The meetings with her mentor continued over Zoom, through a sudden collapse on the street and surgery for a pacemaker. By the time they could meet in person, initially on a bench in Riverside Park, they had made progress. Vasudha was formulating a new way to prove the faithfulness for three-strand braids using a special type of polynomial. It was an unusual choice, because the polynomial had played a role in proving why Burau was unfaithful for braids of five or more strands. She now believed it would help her prove why the four-strand case was faithful. Dr. Birman thought that sounded like a good idea.

It was just an intuition, though, formed after months of visualizing braids in her mind. Not a proof. “They had an argument, but it was not possible yet for someone else to understand it,” said Dan Margalit, a topologist at Vanderbilt University who reviewed their early work. He was skeptical they would succeed, but he encouraged Dr. Birman over their many conversations, seeing no harm in the mathematician and her apprentice’s obsession with this interesting, if perhaps impossible, problem.

It was clear, however, that they could use some backup. One day during Vasudha’s senior year, Dr. Birman invited Tara Brendle, one of her final graduate students at Columbia, to join them in the apartment and take a look at their work. She quickly spotted an error in one of their diagrams — at that moment, Dr. Birman knew they had found a collaborator.

Dr. Brendle, who was then in her late forties and a professor at the University of Glasgow, was hesitant to take on Burau, a notorious rabbit hole that she advised her own Ph.D. students to avoid. But Ms. Bharathram’s approach was so interesting, she recalled later, that “I went home and told my children about it.”

She was surprised, too, by how well the pair got along, with a shared intensity that could easily read as bluntness. Topics that others deem personal, about family or hobbies or music, seemed to live at the polite periphery of their conversations. But they shared a taste in mathematics — and that, Dr. Brendle said, is as personal as anything.