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The question of whether or not the Burau representation of the braid group $B_4$ is faithful is an open problem since the 1930s. It is also related to the question of whether or not the Jones polynomial of a knot detects the unknot.

In this talk, I will present my work on this problem, which includes strong constraints on the kernel of this representation. The key techniques include a new interpretation of the Burau matrix of a positive braid, in terms of a new decomposition of positive braids into subproducts.

I will discuss the relevant background for the problem from scratch, and illustrate my techniques through simple examples.