Distinguishing topologically and smoothly doubly slice knots

The purpose of this talk is to show the existence of an infinite
family of smoothly slice knots that are topologically doubly slice, but not smoothly doubly slice.  After outlining what is known about doubly slice knots, I
will present some sufficient conditions for a knot to be topologically
doubly slice, and show how the knots in question satisfy these criteria.
Then, I will outline how the correction terms coming from Heegaard Floer
homology can be used to obstruct these knots from being smoothly doubly
slice.  Along the way, we will encounter some very interesting open
questions related to the study of doubly slice knots.