Award-winning talks by SMP students

Speaker: Tara Kemp
Affiliation: University of Queensland

Latin hypercubes realizing integer partitions

Abstract

L. Fuchs asked the following question: If n is any positive integer and n=n_1+n_2+…+n_k any fixed partition of n, is it possible to find a quasigroup Q of order n which contains subquasigroups Q_1,Q_2,…,Q_k of orders n_1,n_2,…,n_k respectively, whose set theoretical union is Q? Such a quasigroup is equivalent to a latin square of order n with disjoint subsquares of orders n_1,n_2,…,n_k, and the existence of these latin squares is a partially solved problem. This problem of realizing a partition in a latin square can be extended to latin cubes with disjoint subcubes of the same orders. In this talk, we will discuss the existence problem for latin cubes and how the problem changes as the number of dimensions increases.

Speaker: Rhuaidi Burke
Affiliation: University of Queensland

Developments in Computational 4-manifold Topology

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 triangulations of exotic 4-manifolds, and related objects.