Remembering Michael Atiyah

Sir Michael Atiyah, a giant in mathematics, sadly passed away on 11 January, 2019. Another very distinguished mathematician, Professor M.S. Narasimhan, who has been closely associated with ICTP since its early days in many capacities and is now on our Scientific Council, wrote these reminiscences of Michael Atiyah at our request. Narasimhan had a close mathematical relationship with Atiyah that spanned several decades.

Sir Michael Atiyah was a great mathematician who made path-breaking contributions to various fields in mathematics. He and his work played a crucial role in promoting the resurgence of collaboration between mathematicians and theoretical physicists in the latter half of the twentieth century.

Born on 22 April 1929, Atiyah grew up in Sudan and Egypt, while his father was from Lebanon. His undergraduate and postgraduate studies were at Trinity College, Cambridge, and he was a doctoral student of W.V.D. Hodge, graduating in 1955. He spent most of his academic career in Cambridge, Oxford and Princeton.

He was a truly wonderful and lively person, with an effervescent and inspiring personality. He valued collaboration in Mathematics and loved intense mathematical discussions and exchanges. He seemed to derive many of his insights and ideas in the process of talking.

His many insights made him a distinguished geometer, a Fields Medalist and winner of the Abel Prize, whose work creatively combined Analysis, Differential Geometry and Topology. His major contributions were to Topological K-Theory, Index Theorem for linear elliptic operators, and mathematical aspects of gauge theory arising in theoretical physics.

He developed Topological K-Theory together with Friedrich Hirzebruch, which associates to a topological space X, a topological invariant, an abelian group K(X), using vector bundles on X, in a way reminiscent of associating homology groups of X. K-Theory proved to be a remarkably powerful tool in topology, helping to solve some outstanding problems in the area and giving simpler proofs to some known results.

His best-known result is the celebrated Atiyah-Singer Index Theorem, which is a vast generalisation of famous results like the Riemann-Roch theorem and intimately related to K-theory. Given a linear elliptic operator between sections of two vector bundles on a compact smooth manifold, the index of the operator is defined to be the difference of the dimensions of the space of solutions of the operator and that of its adjoint. The index theorem gives a formula for the index of the operator in terms of topological invariants of the manifold and the vector bundles. A major related work is the Atiyah-Patodi-Singer index theorem, which deals with elliptic operators on manifolds with boundary and non-local boundary conditions.

Motivated by insights from theoretical physics, Atiyah worked extensively on gauge theory as well. From a mathematical point of view, gauge theory is the study of connections on a principal bundle, the related Yang-Mills equation satisfied by connections, a non-linear elliptic analogue of the Maxwell equations, and the geometry of the space of solutions of the Yang-Mills equation. Atiyah and Bott made an extensive study of the Yang-Mills equation on compact Riemann surfaces and its relation to the moduli spaces of holomorphic vector bundles on the Riemann surface. This work was very influential and popularised these moduli spaces among physicists.

I was fortunate enough to know Professor Atiyah well; it was easy to interact with him. I had close and fruitful contact with him over the years, and had several conversations with him on moduli problems and some aspects of the Index theorem.

My memories of my mathematical interactions with him include being asked by Atiyah to give a talk in his seminar on a result by Kotake and myself on fractional powers of linear elliptic operators, which the original proof of the Atiyah-Bott fixed point theorem had utilized.

At that time I also drew his attention to a not so well-known paper of Minakshisundaram (published in the Journal of the Indian Mathematical Society) in which he gives a short and elegant proof of the asymptotic expansion of the heat kernel on a Riemannian manifold. Atiyah immediately went to the library, read the paper and then said to me: "It is short and to the point," the best compliment a mathematical paper can receive!

On another occasion, during an Arbeitstagung in Bonn, I told him about the canonical filtration which Harder and I had introduced on vector bundles on curves and mentioned that it could be of use in the work on Yang-Mills equation on Riemann surfaces he was undertaking at that time with Bott.

Moreover, many of my best students did their post-doctoral study with Atiyah, the best-known example being Vijay K. Patodi. They were all grateful to him for his kindness, inspiration and mentorship. He was always available whenever they wanted to discuss with him. Atiyah and I later edited the Collected Papers of Patodi, and in the Foreword Atiyah recalls some aspects of his collaboration with Patodi. The work of the Atiyah-Patodi-Singer trio, he said, “was a great collaboration, exploiting the different talents of the participants.”

In addition to his monumental research output, he also contributed to advancement of science through several important administrative positions he held. He was the President of the Royal Society between 1990 and 1995, and was involved in the creation of the Isaac Newton Institute for Mathematical Sciences in Cambridge, acting as its first Director. He was master of Trinity College, Cambridge, as well as Chancellor of the University of Leicester. In addition to these important positions, he was a member of the Executive Committee of the International Mathematical Union (IMU), and President of the Pugwash Conferences on Science and world affairs. He also contributed to the formation of the Inter-Academy Panel on International Issues and European Mathematical Society.

Although Atiyah had no formal association with the mathematics department of ICTP, he was aware and appreciative of ICTP's important contributions to the advancement of mathematics in developing countries. For several years, starting from 1994, the mathematics prize of ICTP was named each year after a different eminent mathematician, before it was designated the Ramanujan Prize as it is known today. The first one, in 1994, was named after Atiyah, in view of his stature as a great mathematician and as a motivating figure with roots in the developing world. When I sought his concurrence for the Prize to be named after him, I had long discussions with him about the situation of mathematical research in several developing countries, as well as about the concrete actions ICTP takes in promoting mathematical research in these countries. He believed that ICTP is doing important and commendable work towards advancing the mathematical expertise in the developing world. We also talked in general about the problems and potentialities of scientific research in developing countries and the role of international cooperation in science capacity building.

Atiyah also attended the "Colloquium on Geometry" held at ICTP in 2002, which was held in my honour, and gave a talk on "A look at the history of moduli spaces." He was kind enough to refer to our long association and friendship.   

Atiyah served as a role model for aspiring mathematicians growing up in developing countries, and also directly mentored young mathematicians from some of these countries. His passing is a great loss for the mathematical community.

------ M.S. Narasimhan