The focus of my research is the study of objects called manifolds, of which knots, surfaces, three-dimensional space, and four-dimensional space-time are all examples. I am particular interested in how manifolds of different dimension interact with each other. For example, when we study knots, we are really studying all of the ways to embed a one-dimensional manifold (a circle) in three-dimensional space. Similarly, one can ask interesting questions about how to take a surface (a two-dimensional manifold) and embed it in four-dimensional space. Such knotted surfaces are particularly difficult to conceive of, but can be visualized with the help of moving-picture diagrams and related constructions.
The study of manifolds can be approached from two directions: topology and geometry. Euclidean geometry of the plane is familiar to many of us from our high school studies, but there are other interesting geometries. For example, hyperbolic geometry provides a different take on planar geometry. The geometry of three-dimensional manifolds is unbelievably rich area of study, and the geometry of four-dimensional manifolds is largely unexplored.
Topology is similar to geometry, but is not concerned with distances or angle measures. Instead, when one studies topology, one imagines that the spaces being studied are made of rubber and can be stretch and pulled and deformed. To a topologist, a donut is the same as a coffee mug, since both are roughly globular objects that have a single distinguishing hole in them. Topologist are often interested in understanding all the different ways to cut up these rubbery objects and glue them back together in interesting ways.
Though topology and geometry appear quite different on the surface, there are many important and beautiful relationships between the two approaches. For example, the topology of three-dimensional manifolds can be understood entirely in terms of their geometric properties, as can surfaces.
My research pulls from the vast fields of topology and geometry in an effort to understand the connection between knots, surfaces, and manifolds in dimensions three and four.