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.”
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.”
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