K3 surfaces are amongst the most studied surfaces in algebraic geometry. What makes the geometry of a K3 so interesting is that it carries a nondegenerate holomorphic 2-form: thus any K3 is a compact holomorphic symplectic manifold. Examples of such manifolds are quite rare, and their study and classification is an active area of research.
A particular example of a homolorphic symplectic manifold (discovered by Beauville) is the Hilbert scheme parametrizing subsets (or subschemes) of $n$ points on a K3 surface. While any K3 varies in a 20-dimensional family, has a 21-dimensional space of deformations, and it is an open problem to give a geometric description of these additional deformations. This talk will attempt to explain how this extra modulus may be seen to arise from ``noncommutative’’ deformations of $X$.
The presentation is aimed at a broad audience.