Title: The image of the Burau representation of the braid group is torsion-free
Abstract: Unusually for mathematical objects, the braid group on n strands is exactly what it sounds like: a group whose elements are topological equivalence classes of the ways that n strings can be braided together. A 1935 construction by Werner Burau gives a representation of this group into square matrices of size one fewer than n, whose entries are integer Laurent polynomials in a single variable. It took almost six decades to discover that this representation is not faithful, meaning that there are non-trivial braids (eventually on as few as five strands) that the representation fails to detect as being non-trivial. (On three strands it is faithful; four strands remains an open question.)
There are several elegant proofs of the fact that the braid group is torsion-free, meaning that there is no non-trivial braid that you can repeat that will eventually undo itself into parallel strands. Since the Burau representation is not faithful, the image of the representation is a different group, about which much remains unknown. In particular, it has been an open problem whether the image of the Burau representation is, like the braid group itself, torsion-free.
I will give a broad outline of a proof that the image is torsion free, which involves defining a group whose generators are rational subspaces of (n - 1)-dimensional space. Some mention will be made of positive definite matrices, and perhaps even of spectral graph theory.