Complex plane curves, their intersection with round spheres, and knot concordance

We start by recalling that all smooth algebraic hypersurfaces in CP^n of a given degree are smoothly isotopic. In particular, a smooth algebraic curve of degree d in CP^2 is a genus (d-1)(d-2)/2 surface. The 'Thom Conjecture', proven by Kronheimer and Mrowka, asserts that such complex curves have the following minimizing property: smooth algebraic curves in CP^2 are genus minimizing among smooth surfaces in their second integer homology class. We derive consequences for transversal intersections of algebraic curves with round spheres, known as `quasipositive knots and links', giving precise instances of the sentiment that these intersections define very special elements in the smooth knot concordance group. In contrast, in the topological category, we prove that all knots are topological concordant to a quasipositive one. Based on joint work with Maciej Borodzik.