Abstract: There is a rich and intimate  history, going back to Milnor and Nash, intertwining the topology of the link of and complex analytic germ with an isolated singularity and the singularity information from a log resolution. More recent results of McLean and Budur-Bobadilla-Le-Nguyen deepen these connections to more fully involve arc spaces. In this talk, we'll review this story, its more recent developments, as well as some important tools which give insight into the local structure of arc spaces available in derived algebraic geometry; developed in progress with R. Docampo and C. Eric Overton-Walker. No priori knowledge of arc spaces will be assumed.