Seidel-Thomas twists along a spherical object are certain autoequivalences of the derived category D(X) of an algebraic variety X. Roughly, they are mirror symmetry analogues of Dehn twists along Lagrangian spheres on a symplectic manifold. A classical example of a spherical object is a P^1 on a K3 surface.

In this talk, I will give a quick introduction to these objects, and then report on a recent joint work with Rina Anno (UPitt). Our starting point are examples such as the categorical braid group action constructed by Khovanov and Thomas on cotangent bundles to flag varieties. They cook up this action from certain natural P^1-fibrations in a way which strongly suggests existence of a "family" version of Seidel-Thomas twists. It turns out that these are a special case of a more general notion, which we construct via noncommutative methods inspired by Kontsevich: the twist along a functor into D(X). Geometrically, this corresponds the twist along a subvariety of X fibered over a non-trivial base.