My research is in the area of geometric group theory, which overlaps heavily with low-dimensional topology.

Group theory is the technical mathematical term for the study of symmetry. Geometric group theorists aim to study groups—collections of symmetries—by realising them as symmetries of interesting geometric or topological spaces.

To learn more about geometric group theory, you could look at the blog written by my students at the University of Texas. To find out about my contribution to the subject, you can read my papers.

Jon McCammond keeps a list of geometric group theorists.

A wiki entitled Problems in geometric group theory is under development. For a briefer list of open problems in the area, look at Mladen Bestvina's list.