The most important mathematician you’ve (probably) never heard of

Ask someone to name the most important physicist of the 20^th century, and they will almost certainly say Albert Einstein. Ask the same question about the field of mathematics, however, and you will probably be met by blank stares – which is why I am going to introduce you to Alexander Grothendieck.

Einstein, as both the inventor of the theory of relativity and a key figure in developing quantum mechanics, had a huge influence on physics, and he transcended science to become a genuine global celebrity. Grothendieck arguably played a similar role in transforming mathematics, but he disappeared from academic circles and then society at large before he died, leaving his legacy written only in his revolutionary work.

Before that, both Grothendieck’s work and his personality made him a hard sell as a public figure, compared to the showboating Einstein. Sure, Einstein’s ideas about the nature of space, time and the universe were incredibly complex, not least when expressed mathematically, but he had a knack for storytelling that made his work accessible. Examples like the twin paradox, in which an astronaut travelling at a high speed finds their twin has aged more than they have on return, are a great way into understanding relativity.

By contrast, even describing what Grothendieck got up to requires diving into a mess of abstract and unfamiliar concepts. I’ll do my best to explain some of this, but really I can only ever manage to give a surface impression.

Let’s start at the top level. Grothendieck is most famous among mathematicians for redefining the foundations of algebraic geometry. Very broadly, this is a field concerned with the relationship between algebraic equations and shapes. For example, the values of the equation x² + y² = 1, when plotted on a graph, form a circle of radius 1.

One of the first people to really start formalising this relationship between algebra and geometry was the 17th century philosopher René Descartes, whose Cartesian coordinates we still use to plot equations on graphs today. But this relationship can go much deeper. Mathematicians love to generalise, in the sense of taking an idea and stretching it to be as broad as possible, making connections that weren’t obvious before. Grothendieck was a master at this – indeed, a book about his life and work described “the search for maximum generality” as part of his personal mathematical signature.

Referring back to the circle equation above, the set of points that solve the equation and make up the circle are what mathematicians call an “algebraic variety”. An algebraic variety doesn’t have to be a set of points on the Cartesian plane. It can also be points in 3D space (those that make up a sphere, for example) or even higher dimensions.

This wasn’t enough for Grothendieck. For example, take the equations x² = 0 and x = 0. Both have a single solution, setting x to 0, meaning their set of points – their algebraic varieties – are the same. And yet, clearly the equations are different, so something is getting lost here. In 1960, as part of his search for maximum generality, Grothendieck introduced the concept of a “scheme”, which set out to capture this extra information.

How does it work? Here we need yet another concept: a ring. Confusingly, this has nothing to do with the shape of the circle we’ve been talking about. Instead, what mathematicians call a “ring” is a collection of objects that, when added or multiplied, remain within that collection – in a sense they are enclosed, or circle back on themselves, just like a ring, though the name is only a loose metaphor.

The simplest example of a ring is the integers: all negative whole numbers, all positive whole numbers and zero. No matter how you add or multiply an integer, you will always end up with an integer. Another essential property of a ring is it has a “multiplicative identity”, meaning it is an object that, when multiplied by another object, always produces the second object again. In the integers, this is simple – the multiplicative identity is 1, because any integer multiplied by 1 is unchanged. That also gives us a handy example of something that is not a ring – the set of just the even integers has no 1, so no multiplicative identity.

In introducing schemes, Grothendieck combined the idea of algebraic varieties with that of rings (note: I’m slightly hand waving some extra complexities here!) to encode the missing information for equations like x² = 0 and x = 0. This turned out to be incredibly powerful, because it allowed mathematicians to turn problems from a variety of subdisciplines into geometrical problems without losing any crucial details or structure, and then attack them with geometrical tools.

I’ll highlight two important problems that fell to mathematicians wielding the new sword of schemes. The first are the Weil conjectures, four statements proposed in 1949 by mathematician André Weil that concerned counting the number of solutions for particular kinds of algebraic varieties. Going back to our circle example, there are an infinite number of values that fit the equation x² + y² = 1 (you can roughly think of this as being similar to saying a circle has an infinite number of ‘sides’). But Weil was interested in varieties where only a finite number of solutions are possible. He conjectured, but couldn’t prove, an equation known as a zeta function could count these solutions.

Using schemes, Grothendieck and his colleagues were able to prove three of Weil’s conjectures in 1965, with the fourth proof coming sometime later in 1974 from Pierre Deligne, a former student of his who also used schemes. Deligne’s proof was considered to be one of the greatest results of 20^th century mathematics, answering a challenge that had puzzled mathematicians for 25 years. It also cemented just how powerful Grothendieck’s schemes could be in linking areas of mathematics, in this case number theory and geometry.

Schemes also played a vital role in cracking the infamous Fermat’s last theorem, a problem in number theory that bedevilled mathematicians for more than 350 years until it was solved by Andrew Wiles in 1995. The theorem says there are no positive integers a, b and c that satisfy the equation aⁿ + bⁿ = cⁿ if n is any integer larger than 2 (the case of 2 itself is simply Pythagoras’ theorem, you may notice). It was scribbled by 17^th century mathematician Pierre de Fermat in the margin of a mathematics book, which he said was too small to contain his proof. But Fermat almost certainly didn’t have a proof, given Wiles’ solution relied on mathematics developed much later, including by Grothendieck. For example, Wiles used the tools of algebraic geometry to translate the problem into one concerning elliptic curves, a particularly useful type of algebraic variety that can be studied with the language of schemes, but really his whole approach was inspired by the new way of thinking Grothendieck had introduced.

There is so much more of Grothendieck’s work I haven’t covered that make up the fundamental tools many mathematicians use today. For example, he generalised the idea of “space” (in a mathematical sense) to a “topos”, letting you consider not just the points within space but many extra levels of information when trying to solve a problem. He and his colleagues also wrote two immense tomes on algebraic geometry that still serve as the bible for the subject today.

So with all of this influence, why haven’t you heard of him? As I’ve perhaps demonstrated, his work takes some effort to grasp. But he also shunned the limelight for a variety of reasons. As a committed pacifist, he refused to attend the 1966 ceremony for his awarding of the Fields Medal, one of the highest honours in maths, because it was being held in Moscow and he objected to the military actions of the Soviet Union (he had similar views about US military action, it must be said). “Fertility is measured by offspring, not by honours,” he said, preferring to let his mathematical work speak for itself.

In 1970, he abandoned academia, leaving his position at the Institut des Hautes Études Scientifiques in France in protest of its military funding. Initially, he continued his mathematical work outside of formal academia, but he became increasingly isolated. In 1986 he wrote an autobiography, Harvests and Sowings, about his experience of doing mathematics and his disillusionment with the mathematical community. The following year, he produced a philosophical manuscript called The Key of Dreams, in which he described how God sent him prophetic dreams. Both texts circulated among mathematicians but have only been formally published in recent years.

The following decade, Grothendieck retreated even further from society. Living in a remote French village, having cut all ties with the mathematical community, at one point he attempted to subsist only on dandelion soup until locals intervened. It is believed he continued to write extensively on mathematics and philosophy, but none of the work was published. Indeed, in 2010 he began sending mathematicians a letter demanding that none of it should be. For all the connections he made – and made possible – in the world of mathematics, he ultimately rejected them in his personal life. He died in 2014, leaving behind an immense mathematical legacy.