The Lefschetz formula for elliptic complexes

Let \(0\to\Gamma(E_0)\xrightarrow{d}\Gamma(E_1)\xrightarrow{d}\cdots\to\Gamma(E_N)\to0\) be an elliptic complex on a closed manifold, with cohomology \(H^i\). A smooth map \(f:M\to M\) with lifts \(\varphi_i:f^*E_i\to E_i\) commuting with \(d\) induces maps \(T_i\) on \(H^i\), and the Lefschetz number is \(L(T)=\sum(-1)^i\operatorname{tr}(T_i|H^i)\).

The proof replaces the trace on cohomology by the supertrace of \(T\) on the whole complex, smoothed by the heat kernel or a parametrix, and localizes at the fixed points where the kernel of \(T\) meets the diagonal. Atiyah and Bott used the formula to give new proofs of Hermann Weyl’s character formula for compact Lie groups and of results on actions of finite groups on manifolds.

The eta invariant

For a self-adjoint elliptic operator \(B\) on a closed odd-dimensional manifold, with nonzero eigenvalues \(\lambda\), Atiyah, Patodi and Singer defined

convergent for \(\operatorname{Re}s\) large, with a meromorphic continuation regular at \(s=0\). The number \(\eta_B(0)\) measures spectral asymmetry.

For the signature operator this gives \(\operatorname{sign}(X)=\int_XL(p)-\eta(0)\). The eta invariant later became the global anomaly in quantum field theory and a key ingredient of the Chern–Simons invariants of 3-manifolds.

Moment maps and Yang–Mills on surfaces

For the general reader. In mechanics, a symmetry has a conserved quantity, its momentum. Atiyah found that the possible values of these momenta always fill a convex polygon. With Bott he then applied the same idea in infinite dimensions, treating the curvature of a gauge field as a momentum, and used it to compute the topology of moduli spaces of bundles on surfaces.

Convexity

Let \((M,\omega)\) be a symplectic manifold with an action of a torus \(T\). The action is Hamiltonian if there is a moment map \(\mu:M\to\mathfrak t^*\) with \(d\langle\mu,X\rangle=\iota_{X_M}\omega\) for every \(X\in\mathfrak t\), where \(X_M\) is the generating vector field, and \(\mu\) is \(T\)-invariant.

Atiyah’s proof uses Morse theory: each component \(\langle\mu,X\rangle\) is a Morse–Bott function whose critical submanifolds have even indices, so its level sets are connected. Induction on the dimension of \(T\) gives convexity.

Schur–Horn. Let \(M\) be the set of Hermitian \(n\times n\) matrices with eigenvalues \(\lambda_1,\dots,\lambda_n\), a coadjoint orbit of \(U(n)\), and let the diagonal torus act by conjugation. The moment map takes a matrix to its diagonal. Fixed points are the diagonal matrices, whose diagonals are the permutations of \((\lambda_i)\). So the set of diagonals is the permutohedron, the hexagon shown at the top of this page for \(n=3\).

Yang–Mills over a Riemann surface

Fix a compact Riemann surface \(\Sigma\) and a Hermitian bundle \(P\) of rank \(n\). The space \(\mathcal A\) of unitary connections is an affine space modelled on \(\Omega^1(\Sigma,\mathfrak u(P))\), with symplectic form \(\omega(a,b)=-\int_\Sigma\operatorname{tr}(a\wedge b)\). Atiyah and Bott observed that the gauge group \(\mathcal G\) acts with moment map the curvature, \(A\mapsto F_A\). The symplectic quotient \(F^{-1}(0)/\mathcal G\) is the moduli space of flat connections. By the theorem of M. S. Narasimhan and C. S. Seshadri it is also the moduli space of semistable holomorphic bundles of degree 0, the analogue of the statement that a symplectic quotient equals a complex quotient by stability.

The method was developed by his student Frances Kirwan into a general theory of the cohomology of quotients in algebraic and symplectic geometry. In 1984 Atiyah and Bott also formulated the localization theorem in equivariant cohomology, found independently by Nicole Berline and Michèle Vergne.

Instantons and monopoles

For the general reader. The equations of the strong and weak forces are Yang–Mills equations. Their simplest finite-energy solutions in four Euclidean dimensions are instantons. In 1977–78 Atiyah and his collaborators counted and then constructed all of them, using twistor geometry and linear algebra. Within a few years his student Simon Donaldson used instantons to discover unexpected properties of four-dimensional spaces.

Self-duality

Let \(A\) be a connection on an \(\mathrm{SU}(2)\)-bundle \(P\) over a compact oriented Riemannian 4-manifold \(X\), with curvature \(F_A\in\Omega^2(X,\mathfrak{su}(2))\). The Hodge star splits 2-forms into self-dual and anti-self-dual parts, \(F=F^++F^-\). The instanton number is \(k=c_2(P)[X]=\frac1{8\pi^2}\int_X\operatorname{tr}(F\wedge F)\). With the norm \(\|F\|^2=-\int_X\operatorname{tr}(F\wedge*F)\) one has

So \(\|F\|^2\ge8\pi^2|k|\), with equality exactly for anti-self-dual (\(F^+=0\)) or self-dual (\(F^-=0\)) connections. These absolute minima of the Yang–Mills functional are instantons. Conventions on which sign is called “self-dual” vary between authors.

The dimension comes from the index theorem applied to the deformation complex \(\Omega^0(\mathfrak g_P)\xrightarrow{d_A}\Omega^1(\mathfrak g_P)\xrightarrow{d_A^+}\Omega^+(\mathfrak g_P)\). The curvature hypothesis is used for a vanishing theorem showing the last cohomology group is zero. Through Penrose’s twistor correspondence, which Atiyah learned from Roger Penrose and Richard Ward, instantons on \(S^4\) correspond to holomorphic bundles on \(\mathbb{CP}^3\) with a real structure.

Atiyah and Hitchin found the construction in November 1977; Vladimir Drinfeld and Yuri Manin found it independently, and the four published it jointly. It reduced a nonlinear partial differential equation to quadratic matrix equations. The same pattern, a hyperkähler quotient of a vector space, reappeared in Kronheimer and Nakajima’s quiver varieties. Atiyah and Hitchin went on to study magnetic monopoles, solutions of the Bogomolny equations \(F_A=*d_A\Phi\) on \(\mathbb R^3\). Their 1988 book computed the hyperkähler metric on the moduli space of two monopoles and used it to describe slow monopole scattering, in which two monopoles colliding head-on emerge at right angles.

Topological quantum field theory

For the general reader. In 1988 Edward Witten showed that Donaldson’s invariants of 4-manifolds come from a quantum field theory. Atiyah responded by writing down what such a theory must be, as a mathematical object, in a few axioms. These axioms turned a physical intuition into a definition that mathematicians could build on, and they now organize much of low-dimensional topology and representation theory.

The axioms

Equivalently, \(Z\) is a symmetric monoidal functor from the category of \((d+1)\)-dimensional cobordisms to vector spaces. A closed \((d+1)\)-manifold \(M\) is a cobordism from \(\emptyset\) to \(\emptyset\), so \(Z(M)\in\mathbb C\) is a number, the partition function. Atiyah’s motivating examples were Floer homology in dimension 3+1, Witten’s 2+1-dimensional Chern–Simons theory, whose vector spaces are spaces of conformal blocks and whose invariants of links give the Jones polynomial, and theories in dimension 1+1 coming from conformal field theory as in the axioms of Graeme Segal, his former student.

Two dimensions

Let \(A=Z(S^1)\). The pair of pants gives a commutative multiplication, the disc gives a unit \(1\in A\) and a linear form \(\theta:A\to\mathbb C\), and the cylinder bent into a U gives a non-degenerate pairing \(\langle a,b\rangle=\theta(ab)\). So \(A\) is a commutative Frobenius algebra, and conversely every commutative Frobenius algebra determines a 2-dimensional TQFT.

The Oxford school and public life, 1963–1997

Atiyah received the Fields Medal at the International Congress in Moscow in 1966, for his work on K-theory and the index theorem. In 1969 he moved to the Institute for Advanced Study in Princeton, where he worked with Bott and Patodi on the heat-equation proof of the index theorem and with Singer on the eta invariant. He returned to Oxford in 1972 as a Royal Society Research Professor, a post free of teaching that let him build a research school.

The Oxford geometry group of the 1970s and 1980s became one of the strongest in the world. Atiyah’s doctoral students included Graeme Segal, George Lusztig, Nigel Hitchin, Simon Donaldson, Frances Kirwan, Peter Kronheimer, Lisa Jeffrey and Ruth Lawrence. He worked closely with Roger Penrose’s twistor group, and from the late 1970s he was in regular contact with Edward Witten. Their exchanges ran in both directions: geometry gave Witten tools, and Witten’s quantum field theory gave geometers new invariants to explain, such as Donaldson’s invariants and the Jones polynomial. Of the four Fields Medallists of 1990, three (Vladimir Drinfeld, Vaughan Jones and Witten) worked at the meeting point of geometry and physics that Atiyah had done much to create.

His textbook Introduction to Commutative Algebra (1969), written with Ian Macdonald, became the standard short course in the subject. His K-theory lecture notes (1967) and The Geometry and Physics of Knots (1990) played similar roles.

From 1990 Atiyah took on three major posts at once. He was President of the Royal Society (1990–95), Master of Trinity College, Cambridge (1990–97), and the first Director of the Isaac Newton Institute for Mathematical Sciences in Cambridge (1990–96), which he helped shape as a visiting institute organized around long programmes. He had already served as President of the London Mathematical Society (1974–76). He was knighted in 1983 and appointed to the Order of Merit in 1992. In his final address as President of the Royal Society in 1995 he criticized British nuclear-weapons policy, and from 1997 to 2002 he was President of the Pugwash Conferences on Science and World Affairs. He also helped found the European Mathematical Society and the InterAcademy Panel.

Edinburgh, last years, legacy, 1997–2019

In 1997 Atiyah retired from Trinity and moved to Edinburgh, his wife’s home city, as an honorary professor at the University of Edinburgh. He was Chancellor of the University of Leicester from 1995 to 2005 and President of the Royal Society of Edinburgh from 2005 to 2008. In 2004 he and Singer shared the Abel Prize, cited for the discovery and proof of the index theorem, which brought together topology, geometry and analysis, and for building bridges between mathematics and theoretical physics.

He remained active in research and lecturing into his late eighties. Lily Atiyah died in March 2018. In September 2018, at the Heidelberg Laureate Forum, he presented an argument that he said proved the Riemann hypothesis, based on a function connected with the fine-structure constant. Experts found the argument incomplete, and the hypothesis remains open. He died in Edinburgh on 11 January 2019, aged 89.

Legacy

• Topology: K-theory became a fundamental cohomology theory, extended by others to algebraic K-theory, KO-theory, equivariant K-theory and K-theory of operator algebras.
• Analysis and geometry: the index theorem and its heat-kernel proof created global analysis as a subject; the eta invariant and families index are tools in geometry and physics.
• Symplectic geometry: convexity, equivariant localization and the Atiyah–Bott approach to moduli spaces shaped modern symplectic and algebraic geometry.
• Mathematical physics: the study of instantons and monopoles led to Donaldson theory, and the TQFT axioms gave a precise framework for the quantum invariants of knots and manifolds.
• Honours: Fields Medal (1966), Royal Medal (1968), De Morgan Medal (1980), Feltrinelli Prize (1981), King Faisal Prize (1987), Copley Medal (1988), Benjamin Franklin Medal (1993), Abel Prize (2004), Grande Médaille of the French Academy of Sciences (2010).

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Michael Francis Atiyah, MacTutor History of Mathematics.
2. Michael Atiyah, Wikipedia.
3. N. J. Hitchin, A mathematical biography of Michael Atiyah, Celebratio Mathematica.
4. The Abel Prize 2004, Norwegian Academy of Science and Letters.
5. M. Raussen and C. Skau, Interview with Michael Atiyah and Isadore Singer, Notices of the AMS 52 (2005).

Mathematics

1. M. F. Atiyah, Vector bundles over an elliptic curve, Proc. London Math. Soc. 7 (1957).
2. M. F. Atiyah, Complex analytic connections in fibre bundles, Trans. AMS 85 (1957).
3. M. F. Atiyah and F. Hirzebruch, Vector bundles and homogeneous spaces, Proc. Symp. Pure Math. 3 (1961).
4. M. F. Atiyah, K-theory, Benjamin, 1967.
5. J. F. Adams and M. F. Atiyah, K-theory and the Hopf invariant, Quart. J. Math. 17 (1966).
6. M. F. Atiyah and I. M. Singer, The index of elliptic operators on compact manifolds, Bull. AMS 69 (1963); The index of elliptic operators I, III, Ann. of Math. 87 (1968).
7. M. F. Atiyah and R. Bott, A Lefschetz fixed point formula for elliptic complexes I, II, Ann. of Math. 86 (1967), 88 (1968).
8. M. F. Atiyah, V. K. Patodi and I. M. Singer, Spectral asymmetry and Riemannian geometry I, Math. Proc. Cambridge Philos. Soc. 77 (1975).
9. M. F. Atiyah, Convexity and commuting Hamiltonians, Bull. London Math. Soc. 14 (1982).
10. M. F. Atiyah and R. Bott, The Yang–Mills equations over Riemann surfaces, Phil. Trans. R. Soc. A 308 (1983).
11. M. F. Atiyah, N. J. Hitchin and I. M. Singer, Self-duality in four-dimensional Riemannian geometry, Proc. R. Soc. A 362 (1978).
12. M. F. Atiyah, V. G. Drinfeld, N. J. Hitchin and Yu. I. Manin, Construction of instantons, Phys. Lett. A 65 (1978).
13. M. F. Atiyah and N. J. Hitchin, The Geometry and Dynamics of Magnetic Monopoles, Princeton, 1988.
14. M. F. Atiyah, Topological quantum field theories, Publ. Math. IHÉS 68 (1988).
15. M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, 1969.
16. H. B. Lawson and M.-L. Michelsohn, Spin Geometry, Princeton, 1989.