A tanka that celebrates a mentor-mentee mathematical journey:

The Birman-Vasudha Collaboration

A female version
of Hardy-Ramanujan—
with fellow Brendle,
the trio proved a years-long
unsolved braid-theory knot.

The story of a 92-year-old mathematician mentoring her teenage apprentice shows that math isn’t just a young man’s game, but also a young (and old) woman’s game. It also debunks the myth that if you haven’t produced or published your best math by forty, your productivity is only heading south from then on.

Joan Birman (now 99), Tara E. Brendle, and Vasudha Bharathram (a young mathematician of Indian origin who is beginning a Princeton PhD) recently proved that the Burau representation of the braid group is faithful for n = 4. This resolved a decades-old (roughly 90-year) open case in braid and knot theory and low-dimensional topology that had defeated and humbled the giants.

Nonagenarian Birman’s email reply to the younger Vasudha—and her later guidance of her protégé’s mathematical path—forms a female parallel to G. H. Hardy’s response to a letter from the largely self-taught Indian mathematician Srinivasa Ramanujan.

Mathematical longevity doesn’t stop at forty or even sixty; you can still be mathematically active and productive long after you’ve retired from formal employment.

For most of us who don’t have what it takes to crack a long-unsolved topological nut or untie a stubborn topology knot, we can still mentor a young man or woman who is often stifled or bored by canned school math.

Collaboratively yours

© Yan Kow Cheong, September 13, 2026