A 2-knot is defined to be an embedding of S^2 in S^4. Unlike for knots
in S^3, the theory of concordance of 2-knots is trivial. This talk
will be framed around the related concept of 0-concordance of 2-knots.
It has been conjectured that this is also a trivial theory, that every
2-knot is 0-concordant to every other 2-knot. We will show that this conjecture is false, and in fact there are infinitely many 0-concordance classes.
We'll in particular point out how the concept of 0-concordance is
related to understanding smooth structures on S^4. The proof will
involve invariants coming from Heegaard-Floer homology, and we will
furthermore see how these invariants can be used shed light on other
properties of 2-knots such as amphichirality and invertibility.