At age 99, mathematician Joan Birman and collaborators Tara Brendle and Vasudha Bharathram solved a century-old mystery about the Burau representation, proving that four-strand braids maintain faithful matrix representations while five or more strands do not. The problem, introduced by Werner Burau in the 1930s, asks whether certain algebraic translations of braids lose information, and Birman's 1970s foundational work on the topic ultimately led to its resolution nearly a century later.
Matrix calculus is unnecessary for machine learning and tensor differentiation. Instead of learning complex matrix calculus rules, physicists discovered a simpler approach: write out index notation and use ordinary differentiation, which is faster, more intuitive, and always works.