99-Year-Old Mathematician Solves Century-Old Puzzle

At 99 years old, most people are slowing down. Joan Birman just solved one of mathematics' most stubborn puzzles.

The Columbia University mathematician, working with colleagues Tara Brendle and Vasudha Bharathram, finally answered a question that has haunted mathematicians since the 1930s. Back then, German mathematician Werner Burau figured out how to translate braids into mathematical grids called matrices, making them easier to work with.

But there was a catch. Mathematicians worried these translations might be "unfaithful," meaning one matrix could represent multiple different braids. That would be like having one address for several different houses.

For decades, researchers knew that simple braids with one, two, or three strands stayed faithful. They also discovered that complicated braids with five or more strands all turned unfaithful. But four-strand braids? Nobody could crack that case.

Birman first popularized this puzzle 60 years ago when she wrote a foundational textbook on braids. Her childhood love of knitting with her grandmother had blossomed into a lifelong mathematical obsession. She spent decades watching other mathematicians take different approaches to the problem, each convinced the four-strand group must be unfaithful.

They were all wrong. Bharathram had a hunch that maybe everyone was asking the wrong question. Instead of trying to prove unfaithfulness, what if they tried to prove the opposite?

The team developed a new method using points and loops on paper. Think of drawing dots that represent strands, then drawing circles around them to capture how those strands interact. The insides of those circles tell you how to calculate the matrices.

Here's what they discovered: With three strands, there are so few possibilities that the matrices have to be faithful. Four-strand braids get tricky with some complicated scenarios, but they still work. Once you hit five or more strands, though, the possibilities explode and faithfulness becomes impossible.

Why This Inspires

Yang-Hui He, a mathematician at the London Institute for Mathematical Sciences, calls it "one of the most interesting stories in recent years." He and his team had spent six months trying to solve the problem using artificial intelligence. Then one Wednesday morning, Birman and her collaborators announced they'd done it.

At 99, Birman proved that curiosity doesn't retire. The same passion that made her love knitting as a girl drove her to close a case she'd helped open six decades earlier. She showed that some mysteries just need patience, fresh perspectives, and a willingness to question what everyone assumes is true.

Mathematics just gained clarity on a century-old question, thanks to a researcher who refused to give up.