Robert Langlands enjoys daily walks up Mount Royal. At 78, he climbs with sure steps, weaving along beaten paths to the cemeteries on the mountain’s north side.

He takes those hikes when he is back at his Montreal condo, between semesters at the renowned Institute for Advanced Studies in Princeton, N.J., where Langlands has been a professor for more than 40 years.

His destination, the bright summer day when the Star first caught up with him, was the gravesite of writer Mordecai Richler, on a spot called Rose Hill. It was hot, the air was alive with birds, and tombstones rose and fell as if on a wave of green. Death seemed almost acceptable. But Langlands strolls the graveyards for a different kind of inner peace.

“If you’re lucky, it’s a way to stop thinking,” he says. “The wheels don’t stop so easily after a while.”

Langlands, a Canadian, is one of the world’s great mathematicians. His universe is the outer limits of pure mathematics, a rarefied realm where abstract objects exist, infinity is corralled and symmetry reigns.

In 1967, as a young professor at Princeton University, he revolutionized the ancient discipline. He discovered patterns in highly esoteric objects called automorphic forms and motives, and he restructured mathematics with two dazzling theories.

They indicated what mathematician Edward Frenkel calls “the source code of all mathematics,” and are credited with linking math’s main branches — number theory (once called arithmetic), harmonic analysis, which includes calculus, and geometry.

To mathematicians, this is mind-blowing stuff. The branches deal with completely different things: number theory is about, yes, numbers, harmonic analysis studies motion and geometry deals with shapes. They may as well be different planets.

“Suddenly, you have a teleportation device that enables you to go — boom — from one place to another,” says Frenkel, a Russian-born mathematician at the University of California at Berkeley, describing the math-warping impact of Langlands’ conjectures.

The theories have not been fully proven. But every time they’ve been tested, they work. Frenkel is convinced the conjectures lay the groundwork for a “Grand Unified Theory of Mathematics,” although some aspects remain beyond Langlands’ embrace.

“He was a visionary,” Frenkel adds in a phone interview from California. “He pointed us into a direction where we can go and find the truth, find out what’s really going on. It’s about seeing the world in the right light.”

Resistance from eminent academic quarters was stiff. But Langlands’ new order eventually evolved into a vast area of research known as the Langlands Program.

Frenkel, a driving force in the program, pushed its boundaries further a decade ago. He landed a multi-million-dollar grant from the U.S. Defence Advanced Research Projects Agency (DARPA). The mandate is to apply Langlands’ conjectures to the search for physics’ Holy Grail — a theory that unifies laws governing all known physical interactions in the universe. Physicists in the DARPA research include Edward Witten, a leading “superstring” theorist.

Unifying nature’s fundamental forces — gravity, electromagnetism, the strong and weak nuclear forces — was a goal that eluded Albert Einstein. If Langlands’ theories help make it happen, the cosmic symmetry will come with a poetic touch: Langlands’ office in Princeton is the same one Einstein occupied during the later part of his two decades at the Institute for Advanced Studies, until his death in 1955.

“He’s like a modern-day Einstein,” says Frenkel, author of Love and Math, a book about the Langlands Program. “But everybody knows about Einstein and nobody knows about Langlands. Why is that?”

That kind of talk makes Langlands cringe. He’s sure of his achievements but skeptical about their value in the real world. Pure math is an adventure of the mind, unconcerned with applications. He’s even uncomfortable with the program that bears his name, believing its expansion into the field of differential geometry — a branch that uses calculus to study the properties of geometric shapes — has been poorly thought out.

Most of all, he recognizes his work is complete nonsense to all but a tiny clique.

“What normal person cares whether the square root of two is a rational number?” he asks, referring to numbers than can be written as fractions.

No doubt even high school math is anathema to the many who remember it as an instrument of torture. Yet math-based algorithms increasingly dominate daily life, from stock market trading to Internet searches to online payments. Math geeks are cashing in.

For the initiated few, the attraction is as old as philosophy. Math is the language of absolute truths: in no universe can 2 + 2 equal 5.

For Jim Arthur, a professor at the University of Toronto, math reveals the beauty and symmetry of the universe in ways that “almost overwhelm your sense of esthetics.”

If philosophers can ever swoon, Bertrand Russell came close when he exalted math as “the true spirit of delight.”

Langlands has his feet more firmly on the ground. Since Plato, many mathematicians have considered math objects and equations to be as real as gravity, waiting to be discovered. Langlands instead sees them as human inventions.

A certain distress has therefore entered his life of late. He’s focused on adding what he can toward a complete proof of his conjectures, but age is playing tricks. The objects he once easily manipulated — the equations that produce shapes like weird doughnuts and tubes that curve into infinity — are losing their solidity.

“It happens when you get old,” he says after reaching Richler’s grey tombstone. “You have less assurance about what’s in your head. Did I turn off the tap?”

At times, the opposite happens. He awakes and for some moments finds a world where mathematical entities seem as real as the furniture.

“It’s nightmarish,” he says. “It must have something to do with madness. If in the middle of the night you wake up and there’s some fusion between the mathematical objects and the real world, then you’re mad.

“And if you’re lucky, it’s just fatigue and not some permanent dysfunction.”

Robert Langlands was reluctant when I first contacted him about this story, not least because I know nothing about math. He assigned me homework: if I was “at all serious,” I should consult a collection of his writings on the Institute for Advanced Studies website, and speak with Jim Arthur, “the leading mathematician in Canada.”

After all that, little more than the surface of things made sense to me. At one point, I triggered an email disagreement by connecting motives found in geometric shapes with spectra, the information in waves and frequencies. Langlands dismissed the link; Arthur insisted on it.

“A motive is supposed to come with a product of l-adic Galois representations, whose Frobenius eigenvalues would be related to Hecke eigenvalues by reciprocity?” Arthur wrote. “But could we not argue that the latter are closely related to quantum mechanics-like spectra, being ‘momentum observables’ in the sense that they commute with the Laplacian/Hamiltonian?”

In other words, I would probably understand nothing.

Months later, I stepped into Langlands’ comfortable old condo in Montreal’s leafy Outremont neighbourhood. “I don’t know quite why you’re undertaking this project,” he said, laughing.

Langlands is tall and active for his years. He hikes and cycles regularly with his wife, Charlotte, around Vermont and Quebec, but the years when they did so across Europe are over. He fears the joy nature brought him may be lost to future generations.

“In what sense survival will be possible is by no means clear to me,” he says. “The frivolity of Canadians with respect to climate change is especially shocking.”

He is not afraid to speak his mind. He has described mathematicians as “egotistical creatures,” has lamented an “endemic sense of mediocrity” among Canadians and confesses to being “a little bit caustic” with peers slow to see the full potential of his ideas.

“He’s clearly one of the most important living mathematicians,” Frenkel says. “His legend precedes him. But the question is, ‘Do mathematicians really know what he has done?’ It’s like having a famous writer but no one has read his books.”

Langlands often notes he’s not the modest type. But he’s more inclined to stress the enormous work needed to prove his conjectures than the successes so far.

“Presumably, with time, it will be a big part of mathematics,” he says of his theories. “It has to be said, however, that these questions are not very important for the world.”

We sat in Langlands’ living room, decorated with stone and clay sculptures by his wife. He began with a five-minute primer.

“So, you’ve been to school; you took Euclidean geometry?”

“I took it,” I said, assuming I must have, despite not recalling a thing.

“There are various things in Euclid that remain important to this day . . . It is proved that the square root of 2 is an irrational number. So, what we’ll be dealing with when we talk about these things is the study of irrational numbers.”

He added that ancient Greeks were troubled by the baffling number pi, another irrational number, before asking, “Cartesian geometry — did you learn anything about that? A point in a plane has coordinates x and y? And that you can write a circle with x² + y² = r²?”

“I can recall sitting in that class,” I said, lying.

Newton’s calculus, he then noted, is about velocity and acceleration. Temperatures at various points of the Earth can be treated as functions. And matrixes have something called eigenvalues, numbers detected, for example, when a guitar string is plucked. With that, the lesson was over.

So I told him about Grade 4, the year math died for me. The teacher divided the class into two teams and for whatever reason made me the captain of one. The game was, ostensibly, to learn multiplication.

The daily ritual had the teams line up at different ends of the room. The teacher would flash a card, say, 8 x 4 =, and the first to shout the answer won. The captains squared off first, and every morning the room echoed with my rival’s answer while I stood frozen with my mouth slightly open. At some point, I quite reasonably decided, to hell with math.

Langlands wasn’t moved: “Well, I cried when they wouldn’t accept me in the church choir, so?”

Robert Phelan Langlands was born in New Westminster, B.C. As a child, he wanted to become a Catholic priest and built an altar in his room. When he was 9, the family moved to White Rock, a resort town near the U.S. border. His parents opened a shop that sold construction materials.

It was shortly after Second World War. Saturday evenings, the young Langlands admired the swagger and freedom of bigger boys cruising the town’s main drag. He would return the next morning to collect empty beer bottles and recycle each for a penny or two.

He delivered the Vancouver Sun. He changed the marquee at the local movie theatre three times a week in exchange for watching the films free. By 13, his free time was spent helping out at his parents’ store.

He was smart enough to skip a grade in high school, but frustrated teachers who believed he was coasting. “It was a country school,” he says. “It probably wasn’t too hard to be intellectually outstanding.”

He thought of dropping out. In Grade 12, an English teacher announced to the class that if Langlands didn’t go to university, he would be wasting God-given talents. Flattered, Langlands applied.

By then, he had met Charlotte, with whom he would eventually have four children. Her father owned a book filled with biographies of Marx, Einstein and other great thinkers. It inspired Langlands to become a savant. “In the 18th century sense of the word,” he explains, “the notion of being a learned man.”

He would become fluent in French, Russian, German and Turkish, and well-versed in their literature. Frenkel, who exchanges emails with Langlands in Russian, speculates that his versatility with languages may have had something to do with his ability to see connections in disparate fields of mathematics.

He completed his undergraduate degree at the University of British Columbia. His 1960 PhD thesis at Yale University analyzed the obscure domain of “Semi groups and representation of Lie groups.” It was spotted by a mathematician at Princeton University and, without having applied, Langlands was offered a job as an instructor.

He taught at Princeton and later Yale until 1972, when he became a professor at the Institute for Advanced Studies. He would win almost a dozen major math awards.

The American Mathematical Society, in giving Langlands its first award in 1988, applauded his “revolutionary” insights. His “path-blazing work” was celebrated in 1996 when he became the first Canadian to win the coveted Wolf Prize. The $1 million (U.S.) Shaw Prize, which he shared with British mathematician Richard Taylor in 2007, praised Langlands for having “initiated a unifying vision of mathematics that has greatly extended the legacy of mathematics of previous centuries.” The Langlands Program, the citation added, “has guided mathematicians for the past 40 years and will continue to do so for years to come.”

“I like mathematics to this day,” Langlands says. “I feel at ease with it. I never doubted myself. I never really found a case where somebody else knew better than I did what was going on. Although I didn’t understand everything, by and large I was right.”

In 1966, Langlands almost abandoned mathematics. Deep mysteries in number theory discouraged him. He decided on a change of scenery and applied for a job in Turkey.

“The decision itself freed me and I began to amuse myself with mathematics without any grand hopes or serious intentions,” he said in written answers to a 2010 UBC interview.

Inspiration struck during the Christmas break, in an empty, grand old building on the Princeton campus, as Langlands gazed at a garden through leaded windows.

He described his revelation in a Jan. 16, 1967 letter to André Weil, a giant in the field of number theory: “If you are willing to read it as pure speculation,” he wrote Weil, “I would appreciate that; if not — I am sure you have a waste basket handy.”

Three years later, after he’d returned from Turkey, Langlands published his two theories, called functoriality and reciprocity, under the title “Problems in the Theory of Automorphic Forms.” Math would never be the same again.

Automorphic forms can be thought of as fundamental particles in harmonic analysis, which deals in part with waves and frequencies. In a symphony orchestra, automorphic forms issue instructions and work with eigenvalues — the different speeds a violin string moves when struck, for instance — to produce the notes played.

The number of automorphic forms is infinite and each gives different instructions. The data they produce is spectral, the kind that comes from waves and frequencies. Langlands used a method known as the “theory of continuous groups” to link and arrange automophic forms in ways no one knew possible.

At the same time, he cast his gaze on another branch of mathematics — number theory — and recognized patterns in even more esoteric entities called motives. They’re considered the indivisible building blocks of all geometric forms, from circles to bizarre shapes. They too are infinite, although some mathematicians aren’t convinced they exist. Langlands corralled and classified those, too.

He also discovered a symmetry between automorphic forms and motives, what Frenkel would later describe as the teleportation device between different worlds. Langlands saw that for every automorphic form issuing instructions, there is a motive issuing those same instructions but going about it in an entirely different way.

Langlands’ conjectures work at levels beyond the immediate grasp of most mathematicians. A leading expert in the field is Jim Arthur, who developed the so-called Arthur-Selberg trace formula needed to apply functoriality. At his University of Toronto office, he spent hours trying to explain the conjectures to me, somehow showing not the slightest frustration. Eventually, he asked me, “Can you hear the shape of a drum?”

Imagine standing at a spot where you hear the beat but can’t see the drum. Suddenly you have a formula that lets you use spectral data from the sound wave to figure out the exact shape of the drum that produced it. Likewise, information about the shape of the drum lets you anticipate its sound.

Langlands’ insights work that kind of magic at far more abstract levels, between representations of what are called Galois groups and Lie groups. They point the way to “all the secrets of the arithmetic world,” Arthur says.

If different mathematical objects are suddenly found to share the same identity, calculations and answers from one branch of math can be used in another. “One can think about these objects in an entirely different way,” Langlands says. Theoretical dead ends are overcome, new areas of research are opened and age-old mathematical conundrums appear in a new light.

In 1994, functoriality was used by British mathematicians Andrew Wiles and Richard Taylor to solve a famous, 330-year-old brain teaser known as “Fermat’s Last Theorem” — an achievement celebrated in the mathematical world like a Super Bowl winning touchdown.

Arthur, winner of the 2015 Wolf Prize in math, suspects the conjectures will eventually help solve other fundamental problems, perhaps even the 150-year-old Reimann hypothesis on prime numbers — a whole number that can be divided evenly only by one or itself. The hypothesis gives a remarkably accurate approximation for the number of primes less than any given larger number, such as 10 to the 100th power. But it remains a conjecture.

“People would sell their souls to prove this,” Arthur says.

Yet few mathematicians applauded Langlands’ discovery.

“People didn’t get it,” says Arthur, who had Langlands as his PhD thesis adviser. “It was way ahead of its time.” It took four decades for the theories to be widely recognized, although important resistance persists.

“I’ve had very prominent people — and you might recognize the names — leave a lecture in anger,” says Langlands, naming, when pressed, leading French mathematician Jean-Pierre Serre.

Asking academics to radically change assumptions that built their careers is rarely easy. The conjectures also require thinking on a grand scale, a challenge in a discipline fragmented with siloed specialists. “What the farmer doesn’t recognize, he won’t eat,” Langlands says.

When mathematicians do take a bite, Langlands doesn’t always get his due. He suspects the Shaw Prize he shared was initially intended solely for Taylor, until someone convinced the selection committee that functoriality wasn’t Taylor’s idea. “I was a tag-along, I think. So it rather irritated me.”

André Weil, who died in 1998, used Langlands’ ideas unattributed, to the point where some number theorists began referring to them as the Weil conjecture. Langlands eventually reminded Weil of his 1967 letter.

Those who embrace his theories have expanded them to include geometry and, more recently, quantum physics through Frenkel’s DARPA grant. Frenkel likes to quote Galileo to emphasize the natural bond between math and physics: “The laws of Nature are written in the language of mathematics.”

Spectacular discoveries in physics have been foretold by mathematics. One of history’s foremost mathematicians, the German Carl Friedrich Gauss, calculated in the early 1800s a theorem on the intrinsic curvature of space, echoed in Einstein’s theory of relativity almost a century later.

In the early 1960s, American physicist Murray Gell-Mann used the math principles of group theory to organize composite particles called hadrons and to predict the existence and distribution of sub-atomic quarks. Experiments several years later confirmed both the organizing pattern and the quarks. He received the Nobel Prize in physics for this work in 1969.

“Mathematics allows you to see the invisible,” Frenkel says.

The link currently exciting some physicists involves a “duality” found between the forces of electricity and magnetism. These forces are said to be symmetrical — they affect each other in the same way. In quantum physics, this duality is central to the quest for a theory that unites all known physical interactions in the universe, from the stars to the smallest particles.

The symmetry between electricity and magnetism parallels one that exists in Langlands’ conjectures, involving mathematical groups that associate with automorphic forms. String theorist Edward Witten of the Institute for Advanced Studies describes the existence of this analog as “amazing.”

“We of course don’t know how the story will develop from here,” he says in an email to the Star. “Maybe the physics will shed light on number theory, maybe number theory will produce insights that will help physicists. Maybe for a long time we won’t know and the two parts of the picture will just develop mostly independently of each other.”

Arthur won’t be surprised if physicists eventually discover that math’s elusive motives exist as fundamental particles in the universe. Langlands dismisses that as sentimental fantasy. He’s convinced the majority of physicists, despite Witten’s giant reputation, see little value in applying his work.

He would prefer that mathematicians in the Langlands Program focus on finding a complete proof of functoriality using Arthur’s trace formula. That requires command of high-level mathematics understood “only by a fairly small number of people,” Langlands told me in an email. “The result is that a good deal of the literature accessible to mathematicians as a whole is made up of a little real knowledge held together by guesswork.”

Back on Mount Royal, Langlands doesn’t need the sight of tombstones to remind him he won’t be the one to prove his conjectures.

“I think I know what the decisive step would be, but it would take a very long time to clear out all the underbrush,” he says, estimating he would need 20 years of concentrated work. “It’s a little presumptuous to think I’ll have that much time.”

Instead, he wants to rework the way Frenkel and others have linked his conjectures to differential geometry.

“If they have to use the name Langlands Program,” says the man who remodeled mathematics, “I’d just as soon like it.”