Sir Michael Francis Atiyah was awarded a Fields Medal, the highest honour in mathematics, at the International Congress at Moscow for his seminal work on developing K-Theory. He was just 37 years old.

There is no Nobel prize in mathematics because Alfred Nobel considered physics, and to a lesser extent chemistry, the foremost of all the sciences. The 20^th century however demonstrated just how valuable mathematical tools could be in helping scientists understand the fundamental structure of matter. K-Theory and Atiyah’s later work in collaboration with others on index theorems for example would act as a catalyst for an extraordinarily fruitful interaction between the static shapes of mathematicians and the dynamic flows of theoretical physicists.

Atiyah, born in 1929, was a British-Lebanese mathematician who originally specialised in geometry. At Cambridge he studied topology – the interplay between straight line phenomena and their more complicated counterparts which curve, bend, twist or come back on themselves. Topology is the mathematical study of the properties that are preserved through these contortions. The study of what happens on a larger scale, when you try and go around and back on yourself like Christopher Columbus. In fact, K-Theory has its roots in our navigation of the Earth.

Scientific progress is in general made with incremental advances through partial theories that describe a limited range of happenings. K-Theory is a topologic technique which brought together ideas from algebraic geometry, linear algebra and number theory. It applies ‘vector bundles’ – that are pieces of information (more specifically, moving families of curved lines) that permit an intermediate notion of how to get hold of a curved phenomenon. The application of K-Theory permits the study of their flat regions and then pieces them all together.

The maps human beings used to explore the Earth can also be used to explore both the large-scale universe, going out into space, and the small-scale universe, studying atoms and molecules. What Atiyah was doing was trying to unify all that, and he believed K-Theory was the natural way to do it. Indeed K-Theory has led to the solution of many significant mathematical problems.

Atiyah was himself inspired by the work of Frenchman, Alexander Grothendieck, a leading figure in modern algebraic geometry, whose earlier ideas contributed substantially to the development of K-Theory. Although not by training an algebraic topologist, Atiyah began to involve himself more and more in the subject, believing that a number of problems in topology would become much easier to prove if one could produce a coherent theory based on topological vector bundles. The K-Theory theorem was established by Atiyah in close collaboration with Fredrich Hirzebruch, a mathematician also working in the fields of topology and algebraic geometry, considered the most important German mathematician of the Post-War period.

Atiyah’s subsequent collaboration with American mathematician, Isadore Manuel Singer, built on K-Theory to establish an analytical tool called Index Theory, now considered to be one of the most far-reaching theorems ever.

Index Theory occupied Atiyah and Singer for over 20 years. At its simplest, this was a form of topology used to predict how many solutions a linear equation may have without necessarily knowing any or all the individual solutions. This valuable insight provides a short-cut to getting to know whether such solutions exist or not. Their most notable and far-reaching mathematical achievement, the Atiyah-Singer Index Theorem, won its authors the Abel Prize (a generous financial award made annually by the King of Norway) in 2004 for outstanding scientific work in the field of mathematics.

Today, scientists describe the universe in terms of two basic partial theories – the general theory of relativity which describes the force of gravity and the large-scale structure of the universe and quantum mechanics which deals with phenomena on extremely small scales. Unfortunately, both theories are known to be inconsistent with each other. The key challenge of contemporary physics is to find a universal theory that will incorporate them both – ‘a quantum theory of gravity’ according to Professor Stephen Hawking in his abridged A Briefer History of Time ¹.

The Atiyah-Singer index theorem was key to the exchange of ideas between mathematics and theoretical physics in the 1980s and 1990s. It is now considered one of the most fundamental theorems that can be used to help explain our universe. Mathematical methods derived from the index theorem were used in physics to help develop String Theory, mankind’s most promising attempt yet to find a common explanation for gravitation illustrating how the universe is made.

With a career of such extraordinary breadth and depth, Atiyah has been called the greatest English mathematician since Isaac Newton. He is perhaps more aptly described as a ‘matchmaker’ ², with an intuition for arranging just the right intellectual liaisons over the course of the last half century, helping to bridge gaps between apparently disparate ideas within the fields of mathematics as well as between mathematics and physics. Indeed, it is that interconnection which most intrigued him. “One problem may have a dozen different ways of being looked at in different subjects, a bit of algebra, a bit of geometry, a bit of topology. It’s the interaction and bridges that interest me” ³.

Convivial with a personal charm not always compatible with the abnormality of genius, Atiyah was naturally attuned to the cross-pollination and exploration of ideas without ever losing himself in them. Yet to succeed in cracking open the locked problems of mathematics takes a degree of obsession that may have unhinged some of his contemporaries. The lives of some other mathematical geniuses have seemed to arc from mechanistic to mentalistic obsessions or led to cognitive disinhibition. Examples have included Kurt Godel (one of the most significant logicians in history who had an obsessive fear of being poisoned and eventually starved himself to death in 1977), Paul Edros, (the most prolific producer of mathematical conjectures of the 20^th century whose belongings would fit in a suitcase. Edros was essentially homeless, showing up at people’s houses to “do math”), the aforementioned fellow pioneer Alexander Grothendieck (a leading figure in the creation of modern algebraic geometry, who refused to attend the Fields Medal ceremony in 1966 for political reasons and who in 1991 would live in seclusion in a remote village at the foot of the Pyrenees while writing thousands and thousands of pages of text on spirituality and a “coming day of reckoning” ⁴), and most recently Gregori Perelman (the famed mathematician who turned down the 2006 Fields Prize and in 2010 the $1 million Millennium Prize for the resolution of the famous Poincare conjecture despite being unemployed and living in poverty in Russia).

Atiyah, by fortunate contrast, was grounded, sociable and naturally at ease with the world throughout his career. Happily married for over 60 years to his Scottish wife Lily who bore him three sons, he employed freewheeling conversation as a powerful instrument for creating mathematical alliances. Becoming a well-regarded British establishment figure in later life, Sir Michael, as he became, never forgot his paternal roots in the Middle East.

Following his initial success, 1966 became the first year the Fields Medal was designated for the most outstanding mathematicians under the age of 40. It has been ever since.