Title:  Metric spaces on concordance classes of knots

Abstract:  Knot theory is an essential tool to understand the topology of 3- and 4-dimensional manifolds.  To understand 3-manifolds, we consider knots up to isotopy, where you can stretch and bend the knot without breaking it.   However, to understand 4-manifolds, we wish to understand knot up to concordance.  In this talk, we will introduce the notion of a positive/negative generalized tower associated to a knot up to concordance, built out of immersed disks and gropes. The bottom stage of these can thought of as a generalization of an immersed disk in R^3 \times R, thought of as space time, obtained by performing a crossing change through time (with a positive/negative crossing).  Using the complexity of the tower, we define a metric space on the space of knots and show it is non-discrete. We expect that this metric space restricts to a non-discrete metric on the set of topologically slice knots.  We will not assume any knowledge of knot concordance and will define the necessary background.