In July 2026, Professor Joan Birman and collaborators proved that the Burau representation of the classical braid group is faithful for n=4, resolving the last remaining open case for this longstanding problem.
Here is a recent Scientific American article highlighting Professor Birman and this remarkable achievement: How a 99-year-old mathematician unraveled a century-old braid mystery.
Joan Birman popularized the Burau representation problem decades ago. Now she and two colleagues have a proof that reveals brand new insights into the geometry of braids.
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