I will speak about recent work with Michael Wemyss (arXiv:1309.0698), applying noncommutative deformation theory to study the birational geometry of 3-folds. In particular, I will explain how every flippable or floppable rational curve in a 3-fold has a naturally associated algebra of noncommutative deformations, even in the singular setting. We investigate the properties of this algebra, and indicate how to calculate it in examples using quiver techniques. This gives new information about the (commutative) geometry of 3-folds, and in particular provides a new tool to differentiate between flops.
As a further application, we show how the noncommutative deformation algebra controls the homological properties of a floppable curve. In this setting, work of Bridgeland and Chen yields a Fourier-Mukai flop-flop functor which acts on the derived category of the 3-fold (assuming any singularities are at worst Gorenstein terminal). We show that this functor can be described as a spherical twist about the universal family over the noncommutative deformation algebra.
In the second part, I will talk about further work in progress, and explain some more technical details, such as the use of noncommutative deformation functors, and the categorical mutations of Iyama and Wemyss. If there is time, I will also give some higher-dimensional examples, and discuss situations involving chains of intersecting floppable curves. In this latter case, deformations, intersections and homological properties are encoded by the path algebra of a quiver, generalizing the algebra of noncommutative deformations.