K(pi,1) via contraction algebras

Usually in algebraic geometry settings, such as for a flopping contraction X–>Spec R, we are interested in establishing contractibility of the stability manifold. Actually, this is not strictly true: we are usually interested in proving contractibility of certain components of the stability manifold of certain subcategories of Db(coh X). Depending on which components, and on which subcategories, the difficulty of the problem varies. I will explain how to approach the contractibility in one such case, by ""mirroring"" the stability manifold from the image of a spherical functor, to the source category of the spherical functor. It turns out that contractibility is much easier there. There are three main corollaries: K(pi,1) for all intersection arrangements in ADE root systems (which includes the Coxeter groups I_n with n=3,4,5,6,8), plus faithfulness of group actions in various settings (the first avoiding normal forms), plus contractibility of stability manifolds in some 3-CY settings. If there is time, I will explain some of the issues in the ""infinite"" case. This is joint work with Jenny August.