Srinivasa Ramanujan: The Prodigy of Old That's Still Winning The Race
Hussein Mourtada had long been haunted by a suspicion, one that took root during his doctoral work on special points called singularities—places where curves intersect or twist at abrupt angles. He had completed his dissertation just months before, but he couldn’t shake the feeling that these enigmatic points held secrets beyond their immediate geometry. Singularities, he believed, might conceal a hidden, profound structure.
Then, one afternoon in January 2011, in a Paris office surrounded by fellow graduate students, Mourtada finally saw it—a breakthrough so unexpected it jolted him from his chair. Without a second thought, he leapt onto his desk and started dancing. The mathematician had discovered the proof he’d been searching for, a way to reveal the hidden structure of these singularities. Even more astonishing, buried within this structure were echoes of cryptic mathematical statements first jotted down a century earlier by the legendary Indian mathematician Srinivasa Ramanujan—formulas he claimed had appeared to him in a dream.
Srinivasa Ramanujan is a lasting example of the myth of the self-taught genius, embodying brilliance and hardship. Growing up impoverished and largely uneducated, he pursued his research in isolation in southern India, often struggling to afford even basic necessities. In 1912, at the age of 24, he began reaching out to prominent mathematicians through a series of letters. Most went unanswered, but one caught the attention of G.H. Hardy, a leading English mathematician and number theorist. After corresponding with Ramanujan for nearly a year, Hardy recognized the extraordinary nature of his work and, navigating colonial bureaucracy, arranged for Ramanujan to come to England.
Once in England, Hardy and his colleagues quickly realized that Ramanujan could perceive mathematical truths in ways they could scarcely comprehend. Hardy, a significant figure in his own right, would later claim that discovering Ramanujan was his greatest contribution to mathematics. In his short life—he died in 1920 at only 32—Ramanujan developed thousands of theorems, many strikingly elegant and without formal proof, often saying that his equations were gifts from the gods.
Today, more than a century later, mathematicians continue to find the remnants of Ramanujan’s genius in unexpected realms, as his revelations surface again and again across diverse fields. His work remains a puzzling gift that defies time and explanation.
Ramanujan’s most renowned contributions may lie in his partition identities—equations that reveal the myriad ways to break down whole numbers into smaller components (like 7 = 5 + 1 + 1). But over the years, these identities have taken on a life of their own, branching far beyond simple number theory. By the 1980s, mathematicians were uncovering unexpected connections between Ramanujan’s equations and a range of other fields: statistical mechanics, the physics of phase transitions, knot theory, string theory, and representation theory. The identities seemed to crop up wherever the mathematics of symmetry and complex structures were involved.
Recently, Ramanujan’s influence has extended even further, reaching into the territory of algebraic geometry through the work of Hussein Mourtada and his collaborators. This field, focused on understanding curves and surfaces defined by algebraic equations, has revealed striking parallels to Ramanujan’s identities. For more than a decade, Mourtada and his team have been uncovering new identities that bear an uncanny resemblance to those Ramanujan penned.
“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. “It’s not just a happy coincidence. I don’t want to sound religious, but the mathematical god is trying to tell us something.”
From an early age, Srinivasa Ramanujan stood apart. Without formal training, he devoured mathematics with a rare fervor, immersing himself in advanced—though often outdated—textbooks, and experimenting with numerical properties and patterns. By high school, his aptitude was undeniable, and in 1904, he earned a full scholarship to the Government Arts College in Kumbakonam, his hometown in southern India. Yet Ramanujan’s focus was singular: he ignored every subject except mathematics. Within a year, he lost his scholarship. He later attempted another college in Madras, but again, his obsession with math proved too strong, and he left without completing his studies.
Undeterred, Ramanujan pursued mathematics on his own for years, often while struggling with poor health. To support himself, he tutored students, and in 1912, he found steady work as a clerk at the Madras Port Trust. This modest position allowed him to continue his research, and he managed to publish a few of his findings in Indian journals. Then came the moment that would change his life: Ramanujan decided to reach out to British mathematicians, sending them pages filled with equations he had uncovered. In his letter, he acknowledged his unconventional path: “I have not trodden through the conventional regular course… but I am striking out a new path for myself.”
Among the recipients was G.H. Hardy at the University of Cambridge, one of the foremost authorities in number theory. Hardy, upon seeing Ramanujan’s work, was astonished: the formulas were so unusual, so complex, that they "defeated" him, as he later recalled. “They must be true because, if they were not true, no one would have had the imagination to invent them.” Inspired, Hardy arranged for Ramanujan to join him at Cambridge.
In 1914, Ramanujan arrived in England and began a remarkable five-year collaboration with Hardy. One of his first tasks was to prove a general statement about his continued fractions—those intricate, nested expressions he had independently uncovered in Madras. To do so, he needed to prove two underlying statements. But try as he might, he couldn’t; neither could Hardy nor any of the scholars they approached for help.
As it turned out, those statements had already been proved, nearly twenty years earlier, by the little-known English mathematician L.J. Rogers. Rogers’ work had slipped into obscurity, his proofs gathering dust as he quietly tended to his piano, garden, and other pursuits. When Ramanujan unearthed Rogers’ work in 1917, these findings became known as the Rogers-Ramanujan identities.
Among Ramanujan’s staggering achievements, these identities would prove especially influential. They rippled through mathematical fields, branching into new discoveries across nearly every discipline. The identities became seeds, planted into the fertile ground of modern mathematics, and in the decades that followed, they bloomed with a brilliance that still persists today.
Unfortunately, Ramanujan’s journey ended abruptly. In 1919, he fell ill and returned to India, passing away the following year at just 32. It would fall to others to explore the worlds he glimpsed and left behind.
Hussein Mourtada’s journey into mathematics began with a love for play. Growing up in the 1980s in Baalbek, Lebanon, he was far more interested in games—soccer, billiards, basketball—than in studying. Mathematics, too, held a certain allure. “It looked like a game,” he recalled. “And I liked playing.”
As an undergraduate at Lebanese University in Beirut, Mourtada pursued both law and mathematics, initially envisioning a career in law. Yet while he appreciated the philosophical aspects of legal studies, the practical side left him cold. Mathematics, on the other hand, felt different. He was drawn to the puzzles it offered and to the community that welcomed him. Growing up, it was his teachers and classmates that made school engaging, even if he sometimes fell asleep in class. As he immersed himself in math, he found that mathematicians have a rare honesty. “You need to be honest with yourself to be a mathematician,” he observed. “Otherwise, it doesn’t work.”
Mourtada eventually moved to France to pursue a doctorate in algebraic geometry, a field that explores shapes created by specific mathematical rules. While some shapes are smooth, like lines or circles, others have unique points, called singularities, where they intersect or sharply change direction—imagine the crossing point in a figure-eight shape. These singularities are areas of intense interest for researchers, as they reveal unexpected behaviors and hidden patterns within the structure of these shapes. Algebraic geometers, like Mourtada, study these features to uncover the underlying complexity, especially as these shapes grow more intricate and extend into dimensions that are hard to visualize.
Shortly after completing his graduate studies, Mourtada and two colleagues, Jan Schepers and Clemens Bruschek, turned their attention to the arc space associated with a particular singularity. In an attempt to better understand it, they divided the space into smaller sections, similar to how an archaeologist analyzes separate layers of an ancient site. Yet, they soon hit an impasse. But for Mourtada, something in their findings continued to resonate. As he examined the numbers they had calculated—the counts of polynomials in each section of the arc space—he felt a flicker of recognition. “I kept on repeating them like a child,” he recalled. “And then suddenly I remembered.”
The study of singularities often relies on techniques that allow mathematicians to sense the "texture" of a shape, especially in spaces too complex to be visualized. John Nash had laid some of this groundwork in the 1960s with his exploration of arc spaces—collections of tiny arcs passing through singular points. By examining how these arcs behaved at a singularity, Nash could assess the smoothness of the surrounding structure. “If you want to see if it’s smooth, you want to pet it,” said Gleb Pogudin of the École Polytechnique in France, invoking an almost tactile sense of geometry.
In practical terms, an arc space presents an infinite array of polynomial equations. This is precisely where Mourtada’s expertise shines. “Understanding the meaning of those equations is really the thing Mourtada is expert in,” said Bernard Teissier, a colleague at the Institute of Mathematics of Jussieu in Paris. “These equations can be very complicated. But they have a certain music to them. There is a lot of structure which governs their nature, and he’s just the person who best listens to this music and understands what it means.”
The Rogers-Ramanujan identities are like an neatly cut emerald: intricate, dazzling, and constantly shifting with every new angle. At their core, these identities set complex infinite sums equal to infinite products, forging an unexpected connection between two fundamental mathematical functions—addition and multiplication—that seem worlds apart.
This connection captivated mathematicians across generations. Percy MacMahon, an English mathematician with an unlikely beginning as a soldier in the late 19th century, found his way to mathematics after illness forced him to leave the military. By 1915, he was crafting the first comprehensive textbook on combinatorics, the mathematical study of counting. In the Rogers-Ramanujan identities, MacMahon saw that beneath their surface, both sides of the equal sign spoke the language of counting. He realized, much like the German mathematician Issai Schur, isolated by World War I and arriving at similar conclusions independently, that each identity could be seen as counting specific partitions of integers.
Consider the number 4. There are only a few ways to break it down into smaller numbers: as 4 by itself, as 3 and 1, as two pairs of 2, as 2 with two 1’s, and as four 1’s. These combinations, known as partitions, add up to five, but this simple count hints at a deeper mathematical intrigue. The mathematician Leonhard Euler was the first to systematically explore these partitions, discovering a curious pattern: for any number, the count of partitions using only odd parts matches the count of partitions where all parts are distinct. For mathematicians like Andrew Sills, this unusual symmetry continues to inspire awe, prompting questions like Shashank Kanade’s: “Where did the 5 come from?”
Ramanujan’s garden, as some have called his world of insights, continued to fascinate well into the 20th century. During World War II, physicist Freeman Dyson found solace from the war’s chaos by "wandering in Ramanujan’s garden." But it wasn’t until the late 1970s that new facets of the Rogers-Ramanujan identities cropped-up. Rodney Baxter, an Australian physicist, developed a simplified model of gas behavior to study phase transitions—points at which physical systems abruptly shift, like when water freezes into ice. Through his calculations, Baxter stumbled upon the Rogers-Ramanujan identities within the framework of statistical mechanics, linking them unexpectedly to physical phenomena.
Around this time, mathematicians James Lepowsky and Robert Wilson at Rutgers University uncovered the identities within representation theory, the study of mathematical symmetries. Their findings gave rise to a groundbreaking new field—the theory of vertex operator algebras, essential for modern string theory and instrumental in proving the "monstrous moonshine" conjectures, one of group theory’s most celebrated recent achievements.
Bernard Teissier of the Institute of Mathematics of Jussieu in Paris explained that these connections reveal the identities as part of a much broader mathematical framework. "You begin to see that these identities are natural. They are part of a general frame of things, which is much, much more general than just the partition identities,” he observed.
The Rogers-Ramanujan identities continued to pop up across disciplines in the 1990s and 2000s. In number theory, they surfaced in the study of modular forms, central functions for understanding integer solutions to polynomial equations. In probability theory, they appeared in Markov chains, systems that model random events. In topology, they materialized through knot polynomials, tools used to classify and distinguish knots. Each time, mathematicians wouldboth re-prove the identities using new techniques from these fields and derive from them fresh ones.
MacMahon observed that the Rogers-Ramanujan identities could be understood through partition counting. He noticed that one side of the identity represents ways to break a number down without using any repeated or consecutive parts—such as 4 alone or 3 and 1 for the number 4. The other side, however, counts ways of breaking down the number where each part, when divided by 5, leaves a remainder of 1 or 4—like 4 itself or four 1’s in the case of 4. Remarkably, for any number, these two seemingly different methods of counting yield the same result, revealing a surprising mathematical harmony.
In their depths, they hint at a unified, hidden order within mathematics itself—an order Ramanujan glimpsed and set into motion a century ago.
In 2010, Hussein Mourtada found himself deep in thought, studying the arc space of a singularity known as a fat point. To probe the structure of this singularity, he had broken down its arc space—an intricate system of infinitely many polynomial equations—into layers. As he began counting the polynomials within each layer, something remarkable happened: a pattern. The numbers he saw weren’t random; they bore a striking resemblance to the sum side of the Rogers-Ramanujan identity, which counts partitions without equal or consecutive parts. “It suddenly came to my mind,” he recalled. “Even if I was counting something different from partitions, this was actually exactly what I was counting.” Mourtada, along with collaborators Clemens Bruschek and Jan Schepers, soon proved that their arc space’s structure was indeed captured by this identity. “It’s very surprising that such a simple singularity could have such a deep underlying structure,” Schepers remarked. The realization was thrilling—enough for Mourtada to leap onto his desk in excitement.
Bruschek and Schepers eventually moved away from mathematics, but Mourtada pressed on. “You could say this was the baby case,” explained Bernard Teissier. Over the next decade, Mourtada would broaden the scope of his work, transforming it into “a completely general type of business.”
In 2015, Pooneh Afsharijoo, a young mathematician from Iran, arrived in France to begin her graduate studies with Mourtada. Since then, she and Mourtada have taken on the challenge of understanding increasingly complex singularities and their arc spaces. Their work has led them to uncover a wealth of new identities and even to extend Ramanujan’s oldest identity. The Rogers-Ramanujan identity asserts that two very different counting conditions yield the same number of partitions. Building on this, Afsharijoo discovered a third condition, expanding the framework of Ramanujan’s century-old findings.
Now, Mourtada and Afsharijoo are using graph theory—a method of representing data through networks of points and edges—to map out the structure of arc spaces, opening up new possibilities for exploring partition identities. This approach allows them to link arc spaces to specific configurations of polynomials, a connection that once again ties back to partition identities like those in the Rogers-Ramanujan framework. “It’s the magic inside the integers,” Afsharijoo observed, marveling at the hidden order within mathematics.
In a way, this breakthrough felt inevitable. Mathematicians have long associated polynomial equations with partitions, but Mourtada’s work delves even deeper, capturing subsets of these equations and linking them to specific classes of partitions. By counting these subsets, he and Afsharijoo continue to explore and expand the legacy of partition identities, revealing yet more facets of Ramanujan’s timeless insights.
Every time the Rogers-Ramanujan identities surface in a new context, mathematicians find themselves both astonished and somehow unsurprised. These identities have woven themselves into the fabric of mathematics, creating links between seemingly unrelated fields. “The surprise of it has not worn off,” said George Andrews of Pennsylvania State University. “It still seems unreasonable that all these things are true.” Yet with Ramanujan’s work, the unexpected has come to be expected. “That’s kind of the trademark of Ramanujan’s kind of mathematics,” remarked Ken Ono of the University of Virginia.
Recently, in September, Ono and his colleagues, William Craig and Jan-Willem van Ittersum, published a groundbreaking application of partition identities. Rather than merely tracing new sources where these identities might appear, they used the identities for something entirely different: identifying prime numbers. By constructing a special formula based on functions that count partitions, they created a tool that yields zero when a prime number is plugged into it, and a positive number for any non-prime. “Isn’t that crazy?” Ono mused. Through this method, partition identities reveal an unexpected ability to isolate prime numbers—an insight that adds a new dimension to their reach.
This new discovery led Ono and his team into the realm of modular forms, where they found their formula to be just one part of a broader landscape of prime-detecting functions—infinitely many, in fact. “That’s mind-blowing to me,” he said. “I hope people find it beautiful.” The breakthrough points to a profound relationship between the theory of partitions and the multiplicative structure of primes, a connection mathematicians are eager to explore.
For Andrews, it makes sense that partitions keep cropping up in every corner of mathematics. “The theory of partitions is so basic,” he explained. “Counting stuff and adding stuff up happens in almost every branch of mathematics.” Yet, the depth of these connections remains elusive. “It’s really about getting the perspective right,” Ono noted, hinting at the layered insights still waiting within Ramanujan’s work.
Shashank Kanade of the University of Denver echoed this sentiment, emphasizing that Ramanujan’s findings are never a dead end but rather the tip of an iceberg. “You just have to follow it through,” he said. Mourtada, too, reflects on Ramanujan’s visionary nature, marveling at how Ramanujan could imagine connections that others struggle to see. The development of new fields has allowed today’s mathematicians to trace these identities further, revealing patterns Ramanujan might have discovered through sheer intuition.
“That’s why mathematics is so important,” Mourtada added. “It allows ordinary people like me to find these miracles, too.” Through the persistent search for patterns and hidden truths, Ramanujan’s legacy endures, inviting each generation of mathematicians to uncover a bit more of the iceberg lying just beneath the surface.