Lecture I. The operation of gluing manifolds along their boundaries plays a basic role in all of geometric topology. It is used in existence proofs for hyperbolic structures on 3-manifolds (for example in Thurston's work on his geometrization conjecture) and there is much interest in obtaining quantitative control of such operations. That is, given two known manifolds and an identification of their boundaries that yields a hyperbolic manifold, give good estimates of its geometric features. I will sketch aspects of this question throughout the history of the field, including Dehn surgery, Thurston's Skinning Map and the combinatorial geometry of mapping class groups.
Lecture II. When a hyperbolic 3-manifold fibers over the circle, its geometric features can be read from the fine structure of its monodromy map, specifically the "subsurface projections" of the stable and unstable foliations to the arc complexes of subsurfaces of the fiber. While this correspondence is useful when the topological type of the fiber is fixed, it is not well-understood in general. A good laboratory for studying this is a single 3-manifold that fibers in infinitely many different ways, as organized by Thurston's norm on homology. In this setting there are canonical triangulations due to Agol, which can be studied very explicitly via a construction of Gueritaud. We explore how the subsurface projections of monodromies for all the fibers can be seen in the structure of this triangulation, and how this leads to a nice combinatorial picture with estimates that do not depend on complexity of the fibers. Joint work with Sam Taylor.