Understanding moduli spaces is one of the central questions in
algebraic geometry. This talk will survey one of the main aspects of
research in moduli theory — the compactification problem. Roughly speaking,
most naturally occurring moduli spaces are not compact and so the goal is to
come up with geometrically meaningful compactifications. We will begin by
looking at the case of algebraic curves (i.e. Riemann surfaces) and progress
to higher dimensions, where the theory is usually divided into three main
categories: general type (i.e. negatively curved), Calabi-Yau (i.e. flat),
and Fano (i.e. positively curved, where the theory is connected to the
existence of KE metrics). Time permitting, we will use this motivation to
discuss moduli spaces of K3 surfaces (simply connected compact complex
surfaces with a no-where vanishing holomorphic 2-form).

(A joint seminar with the Geometry & Topology Seminar series.)