Speaker: R.A. Burke (University of Queensland)

Abstract: Dimension 4 is the first dimension in which exotic smooth manifold pairs appear — manifolds which are topologically the same but for which there is no smooth deformation of one into the other. On the other hand, smooth and piecewise-linear manifolds (manifolds which can be described discretely) do coincide in dimension 4. Despite this, there has been comparatively little work done towards gaining an understanding of smooth 4-manifolds from the discrete and algorithmic perspective. In this talk, I will present some developments in this direction: a new software implementation of an algorithm to produce triangulations of 4-manifolds from handlebody diagrams, as well as a new heuristic for simplifying these triangulations. Using these new software tools, we present small triangulations of exotic 4-manifolds, and related objects. The small size of these triangulations benefit us by revealing fine structural features in 4-manifold triangulations, and time permitting I will discuss recent work towards a structure and decomposition theory for such triangulations.