Michael Francis Atiyah

Michael Francis Atiyah Edward Witten Click for larger view View full resolution 22 april 1929 · 11 january 2019 [End Page 97] Michael Francis Atiyah was one of the most influential mathematicians of the last century. His work helped reshape multiple areas of mathematics—notably analysis, geometry, and topology. He forged new and unexpected connections between different fields. Beginning in the 1970s, Atiyah also played a major role in redefining the relationship between mathematics and physics. Beyond his scientific work, Atiyah was prominent in scientific leadership and in public affairs. Among other commitments, he was president of the Royal Society of London from 1990 to 1995, president of the Pugwash Conferences on Science and World Affairs from 1997 to 2002, and president of the Royal Society of Edinburgh from 2005 to 2008. Atiyah's father, Edward Selim Atiyah, was an Anglo-Lebanese author and political activist. His mother, Jean Levens, was Scottish. Atiyah's primary school education was at the Diocesan school in Khartoum, Sudan, and he attended secondary school in Cairo and Alexandria during the World War II years. His undergraduate and graduate education were at the University of Cambridge in the United Kingdom. His doctoral thesis was written at Cambridge under the supervision of William V. D. Hodge, himself a pioneer of some of the connections between topology and analysis. Atiyah's early interests were in applications of topology to algebraic geometry—that is, he was interested in understanding the complicated spaces that can be defined by the solutions of simple equations. In the late 1950s, with Friedrich Hirzebruch, Atiyah created a new tool known as topological K-theory. To do this, they took an idea from algebraic geometry and adapted it for the more flexible purposes of topology. Atiyah went on to show that topological K-theory is a powerful tool that can be used to give simple and illuminating solutions to what previously were rather difficult problems. In its original context of algebraic geometry, K-theory had been invented by Alexander Grothendieck, who had used it as the foundation for a vast generalization of a celebrated 19th-century theorem of Bernhard Riemann and Gustav Roch. Atiyah came to suspect that there should exist an analog in topology of the Grothendieck-Riemann-Roch theorem of algebraic geometry. For several years he searched for such an analog. This search was eventually crowned with success in the Atiyah-Singer index theorem, originally proved in 1963 by Atiyah with Isadore Singer. This was Atiyah's single most important work. Roughly, the index theorem predicts the number of independent solutions of a differential equation in terms of the shape, or more precisely the topology, of the space on which the equation is defined. It links multiple areas of mathematics and even physics in a most [End Page 98] remarkable way. The statement and proof involve calculus and differential equations. An important input comes from physics, because a key discovery of Atiyah and Singer in formulating and proving their theorem was that the Dirac equation of relativistic electron theory plays a central role. And the implications of the theorem are widely felt in different areas of mathematics, especially geometry and topology. In the following decade, Atiyah developed many extensions and applications of the index theorem, in work with Graeme Segal, Raoul Bott, and Vijay Patodi, among others. His papers from this period opened up many new fields of research, with repercussions in multiple directions. In the mid-1970s, Atiyah developed a serious interest in physics, and thereafter he became an influential figure in theoretical physics. What set the stage for Atiyah's interest in physics was the emergence by the mid-1970s of the Standard Model of particle physics. Physicists in the 1950s and 1960s had been preoccupied with problems of quantum theory that seemed far removed from the concerns of modern mathematics. I, for example, as a physics graduate student of the 1970s, did not learn about the Atiyah-Singer index theorem or any other topic in modern mathematics. Conversely, mathematicians of the period were generally not in close touch with developments in physics. For instance, Atiyah and Singer, when they needed the Dirac equation, had to reinvent it, as Atiyah recounted...