Daniel Ruberman (Brandeis) : Slice Knots and the Alexander Polynomial

A knot in the 3-sphere is slice if it bounds an embedded disk in the 4-ball.
The disk may be topologically embedded, or we may require the stronger
condition that it be smoothly embedded; the knot is said to be (respectively)
topologically or smoothly slice. It has been known since the early 1980's that
there are knots that are topologically slice, but not smoothly slice. These
result from Freedman's proof that knots with trivial Alexander polynomial are
topologically slice, combined with gauge-theory techniques originating with
Donaldson. In joint work with C. Livingston and M. Hedden, we show that the
group of topologically slice knots, modulo those with trivial Alexander
polynomial, is infinitely generated. The proof uses Heegaard-Floer theory, and
also applies to problems about link concordance.