The Man Behind the Lonely Runner

When Prof. Dr. Jörg Wills opens the door, he’s not just opening the door to his home, but also to the world of mathematics. And he clearly enjoys this brief tour through his rich life—spanning nearly 90 years—and his academic career. Though his eyesight is failing, he describes things with precision and chats engagingly, well aware that the puzzles of mathematics—which continue to fascinate him—require explanation. Wills enjoys explaining things—and he does it well. Above all, he explains the problem he discovered in 1967, which has achieved a certain degree of fame as the “Lonely Runner” conjecture.

The fact that his name is still associated with the Lonely Runner sometimes seems a bit surreal to Wills himself. But it also pleases him. Yet the idea had seemed unremarkable at first, he says thoughtfully. As a young mathematician at the Technical University of Berlin, he wrote his doctoral dissertation on what’s known as Diophantine approximation. The goal was to approximate irrational numbers as closely as possible. “People were always looking for the best approximation, and then I looked for the worst one,” says Wills. “I simply went in the opposite direction.”

From this shift in perspective emerged the conjecture that later became known as the “Lonely Runner,” because the question can be easily explained using this example. Several runners start at the same time on a circular track. They all run at a constant but different pace. The conjecture states that at some point, every runner is “lonely” for a moment—that is, at least a certain distance away from all the other runners. “It sounds like an easy high school math problem,” says Wills. “Anyone with even a little mathematical talent thinks: I can solve this with my eyes closed. But it doesn’t work that way.”

That’s the appeal of the problem: the question is easy to state, but the solution is extraordinarily difficult. The conjecture has been proven for certain numbers of runners. Wills started with three back then and worked on the idea for several years. When he joined the former Gesamthochschule—now the University of Siegen—as a professor in 1974, his research focus shifted. He concentrated on geometry.

But his old problem didn’t go away. In 1998, it was given its current name. The Canadian mathematician Luis Goddyn proposed the “Lonely Runner Conjecture.” Wills wrote to him at the time: “Wonderful name, very poetic.”

Today, the Lonely Runner is still the subject of international research.  The fact that mathematicians at leading universities such as Princeton, Yale, Oxford, and Cambridge are working on it demonstrates the problem’s significance. Even the world-renowned Fields Medalist Terence Tao has published important work on the subject. It’s always interesting to hear what the “inventor” has to say about it. Just a few years ago, Wills participated online in a conference at Harvard University. Now he has been invited to a conference at the University of Rostock. He will give a lecture and be present as other mathematicians continue to work on the question he posed nearly 60 years ago.

Meanwhile, a new player has entered the picture: artificial intelligence. Wills is following this development with interest—and a certain unease. “Recently, one of the so-called Millennium Problems in mathematics—the Navier-Stokes problem—is said to have been solved by an AI.” Wills fears that this will change the very nature of mathematical research.

He describes mathematics as climbing a mountain. The problem lies at the bottom; the solution at the top. “You can see the summit even from the bottom,” he says. The difficult part is the path to get there. If only AI can find this path, he believes something fundamental will change. Then it could happen that mathematicians no longer seek the path to the summit themselves, but instead “climb after the AI.” “That’s not a pleasant thought,” says Wills.

For him, it’s precisely the creative step that’s crucial. The “Lonely Runner” is his own example of this. It didn’t start with a complicated calculation, but with the idea of thinking differently from everyone else—in other words, of being creative. 

Wills doesn’t know whether AI will eventually solve “his problem” completely. Perhaps it will crack it piece by piece. “But even then,” he says, “one kind of AI—namely, creative intelligence—will remain more important than the other kind of AI, artificial intelligence.”