It was a kind of note that Joan Birman had received many times: a plea from a high school student, asking for help learning mathematics. After enjoying freshman geometry class, the student wrote, she had recently read her first real mathematical proof. It made her think there might be something more to math than what she was learning in school.

Perhaps Dr. Birman could show her what she was missing. Messages like this had arrived with such regularity during Dr. Birman’s career that she had come to suspect they were assigned by high school teachers. They kept arriving via email, though perhaps not as frequently, even after she retired from Columbia University in 2004.

Now, in the summer of 2019, she was 92 years old, and she had no interest in being a babysitter. Still, she messaged the student back and said that she’d think about it. The truth was, she wanted to keep doing math.

Her husband, Joseph Birman, had died two years earlier, leaving her alone in the house in New Rochelle, N. Y., where they had raised their three children. She had decided to move back to the city, choosing an apartment on Riverside Drive because it was not far from the office she still kept at Columbia.

She wanted to be a cab ride away at all times from a mathematical conversation. And here was this email — which, when she thought about it more, seemed unusually sincere, not coerced, as if the girl actually did want to do math. After a few days, Dr. Birman replied and said she was willing to meet.

She told the teenager to find a textbook, I. N. Herstein’s “Topics in Algebra,” a standard torture device for aspiring graduate students.

The girl arrived that July in the lobby of Dr. Birman’s Upper West Side apartment building, a postwar co-op on Riverside Drive. Her name was Vasudha Bharathram, and she had just turned 15 that month. She was a rising sophomore at Riverdale Country School in the Bronx, but it so happened that she and her family also lived on Riverside, just 10 blocks away in Manhattan.

Seeing her for the first time, Dr. Birman was struck by how small and childish she looked: “She hadn’t gone through her growth spurt yet,” she recalled. Her father had escorted her on the walk through the neighborhood. They sat in the lobby armchairs, and Dr. Birman asked Ms. Bharathram about her move from India two years earlier, and what she liked about high school geometry.

Her own subfield of mathematics was called topology, Dr. Birman explained; it, too, involved shapes, but ones that were fluid and stretchy, not pinned down by rigid coordinates. Dr. Birman took out a pencil and drew an example, called a braid, which twirls around just like the hairstyle. She sent the student off with an assignment to read a chapter of the textbook. She was surprised when, a few days later, Ms. Bharathram sent her another email, this time with photographs of her notebook.

There were pages and pages of solutions, written in a messy teenage scrawl that the elder mathematician could barely decipher. Dr. Birman was startled, first and foremost, by the girl’s intrepidness; problems from Herstein were no one’s idea of a good time. Then she looked more closely at the notes and realized that Vasudha had solved the problems, mostly proofs about basic concepts in abstract algebra, by sketching out unusual paths to the right answer. At their next meeting in the lobby, she asked her to explain her thinking.

There was something original here, Dr. Birman realized. She decided right then on their path forward. She had a problem in mind, one that had escaped her own grasp for decades.

In fact, as they moved through the textbook, they were already working to solve it — though her new student had no idea yet. The lessons soon moved to Dr. Birman’s apartment. She had dispensed with most of her things in the move, but a few artifacts remained of her old life in the big house in New Rochelle.

Shelves of scientific biographies, mostly collected by Joe, a prominent physicist. A pair of hooded gar...