One afternoon in January 2011, Hussein Mourtada leapt onto his desk and started dancing. He wasn’t alone: Some of the graduate students who shared his Paris office were there, too. But he didn’t care. The mathematician realized that he could finally confirm a sneaking suspicion he’d first had while writing his doctoral dissertation, which he’d finished a few months earlier. He’d been studying special points, called singularities, where curves cross themselves or come to sharp turns. Now he had unexpectedly found what he’d been looking for, a way to prove that these singularities had a surprisingly deep underlying structure. Hidden within that structure were mysterious mathematical statements first written down a century earlier by a young Indian mathematician named Srinivasa Ramanujan. They had come to him in a dream.

Ramanujan brings life to the myth of the self-taught genius. He grew up poor and uneducated and did much of his research while isolated in southern India, barely able to afford food. In 1912, when he was 24, he began to send a series of letters to prominent mathematicians. These were mostly ignored, but one recipient, the English mathematician G.H. Hardy, corresponded with Ramanujan for a year and eventually persuaded him to come to England, smoothing the way with the colonial bureaucracies.

It became apparent to Hardy and his colleagues that Ramanujan could sense mathematical truths — could access entire worlds — that others simply could not. (Hardy, a mathematical giant in his own right, is said to have quipped that his greatest contribution to mathematics was the discovery of Ramanujan.) Before Ramanujan died in 1920 at the age of 32, he came up with thousands of elegant and surprising results, often without proof. He was fond of saying that his equations had been bestowed on him by the gods.

More than 100 years later, mathematicians are still trying to catch up to Ramanujan’s divine genius, as his visions appear again and again in disparate corners of the world of mathematics.

Ramanujan is perhaps most famous for coming up with partition identities, equations about the different ways you can break a whole number up into smaller parts (such as 7 = 5 + 1 + 1). In the 1980s, mathematicians began to find deep and surprising connections between these equations and other areas of mathematics: in statistical mechanics and the study of phase transitions, in knot theory and string theory, in number theory and representation theory and the study of symmetries.

Most recently, they’ve appeared in Mourtada’s work on curves and surfaces that are defined by algebraic equations, an area of study called algebraic geometry. Mourtada and his collaborators have spent more than a decade trying to better understand that link, and to exploit it to uncover rafts of brand-new identities that resemble those Ramanujan wrote down.

“It turned out that these kinds of results have basically occurred in almost every branch of mathematics. That’s an amazing thing,” said Ole Warnaar of the University of Queensland in Australia. “It’s not just a happy coincidence. I don’t want to sound religious, but the mathematical god is trying to tell us something.”

New Worlds

Ramanujan’s mathematical prowess was obvious to those who knew him. Without formal training, he excelled; by the time he was in high school he had devoured advanced, though often outdated, textbooks, and was doing independent research on different kinds of numerical properties and patterns.

In 1904, he was granted a full scholarship to the Government Arts College in Kumbakonam, the small city where he had grown up, in what is now the Indian state of Tamil Nadu. But he ignored all subjects besides math and lost his scholarship within a year. He later enrolled in another university, this time in Madras (now Chennai), the provincial capital some 250 kilometers north. Again he flunked out.

He continued his research on his own for years, often while in poor health. During that time, he tutored students in math to support himself. Eventually he secured a job as a clerk at the Madras Port Trust in 1912. He pursued mathematics on the side and published some of his results in Indian journals.

Hoping to get some of his work into more prestigious and widely read publications, Ramanujan wrote letters to several British mathematicians, enclosing pages of findings for their review. “I have not trodden through the conventional regular course which is followed in a university course,” he wrote, “but I am striking out a new path for myself.” Among the recipients was Hardy, an expert in number theory and analysis at the University of Cambridge.

Hardy was shocked at what he saw. Ramanujan had identified and then solved a number of continued fractions — expressions that can be written as infinite nests of fractions within fractions, such as:

They “defeated me completely; I had never seen anything in the least like them before,” Hardy later wrote. “They must be true because, if they were not true, no one would have had the imagination to invent them.” The formulas, unproved, were so striking that they inspired Hardy to offer Ramanujan a fellowship at Cambridge. In 1914, Ramanujan arrived in England, and for the next five years he studied and collaborated with Hardy.

One of Ramanujan’s first tasks was to prove a general statement about his continued fractions. To do so, he needed to prove two other statements. But he couldn’t. Neither could Hardy, nor could any of the colleagues he reached out to.

It turned out that they didn’t need to. The statements had been proved 20 years earlier by a little-known English mathematician named L.J. Rogers. Rogers wrote poorly, and at the time the proofs were published no one paid any attention. (Rogers was content to do his research in relative obscurity, play piano, garden and apply his spare time to a variety of other pursuits.) Ramanujan uncovered this work in 1917, and the pair of statements later became known as the Rogers-Ramanujan identities.

Amid Ramanujan’s prodigious output, these statements stand out. They have carried through the decades and across nearly all of mathematics. They are the seeds that mathematicians continue to sow, growing brilliant new gardens seemingly wherever they fall.

Ramanujan fell ill and returned to India in 1919, where he died the next year. It would fall to others to explore the world his identities had revealed.

The Music of the Game

Hussein Mourtada grew up in the 1980s in Lebanon, in a small city called Baalbek. As a teenager, he didn’t like studying and preferred to play: soccer, billiards, basketball. Math, too. “It looked like a game,” he said. “And I liked playing.”

As an undergraduate at the Lebanese University in Beirut, he studied both law and mathematics, with an eye to a legal career. But he soon found that while he enjoyed the philosophical aspects of law, he did not enjoy it in practice. He turned his attention to math, where he was particularly drawn to the community. As a child, his teachers and classmates were what excited him about going to school, even though he often fell asleep during class. As a budding mathematician, “I had the impression that these are beautiful people,” he said. “They are honest. You need to be honest with yourself to be a mathematician. Otherwise, it doesn’t work.”

He moved to France for his doctorate and started to focus on algebraic geometry — the study of algebraic varieties, or shapes cut out by polynomial equations. These are equations that can be written as sums of variables raised to whole-number powers. A line, for instance, is cut out by the equation x + y = 0, a circle by x² + y² = 1, a figure eight by x⁴ = x² − y². While the line and circle are completely smooth, the figure eight has a point where it intersects itself — a singularity.

It’s easy to spot singularities when you’re dealing with shapes that you can draw on a sheet of paper. But higher-dimensional algebraic varieties are far more complicated and impossible to visualize. Algebraic geometers are in the business of understanding their singularities, too.

They’ve developed all sorts of tools to do this. One dates back to the mathematician John Nash, who in the 1960s started studying related objects called arc spaces. Nash would take a point, or singularity, and define infinitely many short trajectories — little arcs — that passed through it. By looking at all these short trajectories together, he could test how smooth his variety was at that point. “If you want to see if it’s smooth, you want to pet it,” said Gleb Pogudin of the École Polytechnique in France.

In practical terms, an arc space provides an infinite collection of polynomial equations. “This is really the thing that Mourtada is expert in: understanding the meaning of those equations,” said Bernard Teissier, a colleague of Mourtada’s at the Institute of Mathematics of Jussieu in Paris. “Because these equations can be very complicated. But they have a certain music to them. There is a lot of structure which governs the nature of these equations, and he’s just the person, I think, who best listens to this music and understands what it means.”