The last time I met Michael Atiyah, who has died aged 89, was at Tate Modern in London; not the most likely place to run into probably Britain’s greatest mathematician since Isaac Newton, but entirely consistent with his wide-ranging enthusiasm for his subject. It was June 2012, and I joined him and the flamboyant French mathematician Cédric Villani in a panel discussion: Mathematics, a Beautiful Elsewhere. The title says it all.

We have sulphuric acid to thank for Atiyah’s decision to become a mathematician. Early in 1940, as Britain and France fought over his homeland of Lebanon, his parents sent him to Victoria college in Cairo. In a 1984 interview he said that while there he got very interested in chemistry, but eventually decided that making “sulphuric acid and all that sort of stuff” was not for him: “Lists of facts, just facts ...” From that time on, mathematics became his passion. “I never seriously considered doing anything else.” Atiyah’s work was to have a profound influence on today’s mathematics.

Atiyah was a geometer, in the sense of visual thinking allied to abstract symbolism, a new attitude that swept through mathematics in the middle of the 20th century. You thought about it like geometry but wrote about it like algebra, and very esoteric algebra at that. His research divides into four main periods, to some extent overlapping – in the 1950s, algebraic geometry; in the 60s and early 70s, K-theory; the 60s to 80s, index theory; and the late 70s to mid-80s, gauge theory, where his ideas became extremely influential in quantum physics.

Algebraic geometry originally developed from a deep link between geometry and algebra promoted in the 1600s by René Descartes. Start with Euclid’s plane and introduce coordinates – pairs of numbers describing the location of a point, much as latitude and longitude determine a point on the Earth’s surface. Geometric properties of curves can then be described by algebraic equations, so questions in geometry can be tackled using algebra, and vice versa.

In the late 1800s and early 1900s, a new kid appeared on the mathematical block: topology, in which geometrical shapes can be deformed as if made of elastic. Classical features such as lengths and angles lose their meaning, and are replaced by concepts such as being connected, knotted, or having a hole like a doughnut.

Topology turned out to be fundamental to many areas of mathematics. Techniques were devised to associate with a topological space various “invariants”, which reveal when spaces can or cannot be deformed into each other.

One of the most powerful invariants, homology, was established by Emmy Noether, the greatest female mathematician of the late 1800s and early 1900s. She reinterpreted, in terms of abstract algebra, rudimentary methods for counting features such as the number of holes in a surface.

In effect, Noether explained that as well as counting holes and associated structures, we can ask how they combine, and extract topological information from the answer.

Atiyah began his research career in algebraic geometry, but under the influence of his supervisor, William Hodge, at Cambridge, he quickly moved into an adjacent field, differential geometry, which studies concepts such as curvature — how a space deviates from the flat plane of Euclid. There he made big advances in the interactions between algebraic geometry, differential geometry and topology.

Euclid’s investigations of a circle includes its tangents: straight lines that touch it at one point, like a road supporting a bicycle wheel. Similarly, a sphere has a family of tangent planes, one for each point on its surface. A general family of this kind is called a vector bundle: “bundle” because the sphere ties all the planes together, and “vector” because higher-dimensional analogues of lines and planes are called vector spaces.

The topology of a vector bundle provides information about the underlying space. The tangents to a circle, for example, form a cylinder. As proof: rotate each tangent line through a right angle, out of the plane of the circle, and you get a cylinder. There is another vector bundle associated with a circle, in which the lines are twisted to form the famous Möbius band, a surface that differs topologically from a cylinder since it has only one side. Atiyah applied these ideas to “elliptic curves”, actually doughnut-shaped surfaces with interesting number-theoretic properties.

His next topic, K-theory, is a far-reaching extension of Noether’s homology invariant. A cylinder and a Möbius band are topologically distinct because their associated bundles have different twists. K-theory exploits vector bundles to capture higher dimensional analogues of such twists.

The topic underwent a period of rapid development in the 60s, stimulated by remarkable links to other major areas of mathematics, and it provided topologists with a powerful toolkit of invariants.

Atiyah, often jointly with other leading mathematicians, was a driving force behind these developments. Important themes were the cobordism theory of René Thom (how one circle splits into two as you move down a pair of trousers from the waist to the leg holes, only done for multidimensional spaces) and the periodicity theorem, first proved by Raoul Bott, showing that higher K-groups repeat in a cycle of length eight.

Index theory has its origins in the observation that topological features of a landscape, such as the numbers of mountain peaks, valleys and passes, are related to each other. To get rid of a peak by flattening it out you must also get rid of a pass, for example. The index organises such phenomena, and can be used, in suitable circumstances, to prove that a peak must exist in some region.

A landscape is a metaphor for the graph of a mathematical function, and a sweeping generalisation relates the number of solutions of a differential equation to a more esoteric topological index.

Differential equations relate rates of change of various quantities to each other, and are ubiquitous in mathematical physics; the Atiyah-Singer Index Theorem, proved jointly with the American mathematician Isadore Singer in 1963, reveals a highly significant link between a topological index and the solutions of a differential equation.

In an appropriate mathematical setting this can lead to a proof that a solution must exist, so the Atiyah-Singer index has widespread applications to physics. Forty years after their discovery, the pair were jointly awarded the Abel prize of the Norwegian Academy of Science and Letters, in 2004.

Gauge theory arose in physics, formalising certain symmetries of quantum fields and particles. The first example arose from James Clerk Maxwell’s equations for the electromagnetic field (1861), where certain mathematical transformations can be applied without changing the physics.

In 1954 Chen Ning Yang and Robert Mills extended this idea to the strong interaction, which holds together each quantum particle in the atomic nucleus. Symmetry turned out to be vital to quantum mechanics – for example, the recently discovered Higgs boson, which endows particles with mass, acts by breaking certain symmetries – and gauge symmetries have huge importance.

Atiyah contributed key ideas to their mathematics, using his index theory to study instantons (particles that wink into existence and immediately wink out again) and magnetic monopoles (particles like a north magnetic pole without any corresponding south pole).

In 1983 his PhD student Simon Donaldson used these ideas to prove a remarkable theorem: contrary to what almost all topologists expected, four-dimensional space has infinitely many distinct differentiable structures – utterly different in this respect from any other dimension. The broader context for all this work is superstring theory, a conjectured unification of quantum theory and Albert Einstein’s relativity.

Atiyah was born in London, one of four children of Edward, a Lebanese civil servant, and his wife, Jean (nee Levens), who was born in Yorkshire of Scottish descent. The family moved to Khartoum, Sudan, where Michael went to school before boarding at Victoria College in Cairo and then moving to Manchester Grammar school at 16 to prepare for Cambridge. He was always keen on mathematics. An inspiring teacher introduced him to projective geometry and William Rowan Hamilton’s algebra of quaternions, and he read about number theory and group theory – all of which clearly influenced his later mathematical interests.

In 1949, after two years of national service, he studied at Trinity College, Cambridge, remaining there for his PhD. He held positions at the Institute for Advanced Study in Princeton (including a professorship 1969-72), and at Cambridge and at Oxford, where he was Savilian professor of geometry 1963-69 and Royal Society research professor 1973-90. He became a fellow of the Royal Society in 1962, and was the society’s president from 1990 to 1995. In 1966 he won a Fields medal, the highest honour for any mathematician.

In 1990 he became master of Trinity College, Cambridge, and director of the Isaac Newton Institute for Mathematical Sciences, Cambridge. He was knighted in 1983 and made a member of the Order of Merit in 1992. After retiring from Trinity in 1997 he moved with his wife, Lily (nee Brown), whom he had married in 1955, to Edinburgh.

Atiyah was always a keen advocate of public engagement, giving popular talks on the beauty of mathematics and his lifelong passion for the subject. Small and compact, with a quiet, precise delivery, he could nevertheless hold an audience spellbound. That is how I remember him, on that day in Tate Modern, telling non-mathematicians why we do it, what it is for, and what it feels like.

He and Lily had three sons: John, David and Robin. John died in a climbing accident in 2002; Lily died last year. Michael is survived by David and Robin.

• Michael Francis Atiyah, mathematician, born 22 April 1929; died 11 January 2019