Abstract:In this talk, we first introduce various models to study what a generic 3-manifold looks like. We then focus on the Heegaard splitting model of 3-manifolds, equipped with geometric complexity using Teichmuller metric. The main result is that the Hempel distance of a generic Heegaard splitting goes linearly to the infinity. In particular, generic 3-manifolds are hyperbolic in this mode...
Abstract:The volume of a divisor on an algebraic variety measures the growth rate of the dimension of the space of sections of tensor powers of the associated line bundle. In the case of certain Calabi-Yau N-folds possessing a large group of pseudo-automorphisms, we show that the behavior of the volume can be highly oscillatory as the divisor class approaches the boundary of the pseudo-effecti...