Curve contractibility via non-commutative deformations

Deciding whether a subvariety of an algebraic variety is contractible is a deep problem of algebraic geometry. Even when the subvariety is a single smooth rational curve C, the question is extremely subtle. In this talk, I will assume moreover that the ambient variety is a Calabi-Yau threefold. When C is contractible, its Donovan-Wemyss contraction algebra (which pro-represents the deformation theory of C) governs much of the geometry. Our expectation is that deformation theory not only controls contractibility but detects it, even when C is not known to contract. To investigate the deformation theory of C, we use technology developed by Brown and Wemyss to describe a local model for C. I will introduce the key ideas and tools appearing in this problem, the leading conjectures, and I will describe the (partial) results I obtained so far in collaboration with G. Brown and M. Wemyss.