Research

I work in the field of low-dimensional topology, which includes the study of manifolds of dimension up to four (and occasionally higher) and related questions. In particular, I enjoy thinking about knots (tangled loops in three dimensions) and knotted surfaces (tangled spheres, tori, etc. in four dimensions). Some major conjectures motivating my work are:

• The Four-Dimensional Smooth Poincaré Conjecture: Every 4-manifold homeomorphic (or homotopic) to the standard 4-sphere is diffeomorphic to the standard 4-sphere.

• The Slice-Ribbon Conjecture: Every knot which bounds a disk in the 4-ball bounds a ribbon disk.

• The Smooth Unknotting Conjecture: Every knotted surface in the 4-sphere whose exterior has fundamental group Z is unknotted.

Recently, I have defined weak reducibility for bridge-trisected knotted surfaces, giving a partial classification of 4-bridge trisections in the 4-sphere.