This is joint work with Marco Franciosi and Rita Pardini.

Godeaux surfaces, with $K^2=1$ and $p_g=q=0$, are the (complex projective) surfaces of general type with the smallest possible invariants. A complete classification, i.e. an understanding of their moduli space, has been an open problem for many decades.

The KSBA (after Kollár, Sheperd-Barron and Alexeev) compactification of the moduli includes so called stable surfaces. Franciosi, Pardini and Rollenske classified all such surfaces in the boundary which are Gorenstein (i.e., not too singular).

We prove that most of these surfaces corresponds to a point in the moduli which is nonsingular of the expected dimension 8. We expect that the methods used (which include classical and recent infinitesimal deformation theory, as well as algebraic stacks and the cotangent complex) can be applied to all cases, and to more general moduli as well.

The talk is aimed at a non specialist mathematical audience, and will focus on the less technical aspects of the paper.

Critical points having infinite Morse index and co-index are invisible to homotopy theory, since attaching an infinite dimensional cell does not produce any change in the topology of sublevel sets. Therefore, no classical Morse theory can possibly exist for strongly indefinite functionals (i.e. functionals whose all critical points have infinite Morse index and co-index). In this talk, we will briefly explain how to instead construct a Morse complex for certain classes of strongly indefinite functionals on a Hilbert manifold by looking at the intersection between stable and unstable manifolds of critical points whose difference of (suitably defined) relative indices is one. As a concrete example, we will consider the case of the Hamiltonian action functional defined by a smooth time-periodic Hamiltonian $H: S^1 \times T^*Q \to \mathbb R$, where $T^*Q$ is the cotangent bundle of a closed manifold $Q$. As one expects, in this case the resulting Morse homology is isomorphic to the Floer homology of $T^*Q$, however the Morse complex approach has several advantages over Floer homology which will be discussed if time permits. This is joint work with Alberto Abbondandolo and Maciej Starostka.