Inanc Baykur: Topological complexity of symplectic 4-manifolds and Stein fillings

Abstract: Following the ground-breaking works of Donaldson and Giroux, Lefschetz pencils and open books have become central tools in the study of symplectic 4-manifolds and contact 3-manifolds. An open question at the heart of this relationship is whether or not there exists an a priori bound on the topological complexity of a symplectic 4-manifold, coming from the genus of a compatible Lefschetz pencil on it, and a similar question inquires if there is such a bound on any Stein filling of a fixed contact 3-manifold, coming from the genus of a compatible open book. We will present our solutions to both questions, making heroic use of positive factorizations in surface mapping class groups of various flavors. This is joint work with J. Van Horn-Morris.