[7] He returned to England and Manchester Grammar School for his HSC studies (1945–1947) and did his national service with the Royal Electrical and Mechanical Engineers (1947–1949).

He took up a three-year professorship at the Institute for Advanced Study in Princeton after which he returned to Oxford as a Royal Society Research Professor and professorial fellow of St Catherine's College.

Some of his more recent work was inspired by theoretical physics, in particular instantons and monopoles, which are responsible for some corrections in quantum field theory.

[19] His other collaborators included; J. Frank Adams (Hopf invariant problem), Jürgen Berndt (projective planes), Roger Bielawski (Berry–Robbins problem), Howard Donnelly (L-functions), Vladimir G. Drinfeld (instantons), Johan L. Dupont (singularities of vector fields), Lars Gårding (hyperbolic differential equations), Nigel J. Hitchin (monopoles), William V. D. Hodge (Integrals of the second kind), Michael Hopkins (K-theory), Lisa Jeffrey (topological Lagrangians), John D. S. Jones (Yang–Mills theory), Juan Maldacena (M-theory), Yuri I. Manin (instantons), Nick S. Manton (Skyrmions), Vijay K. Patodi (spectral asymmetry), A. N. Pressley (convexity), Elmer Rees (vector bundles), Wilfried Schmid (discrete series representations), Graeme Segal (equivariant K-theory), Alexander Shapiro[20] (Clifford algebras), L. Smith (homotopy groups of spheres), Paul Sutcliffe (polyhedra), David O.

[22] If you attack a mathematical problem directly, very often you come to a dead end, nothing you do seems to work and you feel that if only you could peer round the corner there might be an easy solution.

[2] Other contemporary mathematicians who influenced Atiyah include Roger Penrose, Lars Hörmander, Alain Connes and Jean-Michel Bismut.

[28] As an undergraduate Atiyah was interested in classical projective geometry, and wrote his first paper: a short note on twisted cubics.

[29] He started research under W. V. D. Hodge and won the Smith's prize for 1954 for a sheaf-theoretic approach to ruled surfaces,[30] which encouraged Atiyah to continue in mathematics, rather than switch to his other interests—architecture and archaeology.

[31] His PhD thesis with Hodge was on a sheaf-theoretic approach to Solomon Lefschetz's theory of integrals of the second kind on algebraic varieties, and resulted in an invitation to visit the Institute for Advanced Study in Princeton for a year.

[34] He also studied double points on surfaces,[35] giving the first example of a flop, a special birational transformation of 3-folds that was later heavily used in Shigefumi Mori's work on minimal models for 3-folds.

Atiyah and Todd[43] used K-theory to improve the lower bounds found using ordinary cohomology by Borel and Serre for the James number, describing when a map from a complex Stiefel manifold to a sphere has a cross section.

Atiyah showed how primary operations in K-theory could be used to give a short solution taking only a few lines, and in joint work with Adams[45] also proved analogues of the result at odd primes.

However a simpler and more general proof was produced by introducing equivariant K-theory, i.e. equivalence classes of G-vector bundles over a compact G-space X.

With Hirzebruch he extended the Grothendieck–Riemann–Roch theorem to complex analytic embeddings,[52] and in a related paper[53] they showed that the Hodge conjecture for integral cohomology is false.

[54] The Bott periodicity theorem was a central theme in Atiyah's work on K-theory, and he repeatedly returned to it, reworking the proof several times to understand it better.

[citation needed] The most useful piece of advice I would give to a mathematics student is always to suspect an impressive sounding Theorem if it does not have a special case which is both simple and non-trivial.

In general it gives the index as a sum over fixed point submanifolds of the group G.[71] Atiyah[72] solved a problem asked independently by Hörmander and Gel'fand, about whether complex powers of analytic functions define distributions.

[73] As an application of the equivariant index theorem, Atiyah and Hirzebruch showed that manifolds with effective circle actions have vanishing Â-genus.

The fundamental solutions of linear hyperbolic partial differential equations often have Petrovsky lacunas: regions where they vanish identically.

This version is called the L2 index theorem, and was used by Atiyah and Schmid[80] to give a geometric construction, using square integrable harmonic spinors, of Harish-Chandra's discrete series representations of semisimple Lie groups.

With Hitchin and Singer[84] he calculated the dimension of the moduli space of irreducible self-dual connections (instantons) for any principal bundle over a compact 4-dimensional Riemannian manifold (the Atiyah–Hitchin–Singer theorem).

[86] Atiyah reformulated this construction using quaternions and wrote up a leisurely account of this classification of instantons on Euclidean space as a book.

[93] Atiyah and R. Bott used Morse theory and the Yang–Mills equations over a Riemann surface to reproduce and extending the results of Harder and Narasimhan.

Atiyah and Bott[97] showed that this could be deduced from a more general formula in equivariant cohomology, which was a consequence of well-known localization theorems.

Atiyah showed[98] that the moment map was closely related to geometric invariant theory, and this idea was later developed much further by his student F. Kirwan.

Witten shortly after applied the Duistermaat–Heckman formula to loop spaces and showed that this formally gave the Atiyah–Singer index theorem for the Dirac operator; this idea was lectured on by Atiyah.

This gives another example where the moduli spaces of solutions of two apparently unrelated nonlinear partial differential equations turn out to be essentially the same.

Atiyah continued to publish subsequently, including several surveys, a popular book,[106] and another paper with Segal on twisted K-theory.

Several papers[112] were inspired by a question of Jonathan Robbins (called the Berry–Robbins problem), who asked if there is a map from the configuration space of n points in 3-space to the flag manifold of the unitary group.

[119][120] At the 2018 Heidelberg Laureate Forum, he claimed to have solved the Riemann hypothesis, Hilbert's eighth problem, by contradiction using the fine-structure constant.