My field of research is
low dimensional topology,
more specifically, I am interested in (smooth) 4-manifolds and everything that comes with the territory.
Recently, I have been studying certain maps from 4-manifolds to the 2-sphere and their combinatorial descriptions in terms of curve configurations of surfaces.
This has sparked my interest in singularity theory and mapping class groups of surfaces.
Vanishing Cycles and Homotopies of Wrinkled Fibrations (with K. Hayano)
Wrinkled fibrations are certain smooth maps from 4-manifolds to surfaces. To a wrinkled fibration one can associate its base diagram and its vanishing cycles. While it is rather well understood how the base diagram evolves under homotopies between wrinkled fibrations, the behavior of the vanishing cycles is less obvious. We study this problem for merge homotopies and give various applications.
On 4-Manifolds, Folds and Cusps (to appear in Pacific J. Math.)
We study simple wrinkled fibrations, a variation of the simplified purely wrinkled fibrations introduced by Williams, and their combinatorial description in terms of surface diagrams. We show that simple wrinkled fibrations induce handle decompositions on their total spaces which are very similar to those obtained from Lefschetz fibrations. The handle decompositions turn out to be closely related to surface diagrams and we use this relationship to interpret some cut-and-paste operations on 4-manifolds in terms of surface diagrams. This, in turn, allows us classify all closed 4-manifolds which admit simple wrinkled fibrations of genus one, the lowest possible fiber genus.