Post #4130201

2026-07-27 05:37 UTC

While the Fields Medal celebrates the under-40s, the mathematician Joan Birman has, at the age of 99, solved a major open problem in representations of the Braid groups, a topic she has worked on for more than 60 years. An element of the Braid group 𝐵ₙ is a way of joining two sets of n points with strings, where two elements are the same if the strings can be moved around and/or stretched to follow the same paths in 3-dimensional space while keeping their endpoints fixed. The Burau representation of 𝐵ₙ associates a matrix with each element of the group, in such a way that multiplying these matrices is the same as joining braids together to make new ones. A representation is “faithful” if different elements of the group correspond to different matrices. The Burau representation was known to be faithful for n=1,2,3, and *not* faithful for n≥5, but the n=4 case has only now been settled: “The Burau representation of the braid group is faithful for n = 4” Vasudha Bharathram, Joan S. Birman, Tara E. Brendle https://arxiv.org/abs/2607.05283 I learned this from: https://www.scientificamerican.com/article/how-a-99-year-old-mathematician-unraveled-a-century-old-braid-mystery/ See also: https://en.wikipedia.org/wiki/Braid_group https://en.wikipedia.org/wiki/Burau_representation And in case you were wondering, another group tried using chatbots on this, and failed: “I was introduced to the problem,” He says, “by Emmanuel Breuillard and Sasha [Oleksandr] Kosyak,” two well-known mathematicians who have worked on the quandary for more than 20 years. The group spent six futile months trying to prompt various artificial intelligence chatbots using Birman’s approach. “Then, boom, on a Wednesday morning, Joan Birman herself with her two talented collaborators, claimed that they had solved the problem,” He says. “And they had.”

Replies (5)

@gregeganSF@mathstodon.xyz 2026-07-27 08:27

The formal definition of the Burau representation is a bit daunting, but Vaughan Jones gives a vivid intuitive description. If you think of a braid as a bowling alley with n lanes that can cross over and under each other, and at every crossing there is a probability t of a ball in the upper lane falling down to the lower lane, the (i,j) entry in the Burau matrix is just the probability that a ball starting in lane i exits in lane j. [This metaphor only works for braids that can be built by stringing together any sequence of the “standard generators” where strand i crosses over strand i+1. To get the whole group you need to throw in the inverses of those generators as well.]

Open ##4292318
@highergeometer@mathstodon.xyz 2026-07-27 06:26

@gregeganSF@mathstodon.xyz Nice to hear at least one definitive negative result from people trying to use an AI tool to solve a famous open problem (the idea was in the air, AI couldn't get it over an extended period of time, then the human team solved it)

Open ##4292320
@PetraSchwer@mathstodon.xyz 2026-07-27 06:43

@gregeganSF@mathstodon.xyz this is exciting news!

Open ##4292322
@dbat@mastodon.gamedev.place 2026-07-27 08:07

@gregeganSF@mathstodon.xyz How satisfying that must be for her! Wonderful news.

Open ##4299852
@svat@mathstodon.xyz 2026-07-27 18:20

@gregeganSF@mathstodon.xyz Note for anyone who may not know: > Joan Birman went back to grad school in math in her forties, and is now one of the top researchers in knot theory. — the top answer at https://mathoverflow.net/questions/3591/mathematicians-who-were-late-learners-list (According to Wikipedia she got her PhD in 1968 at age 41.)

Open ##4299856