This project deals with invariants in low-dimensional topology which come from gauge theory and symplectic geometry. The focus will be the relation between these invariants and fibered manifolds, in particular, whether these invariants detect fibered 3-manifolds. It is hoped to prove that Monopole Floer homology detects fibered 3-manifolds, which is the analogue of a known result in Heegaard Floer homology. Another goal of this project is to study a problem of Boileau concerning surgeries on null-homotopic knots.
The shape of the universe is modelled on three and four dimensional manifolds, which are the objects studied in this project. Understanding such shapes is an important step towards the understanding of the universe we live in. In the microscopic world, macromolecules are often visualized as knots and links in the three space. The knot invariants studied in this project thus provide important tools in analyzing the structures of macromolecules, which turn out to be extremely significant in pharmacology.