Curtis McMullen, a mathematician at Harvard University, revisits Niels Abel's proof that the general quintic equation cannot be solved by radicals, delivered at the Institute for Advanced Study. McMullen reframes Abel's result through modern tools: the braid group generated by the roots of a polynomial as its coefficients vary, and the dynamics of billiard trajectories in polygons. He traces how the impossibility of a radical solution for degree five and higher connects to the topology of these braids, showing why no finite sequence of root extractions can unwind the monodromy involved. The talk moves from classical algebra into geometric group theory and dynamical systems, linking Abel's nineteenth century argument to contemporary research questions. Delivered as part of a session on Abel at IAS, the lecture assumes some background in algebra and topology but builds its argument carefully from the historical problem outward.