Joan Birman, a renowned mathematician who retired in 2004 from Columbia University, found renewed purpose in her 90s after an unexpected outreach from a teenage student reignited her engagement with advanced mathematics. In the summer of 2019, at age 92, Birman, then living in New York City, responded to an earnest email from Vasudha Bharathram, a 15-year-old rising sophomore at Riverdale Country School, seeking guidance beyond the standard high school geometry curriculum.

The initial meeting took place in the lobby of Birman’s Riverside Drive apartment building, where the two discussed foundational concepts in topology, Birman’s field of expertise. She assigned Bharathram chapters from I.N. Herstein’s “Topics in Algebra,” challenging material even for graduate students. Shortly thereafter, Bharathram impressed Birman by submitting detailed, although handwritten, solutions to complex proof problems, demonstrating a rare originality and dedication. This marked the beginning of a close mentorship that quickly evolved into collaboration.

Birman, whose significant contributions to knot theory and braid groups date back to the 1960s, saw in Bharathram an opportunity to tackle a longstanding mathematical problem involving the Burau representation—a key algebraic tool used to study braid groups. The question of whether the Burau representation is “faithful” (preserving all information without loss) for braids on four strands had remained unresolved for nearly a century, with faithfulness established for three strands and disproven for five or more.

Over subsequent months and years, their work intensified despite the challenges posed by the COVID-19 pandemic, during which Bharathram continued her studies remotely while maintaining regular virtual meetings with Birman. As their research progressed, they enlisted the assistance of Tara Brendle, a professor at the University of Glasgow and one of Birman’s former graduate students, who identified key issues and helped refine their approach.

Their collaboration led to a novel method involving intricate polynomial calculations that addressed cancellations in the Burau representation, a crucial step toward proving faithfulness for the four-strand case. The breakthrough involved conceptualizing four-stranded braids as five-stranded entities to simplify the algebraic complexities, which Brendle described as “the most satisfying moment” of her career. Following careful refinement, the team published their findings in July 2026, marking a significant advancement in knot theory.

Now 99 years old, Birman remains intellectually active despite health challenges, including hearing loss and heart valve issues that have limited her mobility. She continues to mentor Bharathram, who recently began graduate studies and remains engaged in extending their work into higher-dimensional topology—a field where many questions still lack answers.

Birman’s decades-long career has been marked by pioneering contributions to the study of braids and knots, helping to establish connections between algebraic methods and topological properties. Her mentorship has fostered a lineage of “mathematical daughters,” a role she once hesitated to resume after the loss of an earlier student. This latest collaboration, however, has not only reignited her passion for mathematics but also contributed a major solution to a problem that had eluded experts for generations.

The partnership between Birman and Bharathram highlights the enduring nature of mathematical inquiry and the importance of mentorship across generations. Their work exemplifies how foundational problems continue to inspire collaboration, innovation, and discovery in the field.