On a Saturday afternoon in February, John Danaher stood in the middle of Columbia’s Andrew F. Barth Wrestling Room, teaching a seminar on his front-headlock system. Fifty members of the university’s Brazilian-jujitsu club watched as his disciples Garry Tonon and Nicky Ryan demonstrated the first step. The headlock is a position familiar from childhood roughhousing and Olympic-style wrestling, in which it is used for takedowns; in jujitsu, where the goal is not to pin the opponent but to force him to concede the match, the headlock is an opportunity for strangulation. With your opponent on hands and knees, snake your arm under his neck and across both carotid arteries, then connect your hands in a five-finger grip. Throwing one leg over his torso, fall to your hip and squeeze, cutting off the blood supply to his brain until he taps out in submission. Easy.

But what if he uses his hand to block your leg? Then include his arm in the guillotine and proceed much as before. Gordon Ryan, Nicky’s older brother, showed how. And what if, and what if, and what if? In ninety minutes, Danaher and his team ran through only half of the headlock’s contingencies. “The problem with most approaches to the sport is that they offer a simple solution to a simple problem,” he told the class. “If the solution doesn’t work because the opponent is resisting, there’s nothing else to offer.” Danaher’s approach is different. “If there’s a failure in one part of the system, other parts of the system can be brought in to overcome that failure,” he said. Indeed, his headlock system is just one among many, with decision trees of control and submission organized across the entire body—a comprehensive new paradigm for the ancient sport of grappling.

Brazilian jujitsu is a key element in mixed martial arts, and it was Danaher’s work with M.M.A. fighters that first brought him renown. Firas Zahabi, who coaches Georges St-Pierre, perhaps the greatest all-around fighter ever, compared Danaher to Hannibal Lecter, calling him “scary smart, superbly calculated and logical.” But in recent years Danaher has turned his attention from M.M.A. to the development of pure, submission-only jujitsu—no punches or kicks, no heavy gi or kimono, no baroque point scoring or judges. Just a quicksilver struggle for dominance.

Danaher instructed Gordon Ryan to play defense against the Columbia students. As Danaher looked on, silent except for pinpoint compliments, Ryan escaped strangles and arm locks in seconds. He was a week away from his third Eddie Bravo Invitational, a sixteen-man tournament that’s become a showcase for the Danaher Death Squad and its pioneering method. In Ryan’s first E.B.I., a year ago, he was twenty years old and had been a black belt for less than three months. He entered as a last-minute replacement for his injured teammate Eddie (Wolverine) Cummings. There was no weight limit, and Ryan, at a hundred and eighty-eight pounds, was one of the smallest men in the field. Nevertheless, he won the tournament, beating a former world champion who outweighed him by forty pounds and the imposing wrestler Rustam Chsiev, nicknamed the Russian Bear.

Danaher, who is fifty and looks like a bulky, bald Robin Williams, trains the Death Squad in the humid blue-and-gray basement of Renzo Gracie Academy, around the corner from Madison Square Garden. Some sixty grapplers show up on weekdays, at seven-thirty and noon, for the most innovative jujitsu class in the country. But the competitive core of the squad, its R. & D. department, is much smaller. This group is conducting a research program dedicated to systematizing “the art and science of control that leads to submission,” as Danaher likes to put it. Cummings, a former physicist, is a calculated leg locker; Tonon thrives in chaotic scrambles; and Nicky Ryan, just sixteen years old, already beats adult black belts with preternatural calm.

“The research has a similar feel to experimental physics,” Cummings told me. “You cheat, you look for ways to cut corners, make approximations here or there, ask yourself how you can play with the system, what if I lose this grip or that wedge, how does it change? Same sort of feel. But I feel like the field is ultimately rudimentary right now. I worry sometimes if John dies I’ll have no one to talk to. I’ll be in a room writing on a wall.”

“Classical jujitsu, it’s pretty simple, O.K.? It’s basically a four-step program,” Danaher told the Columbia students. “You put your opponent down on the ground, you get past his legs, you work your way through a hierarchy of pins, and you look for a submission. It’s a great system, and it works very, very well at beginner levels. At higher levels of the sport, you’ve got to go further than that. If you run into expert resistance, you’ve got to have ways of overcoming expert resistance. And the way to do that is to build subsystems within systems, so that there’s a knowledge asymmetry. You have so much more knowledge about a given position than your opponent does that, inevitably, over time, you’re going to break through.”

Danaher’s systems have a long lineage. Jujitsu was developed in Japan, in the fifteenth century, as a no-holds-barred samurai art. But in the late eighteen hundreds, it was superseded by Kanō Jigorō’s sport of judo, which eliminated the dirtier moves, like eye gouging and groin strikes. In the early twentieth century, the judoka Mitsuyo Maeda went on a world tour, picking up tricks from the Western boxers and wrestlers he fought. In Belém, Brazil, he taught the teen-ager Carlos Gracie what he had learned, and Carlos, in turn, taught his brother Hélio. After a few years, the Gracies began competing in no-rules challenge matches. Slight young men, they developed a system that relied on leverage rather than size or strength. Wrestling and judo prized pinning or throwing an opponent on his back. The Gracies realized that, in a real fight, the opposite is often more effective—control from behind, ideally with the opponent belly-down, so that he can be strangled into submission.

Unlike wrestling and judo, Brazilian jujitsu isn’t yet in the Olympics or American high schools, but it is growing rapidly. Schools have sprung up nationwide, attracting both civilian and celebrity practitioners, such as Keanu Reeves, who trained in the martial art for his role in “John Wick: Chapter 2.” On the Showtime series “Billions,” Paul Giamatti’s character studies under Danaher. The sport has evolved technically as well, spawning hundreds, even thousands, of potential moves and countermoves. (The human body in motion is a complicated thing, and two of them in antagonistic combination exponentially more so.) For a novice, or even for an expert, this can be overwhelming; it’s not necessarily clear what to learn, or why, nor is it always obvious that modern-day jujitsu is the qualitative advance over the Gracie style that the Gracie style was over judo. Danaher’s response is to clear away the complexity, by continually reorganizing and refining only the most efficient, consistently effective moves.

The Columbia club’s co-instructor, a one-handed new black belt named Andrius Schmid, thanked Danaher for coming. Then Schmid asked his team if they knew about Danaher’s history at the school. “I was kicked out in disgrace,” Danaher said. “I was voted, by the entire body of Columbia University, Columbia’s most stupid student.”

Danaher was born in Washington, D.C. His father was a pilot in the New Zealand air force and an attaché to the U.S. military during the Vietnam War. When Danaher was young, the family returned home to Whangaparaoa, near Auckland, where he learned to kickbox. He came back to the States in 1991, in his mid-twenties, to do a Ph.D. in epistemology at Columbia. He weighed two hundred and thirty-five pounds, the result of weight lifting, and had long hair. While teaching and writing his dissertation, he moonlighted as a bouncer; his co-workers traded stories of a scrawny Brazilian guy who, in the first Ultimate Fighting Championship tournaments, had dominated much bigger men, trained in boxing, Tae Kwon Do, kung fu, and wrestling. Danaher got a V.H.S. tape of Royce Gracie, Hélio’s son, at U.F.C. 2, winning four fights in less than ten minutes. Soon after, a Columbia classmate little more than half Danaher’s size, who had been training in the Gracies’ “crazy Brazilian wrestling” for only two weeks, challenged Danaher to a bout in the philosophy department. Danaher locked the door, grabbed him in a headlock, and threw him to the ground, a tactic he’d had success with in bar fights. But the grad student wrapped his legs around Danaher and started rotating onto his back for a choke. Danaher’s arms got tired; he released the headlock and scrambled away. “I was absolutely stunned that, despite a considerable size difference, I could do nothing to him on the floor,” Danaher said.