Michael Atiyah

Michael Atiyah
Born	Michael Francis Atiyah 22 4, 1929
Birthplace	Hampstead, London, England
Died	Template:Death date and age Edinburgh, Scotland
Nationality	British
Occupation	Mathematician
Known for	Atiyah–Singer index theorem, topological K-theory
Education	PhD, University of Cambridge (1955)
Awards	Fields Medal (1966), Abel Prize (2004), Order of Merit (1992)

Michael Francis Atiyah (22 April 1929 – 11 January 2019) was a British-Lebanese mathematician whose work reshaped the landscape of modern geometry and topology, forging connections between pure mathematics and theoretical physics that had not been seen since the era of Isaac Newton. Born in London to a Lebanese father and a Scottish mother, Atiyah spent much of his career at the University of Cambridge, the University of Oxford, and the University of Edinburgh, becoming one of the most influential mathematicians of the twentieth century. He is best known for the Atiyah–Singer index theorem, proved in 1963 with Isadore Singer, which established a deep relationship between geometry, topology, and analysis and had far-reaching implications across mathematics and physics. Atiyah was also a co-founder of topological K-theory, a branch of algebraic topology that became a central tool in modern mathematics. He received the Fields Medal in 1966 for his work in topology and the Abel Prize in 2004 for his lifetime contributions, making him one of only a handful of mathematicians to have received both honours. He served as President of the Royal Society from 1990 to 1995, Master of Trinity College, Cambridge, and the first director of the Isaac Newton Institute for Mathematical Sciences. He was appointed to the Order of Merit in 1992 and was knighted in 1983.

Early Life

Michael Francis Atiyah was born on 22 April 1929 in Hampstead, London. His father, Edward Selim Atiyah, was a Lebanese writer and academic who had studied at the University of Oxford, while his mother, Jean Atiyah (née Levens), was Scottish. The family's multicultural background — spanning the Middle East and Britain — gave Atiyah a cosmopolitan upbringing and an international outlook that remained with him throughout his life. His father, who was of Antiochian Orthodox Christian background, had established himself in Britain as an author and commentator on Arab affairs.

Atiyah spent parts of his childhood in the Sudan and in Egypt, where his father held various positions, before the family settled in England. He attended schools in the Middle East before completing his secondary education in England. Even as a young student, Atiyah demonstrated a strong aptitude for mathematics. He later described how the subject attracted him through its combination of clarity and surprise, and he found himself drawn to geometry in particular from an early age.

His brother, Patrick Atiyah, went on to become a distinguished legal scholar and professor of English law at Oxford. The family environment, shaped by intellectual achievement and cross-cultural experience, proved formative in developing Atiyah's broad curiosity and his ability to draw connections between seemingly disparate fields — a quality that would define his mathematical career.

Education

Atiyah undertook his undergraduate and graduate studies at the University of Cambridge, where he became a student at Trinity College. For his doctoral work, he studied under W. V. D. Hodge, one of the leading geometers of the period, whose own contributions to algebraic geometry and the theory of harmonic integrals exerted a strong influence on Atiyah's mathematical development. He completed his PhD in 1955 with a thesis entitled "Some Applications of Topological Methods in Algebraic Geometry," which already signalled his interest in the interplay between topology and geometry that would become the defining theme of his research.

During his time at Cambridge, Atiyah immersed himself in the rich mathematical culture of the university, interacting with many of the figures who shaped British mathematics in the post-war era. The training he received under Hodge provided him with a deep grounding in the geometric traditions of the Cambridge school, while also exposing him to the emerging continental European approaches to algebraic geometry and topology.

Career

Early Academic Career and K-Theory

After completing his doctorate, Atiyah held positions at the Institute for Advanced Study in Princeton, New Jersey, and at the University of Cambridge. During the late 1950s and early 1960s, he began developing the mathematical framework that would become topological K-theory, working in collaboration with Friedrich Hirzebruch. K-theory provided a new algebraic approach to classifying vector bundles over topological spaces, and it rapidly became one of the central tools in algebraic topology. The theory drew on ideas from algebraic geometry and functional analysis, exemplifying Atiyah's ability to synthesise methods from different branches of mathematics into a unified framework.

Topological K-theory proved to have applications far beyond its original setting. It played a role in the solution of classical problems in topology and became a precursor to developments in algebraic K-theory and, much later, in string theory and other areas of theoretical physics.

The Atiyah–Singer Index Theorem

The achievement for which Atiyah is most celebrated is the Atiyah–Singer index theorem, proved in 1963 in collaboration with the American mathematician Isadore Singer of the Massachusetts Institute of Technology. The theorem established a profound relationship between analysis, topology, and geometry by showing that the analytical index of an elliptic differential operator on a compact manifold could be computed from topological data alone. In essence, it connected the number of solutions of a system of differential equations to the shape and structure of the underlying space on which the equations are defined.

The index theorem unified several important earlier results, including the Gauss–Bonnet theorem, the Hirzebruch–Riemann–Roch theorem, and the Hirzebruch signature theorem, revealing them all to be special cases of a single, overarching principle. This insight demonstrated that problems previously studied separately in different areas of mathematics were in fact manifestations of the same underlying phenomenon.

The proof itself went through several refinements and generalisations over the subsequent decades. Atiyah and Singer, along with other collaborators, developed extensions of the theorem using heat equation methods and K-theoretic techniques. Each new proof illuminated different aspects of the result and extended its range of applicability. The Atiyah–Singer index theorem became one of the most important results of twentieth-century mathematics, influencing fields ranging from differential geometry and algebraic topology to number theory and quantum field theory.

When Atiyah was awarded the Fields Medal in 1966, the citation highlighted his work on K-theory and the index theorem as having opened new vistas in mathematics. The index theorem also served as a bridge between mathematics and physics: it provided essential mathematical tools for understanding gauge theories, anomalies in quantum field theory, and the geometry of moduli spaces — all topics that became central to theoretical physics from the 1970s onward.

Connections Between Mathematics and Physics

From the mid-1970s onward, Atiyah became increasingly interested in the connections between geometry and theoretical physics, particularly quantum field theory and gauge theory. He played a central role in bringing together mathematicians and physicists at a time when the two communities had been largely working in isolation from one another. His work on instantons — solutions to the Yang–Mills equations in four dimensions — drew on ideas from algebraic geometry and topology, and contributed to the development of what became known as mathematical physics in its modern form.

Atiyah's collaborations during this period were wide-ranging. He worked with physicists such as Edward Witten, who was a notable student influenced by Atiyah's ideas, and the interaction between the two proved mutually enriching. Witten's work on topological quantum field theory, which earned him a Fields Medal in 1990, was deeply influenced by conversations with Atiyah and by the mathematical structures that Atiyah had helped to develop.

Atiyah himself described mathematics and physics as engaged in a kind of dialogue, with each discipline posing problems and providing insights that enriched the other. In an interview with the Simons Foundation, he spoke about how the index theorem, which had been conceived as a result in pure mathematics, found unexpected applications in physics, and how physical intuition in turn suggested new mathematical directions. He maintained this perspective throughout his career, arguing that the most fruitful mathematics often arose at the boundaries between established fields.

Leadership and Institutional Roles

Atiyah held a succession of distinguished positions in British academic life. He was the Savilian Professor of Geometry at the University of Oxford from 1963 to 1969, and subsequently held other chairs at Oxford and elsewhere. In 1990, he became Master of Trinity College, Cambridge, one of the most prestigious academic positions in Britain, a role he held until 1997.

He was the founding director of the Isaac Newton Institute for Mathematical Sciences in Cambridge, which was established in 1992 as an international research centre for the mathematical sciences. The institute became a leading venue for collaborative research programmes and workshops, and Atiyah's leadership helped to establish its reputation.

From 1990 to 1995, Atiyah served as President of the Royal Society, the United Kingdom's national academy of sciences. During his tenure, he advocated for the importance of fundamental research and for the role of science in public life. He was first elected a Fellow of the Royal Society in 1962, at the age of 32.

Later in his career, Atiyah moved to the University of Edinburgh, where he held an honorary professorship and continued active research well into his eighties. He was also associated with the International Centre for Mathematical Sciences in Edinburgh, contributing to its development as a hub for mathematical research.

Doctoral Students and Mathematical Influence

Atiyah supervised a large number of doctoral students, many of whom became leading mathematicians in their own right. His students included Simon Donaldson, who received the Fields Medal in 1986 for his work on the topology of four-manifolds; Nigel Hitchin, who made foundational contributions to differential geometry; Frances Kirwan, who became a leading figure in algebraic geometry; Graeme Segal, known for his work in topology and mathematical physics; George Lusztig, a major figure in representation theory; Lisa Jeffrey, who contributed to symplectic geometry and mathematical physics; Peter Kronheimer, known for his work in gauge theory; and Ruth Lawrence, a notable figure in knot theory and quantum topology.

This roster of students illustrates the breadth of Atiyah's mathematical interests and his capacity to inspire research across a wide spectrum of topics. His influence extended beyond formal supervision: through lectures, collaborations, and personal interactions, he shaped the thinking of an entire generation of mathematicians and physicists.

Later Work and the Riemann Hypothesis Claim

Atiyah remained mathematically active into his late eighties. In 2016, at the age of 86, he spoke about continuing to work on fundamental questions linking quantum mechanics and gravitation, expressing a desire to find a mathematical framework that could unify the two.

In September 2018, at the age of 89, Atiyah attracted worldwide attention when he presented a lecture at the Heidelberg Laureate Forum in which he claimed to have proved the Riemann hypothesis, one of the most famous unsolved problems in mathematics and one of the seven Millennium Prize Problems carrying a prize of one million dollars. The claim was met with considerable scepticism by the mathematical community, and the proof was not accepted by experts as valid. Nevertheless, the episode reflected Atiyah's enduring ambition and willingness to engage with the deepest questions in mathematics, even in the final months of his life.

Personal Life

Atiyah married Lily Brown in 1955; she was a mathematician who had studied at the University of Edinburgh. The couple had three sons together. One of their sons, also named Michael, predeceased him.

Atiyah was known for his engaging personality, his breadth of interests beyond mathematics, and his gift for communication. Colleagues and students frequently described his ability to see unexpected connections and to convey complex mathematical ideas with clarity and enthusiasm. In his interview with Quanta Magazine in 2016, he spoke about the role of imagination in mathematics and described how he often began with an intuitive sense of what should be true before working out the formal details.

He lived for many years in Edinburgh with his wife and remained active in mathematical life until shortly before his death. Atiyah died on 11 January 2019 in Edinburgh, Scotland, at the age of 89.

Recognition

Atiyah received an exceptional number of honours and awards over the course of his career. He was awarded the Fields Medal in 1966, the highest honour for a mathematician under the age of 40, for his contributions to topology and K-theory. In 2004, he shared the inaugural Abel Prize with Isadore Singer for their work on the index theorem; the Abel Prize is awarded by the Norwegian Academy of Science and Letters and is considered one of the most prestigious prizes in mathematics.

He was elected a Fellow of the Royal Society in 1962 and served as its President from 1990 to 1995. He was knighted in 1983 and appointed to the Order of Merit in 1992, an honour limited to 24 living members and in the personal gift of the monarch.

Atiyah received the Royal Medal (1968) and the Copley Medal (1988) from the Royal Society, the latter being the Society's oldest and most prestigious award. He also received the De Morgan Medal from the London Mathematical Society and numerous honorary degrees from universities around the world.

The Michael Atiyah Building at the University of Leicester was named in his honour, recognising his contributions to mathematics and to British academic life.

He held memberships in numerous national academies of science, including those of the United States, France, Sweden, and India, among others.

Legacy

Atiyah's mathematical legacy is defined by the scope and depth of his influence across multiple areas of mathematics and theoretical physics. The Atiyah–Singer index theorem remains one of the central results of modern mathematics, with applications continuing to emerge in geometry, topology, number theory, and quantum physics. Topological K-theory, which he co-developed, became a foundational tool in algebraic topology and influenced the development of algebraic K-theory and related areas.

His work in bringing together mathematics and physics had a transformative effect on both disciplines. The dialogue he fostered between mathematicians and theoretical physicists during the 1970s and 1980s led to the emergence of new fields of research, including topological quantum field theory and the mathematical study of gauge theories, string theory, and related areas. His influence on Edward Witten and other physicists contributed to a period of unprecedented cross-fertilisation between the two subjects.

As a teacher and mentor, Atiyah's impact was multiplied through the careers of his doctoral students, many of whom — including Simon Donaldson, Nigel Hitchin, Frances Kirwan, Graeme Segal, and George Lusztig — became leaders in their respective fields. His approach to mathematics, which emphasised the importance of ideas and connections over technical machinery, set a style that influenced a generation of researchers.

In a tribute published after his death, the Royal Society described Atiyah as "one of the greatest mathematicians of the last century" and acknowledged his contributions not only to research but also to scientific leadership and public engagement with mathematics. A tribute in Nature noted that his career exemplified "an era in which the boundaries between pure mathematics and theoretical physics dissolved," and that his work "changed how mathematicians and physicists think about the world."

The CERN Courier described him as "one of the giants of mathematics whose work influenced an enormous range of subjects." His contributions to institutional life — as Master of Trinity College, founding director of the Isaac Newton Institute, and President of the Royal Society — ensured that his influence extended well beyond his own research, shaping the infrastructure and culture of mathematical science in the United Kingdom and internationally.