Richard Webb

Equators of the two-sphere and area-preserving homeomorphisms

The curve complex is an important tool in the study of mapping class groups of surfaces (and several other areas of low-dimensional topology and geometry). In this talk, we introduce analogues of the curve complex in order to study the group of area-preserving homeomorphisms (and/or Hamiltonian diffeomorphisms) of the two-sphere. How this relates to the dynamics of area-preserving homeomorphisms, and symplectic geometry, is particularly curious, especially because the original curve complex tells us lots of dynamical and geometric information about mapping classes of surfaces. Using these new tools, we are able to construct new quasimorphisms on the group of Hamiltonian diffeomorphisms of the two-sphere, which are C^0 continuous and vanish on the stabiliser of the standard equator. I will discuss an application of this regarding the geometry of simple closed curves that separate the sphere into two components of equal area (i.e. equators of the sphere), and how this relates to the Equator Conjecture. Joint work with Yongsheng Jia.