### Sir Michael Francis Atiyah †

Born 22 April 1929, London, England, UK

Died 11 January 2019

Fields Medal (1966), “particularly for his work in algebraic topology, including the proof of the index theorem of Atiyah-Singer.”

Abel Prize (2004) along with Isadore M. Singer “for their discovery and proof of the index theorem, bringing together topology, geometry and analysis, and their outstanding role in building new bridges between mathematics and theoretical physics.”

Sir Michael Atiyah spent his childhood in Sudan, where his grandfather, a physician, had emigrated from Lebanon. His father had attended Brasenose College, Oxford and was for a long time in the service of the Sudanese government. As a teenager, Sir Michael Atiyah attended the prestigious Victoria College in Cairo where he apparently learned to count in ten languages. After the family emigrated to England in 1945, he attended Manchester Grammar School, to pursue his love of maths – especially geometry. After military service, he studied at Trinity College, Cambridge (1949-55), where he received his doctorate with a thesis on “Some Applications of Topological Methods in Algebraic Geometry”, supervised by William VD Hodge. In 1955 he married Lily Brown, with whom he had three sons; the eldest one died at the age of 45. Atiyah then undertook research at the Institute for Advanced Study in Princeton (1955-56), and afterwards taught at Pembroke College, Cambridge (1957- 58). He then moved to Oxford as Reader (1961-63). In 1963 he became Savilian Professor of Geometry and Professorial Fellow at New College, Oxford. From 1969 he was a professor at the Institute for Advanced Study in Princeton. In 1972 he returned to St Catherine’s as a Royal Society Research Professor and Professorial Fellow. In addition to research and teaching, Sir Michael Atiyah was also active in science management and the management of research institutions: he was President of the Royal Society (1990-1995), Master of Trinity College, Cambridge (1990-1997) and initiated the establishment of the Isaac Newton Institute for Mathematical Sciences in Cambridge, which he headed from 1990 to 1996.

Besides the Abel Prize and Fields Medal he has received numerous other awards including the Royal Medal of the Royal Society (1968) and the Copley Medal of the 8 Royal Society (1988). He was made a Knight Bachelor (1983) and is a member of the Order of Merit (1992).

Sir Michael Atiyah enjoys gardening, hiking in the Scottish mountains, cooking, reading history and listening to classical music.

In his first decades of research, Sir Michael Atiyah worked mainly in theoretical mathematics – especially in geometry, which he had always been attracted to. But in the 1970s, theoretical physicists began to suspect that Atiyah’s ideas from geometry – specifically in topology – could possibly be relevant to string theory. The focus of his work therefore shifted naturally to physics, and he is today regarded as an important thinker in quantum field theory. However, Atiyah became famous with his mathematical proof of the index theorem, which he himself has called the best thing he has ever done. What is it? The theorem of Atiyah and Singer is built around an entire theory, which has to do with the description of the solutions of systems of differential equations, and it builds a bridge between the theory of differential equations and geometry. Differential equations are a way to describe functions indirectly. Instead of specifying the functions directly, the equations define for example how they change over time. Several such equations together form a system, and the two key issues that arise for each system are: Are there in fact functions that satisfy all the conditions described in the differential equations, and, if so, what do they look like? The functions are also called the solutions of the system, and to describe the number of solutions, each system of differential equations is given a number, known as its analytical index. An analytical index greater than zero means that it is possible to solve the system of differential equations, but calculating the analytical index is usually a difficult task.

However, for a fairly general class of systems of differential equations, one can compute a second number, based on properties of the space on which the system of differential equations lives. Atiyah and Singer showed that under certain conditions this second number is equal to the analytical index. Since these spatial properties have much to do with the structure of geometry, or the topology of the space, the second number is called the topological index – and is much easier to calculate than the analytical index. This fact became particularly important for theoretical physics. Michael Atiyah is known not only for his mathematical research, but also for his philosophical views on science. He has explored in many essays and interviews what mathematics is, what it actually means to do mathematics, and where the mathematics of the 20th and 21st centuries is leading.