Grant: Canada Research Chairs

The purpose of this programme is to investigate questions relating to knot theory and low dimensional manifolds. Manifolds of all dimensions have long been objects of fundamental importance in mathematics. However, the behaviour of manifolds and their topology is heavily dependent on their dimension, with the tools required to study questions in low (less than four) dimensions being substantially different to those used in higher dimensions. My work is primarily aimed at advancing our understanding of a number of questions in low dimensions. The first and largest component of the programme is related to studying Dehn surgery. A knot is a tangled up circle in 3-dimensional space. We perform Dehn surgery on a knot by fattening it up to form a solid tube, cutting out the interior of this tube and replacing it with a new solid tube. Despite the simple nature of this operation, there is still much that we do not understand about how it can change the topology and geometry. I will be studying questions of the form: 1. Which manifolds arise by surgery on a knot? 2. Can we classify all knots which surger to a given 3-manifold? Questions of this form naturally arise throughout low-dimensional topology and Dehn surgery results frequently have applications to other areas of low dimensional topology such as classical knot theory. Secondly, the programme will be studying how the topology of a 4-manifold is constrained by its boundary. An important invariant of 4-manifolds are their intersection forms, which is an object that counts intersection points between 2-dimensional surfaces contained in the 4-manifold. I will be studying questions about the relationship between the intersection form of a 4-manifold and the 3-manifold on its boundary. In particular, I will be studying when the intersection form can be definite, as this is a condition that arises naturally in many applications. For this proposal, the principal applications will be to the theory of singularities.