Philip Engel, Harvard University

Title: Cusp Singularities

Abstract: In 1884, Klein initiated the study of rational double points (RDPs), a special class of surface singularities which are in bijection with the simply-laced Dynkin diagrams. Over the course of the 20th century, du Val, Artin, Tyurina, Brieskorn, and others intensively studied their properties, in particular determining their adjacencies---the other singularities to which an RDP deforms. The answer: One RDP deforms to another if and only if the Dynkin diagram of the latter embeds into the Dynkin diagram of the former. The next stage of complexity is the class of elliptic surface singularities. Their deformation theory, initially studied by Laufer in 1973, was largely determined by the mid 1980's by work of Pinkham, Wahl, Looijenga, Friedman and others. The exception was a conjecture of Looijenga's regarding smoothability of cusp singularities---surface singularities whose resolution is a cycle of rational curves. I will describe a proof of Looijenga's conjecture which connects the problem to symplectic geometry via mirror symmetry, and summarize some recent work with Friedman determining adjacencies of a cusp singularity.