Minimality, equivariant Gromov-Taubes, and smooth classification of G-Hirzebruch surfaces

Minimality is a basic concept in four-manifold topology, which
can be defined in four different categories: holomorphic, symplectic, smooth, and topological. An important corollary
of Taubes' work on the Seiberg-Witten theory of symplectic four-manifolds asserts that symplectic minimality and smooth
minimality are equivalent. In this talk we shall discuss some recent results concerning minimality of a four-manifold
equipped with an action by a finite group G. At a technical level, one encounters the problem of finding  G-invariant,
embedded J-holomorphic two-spheres of negative  self-intersections, which turns out to be quite delicate.
The invariant defined by counting such J-holomorphic two-spheres is an equivariant analog of the Taubes's Gromov invariant. As an application, we will show how the equivariant Gromov-Taubes
invariant can be used to give a classification of Hirzebruch surfaces equipped with a holomorphic cyclic action up to equivariant diffeomorphisms.