One afternoon in January 2011, Hussein Mourtada leapt onto his desk and begined dancing. He wasn’t alone: Some of the graduate students who separated his Paris office were there, too. But he didn’t nurture. The mathematician authenticized that he could finpartner validate a sneaking suspicion he’d first had while writing his doctoral dissertation, which he’d finished a restricted months earlier. He’d been studying distinctive points, called singularities, where curves pass themselves or come to keen turns. Now he had unanticipateedly set up what he’d been seeing for, a way to validate that these singularities had a astonishingly proset up underlying structure. Hidden wilean that structure were cryptic mathematical statements first written down a century earlier by a youthful Indian mathematician named Srinivasa Ramanujan. They had come to him in a dream.
Ramanujan conveys life to the myth of the self-taught genius. He grew up necessitatey and unteachd and did much of his research while isotardyd in southern India, nakedly able to afford food. In 1912, when he was 24, he began to sfinish a series of letters to famous mathematicians. These were mostly disseed, but one recipient, the English mathematician G.H. Hardy, correplyed with Ramanujan for a year and eventupartner swayd him to come to England, smooleang 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 spropose could not. (Hardy, a mathematical enormous in his own right, is shelp to have quipped that his wonderfulest 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 astonishing results, normally without proof. He was fond of saying that his equations had been bestowed on him by the gods.
More than 100 years tardyr, mathematicians are still trying to catch up to Ramanujan’s divine genius, as his visions materialize aget and aget in disparate corners of the world of mathematics.
Ramanujan is perhaps most honord for coming up with partition identities, equations about the contrastent ways you can shatter a whole number up into minusculeer parts (such as 7 = 5 + 1 + 1). In the 1980s, mathematicians began to discover proset up and astonishing joinions 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 recurrentation theory and the study of symmetries.
Most recently, they’ve materializeed in Mourtada’s labor on curves and surfaces that are clear upd 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 join, and to utilize it to uncover rafts of brand-new identities that mimic those Ramanujan wrote down.
“It turned out that these charitables of results have basicpartner occurred in almost every branch of mathematics. That’s an amazing leang,” shelp Ole Warnaar of the University of Queensland in Australia. “It’s not equitable a satisfyed coincidence. I don’t want to sound religious, but the mathematical god is trying to tell us someleang.”
New Worlds
Ramanujan’s mathematical prowess was clear to those who knew him. Without createal training, he excelled; by the time he was in high school he had devoured progressd, though normally outdated, textbooks, and was doing self-reliant research on contrastent charitables of numerical properties and patterns.
In 1904, he was granted a filled scholarship to the Government Arts College in Kumbakonam, the minuscule city where he had grown up, in what is now the Indian state of Tamil Nadu. But he disseed all subjects besides math and lost his scholarship wilean a year. He tardyr enrolled in another university, this time in Madras (now Chennai), the provincial capital some 250 kilometers north. Aget he flunked out.
After flunking out of college, Ramanujan ran away from home, prompting his mother to post a missing-person see in The Hindu.
He progressd his research on his own for years, normally while in necessitatey health. During that time, he tutored students in math to help himself. Eventupartner he safed a job as a clerk at the Madras Port Trust in 1912. He pursued mathematics on the side and unveiled some of his results in Indian journals.
Hoping to get some of his labor into more prestigious and expansively read accessibleations, Ramanujan wrote letters to cut offal British mathematicians, enclosing pages of discoverings for their verify. “I have not trodden thcdisesteemful the traditional normal course which is pursueed 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.
Ramanujan’s first letter to G.H. Hardy joind createulas (5), (6) and (7), strange nested fractions that Hardy shelp “lossed me finishly; I had never seen anyleang in the least appreciate them before.”
Hardy was shocked at what he saw. Ramanujan had identified and then settled a number of progressd fractions — conveyions that can be written as infinite nests of fractions wilean fractions, such as:
They “lossed me finishly; I had never seen anyleang in the least appreciate them before,” Hardy tardyr wrote. “They must be real becaemploy, if they were not real, no one would have had the imagination to invent them.” The createulas, unvalidated, were so striking that they backd Hardy to propose Ramanujan a fellowship at Cambridge. In 1914, Ramanujan reachd in England, and for the next five years he studied and collaborated with Hardy.
One of Ramanujan’s first tasks was to validate a vague statement about his progressd fractions. To do so, he necessitateed to validate two other statements. But he couldn’t. Neither could Hardy, nor could any of the colleagues he accomplished out to.
It turned out that they didn’t necessitate to. The statements had been validated 20 years earlier by a little-understandn English mathematician named L.J. Rogers. Rogers wrote necessitateyly, and at the time the proofs were unveiled no one phelp any attention. (Rogers was satisfyed to do his research in relative obscurity, carry out piano, garden and utilize his spare time to a variety of other pursuits.) Ramanujan uncovered this labor in 1917, and the pair of statements tardyr became understandn as the Rogers-Ramanujan identities.
Amid Ramanujan’s prodigious output, these statements stand out. They have carried thcdisesteemful the decades and apass proximately all of mathematics. They are the seeds that mathematicians progress to sow, growing luminous new gardens seemingly wherever they drop.
Ramanujan fell ill and returned to India in 1919, where he died the next year. It would drop to others to verify the world his identities had discneglected.
The Music of the Game
Hussein Mourtada grew up in the 1980s in Leprohibiton, in a minuscule city called Baalbek. As a teenager, he didn’t appreciate studying and pickred to carry out: soccer, billiards, basketball. Math, too. “It seeed appreciate a game,” he shelp. “And I appreciated carry outing.”
As an undergraduate at the Leprohibitese University in Beirut, he studied both law and mathematics, with an eye to a lhorrible nurtureer. But he soon set up that while he endelighted the philosophical aspects of law, he did not endelight it in train. 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 normally fell asleep during class. As a budding mathematician, “I had the astonishion that these are pretty people,” he shelp. “They are truthful. You necessitate to be truthful with yourself to be a mathematician. Otherteachd, it doesn’t labor.”
Hussein Mourtada has been conveying Ramanujan’s labor into the 21st century.
He shiftd to France for his doctorate and begined to concentrate 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 liftd 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 finishly brittle, the figure eight has a point where it intersects itself — a singularity.
It’s effortless to spot singularities when you’re dealing with shapes that you can draw on a sheet of paper. But higher-unwiseensional algebraic varieties are far more complicated and impossible to envision. Algebraic geometers are in the business of comfervent their singularities, too.
They’ve broadened all sorts of tools to do this. One dates back to the mathematician John Nash, who in the 1960s begined studying roverdelighted objects called arc spaces. Nash would consent a point, or singularity, and clear up infinitely many low trajectories — little arcs — that passed thcdisesteemful it. By seeing at all these low trajectories together, he could test how brittle his variety was at that point. “If you want to see if it’s brittle, you want to pet it,” shelp Gleb Pogudin of the École Polytechnique in France.
In pragmatic terms, an arc space supplys an infinite accumulateion of polynomial equations. “This is repartner the leang that Mourtada is expert in: comfervent the uncomferventing of those equations,” shelp Bernard Teissier, a colleague of Mourtada’s at the Institute of Mathematics of Jussieu in Paris. “Becaemploy these equations can be very complicated. But they have a certain music to them. There is a lot of structure which administers the nature of these equations, and he’s equitable the person, I leank, who best hears to this music and understands what it uncomfervents.”