My research program investigates the interplay between the geometric
and topological properties of certain hyperbolic 3-manifolds; a recent
collaboration with Peter Shalen will hopefully lend insight into an
improved volume bound for this class of manifolds as well as inform
techniques for arithmetic manifolds.

There are compelling reasons to be interested in hyperbolic 3-manifold topology and the study of Kleinian Groups. To name one: closed, orientable, hyperbolic 3-manifolds are completely determined, up to isometry, by their fundamental group. To name one more, volume is a topological invariant! A comparable statement in Euclidean 2-space would be that any two similar triangles have the same area, which is clearly false. This rigidity afforded to certain hyperbolic structures that is hinted at in both remarks above----namely Mostow Rigidity---sets the stage for a very rich and interesting field of inquiry.

OTHER RESEARCH:

Undergraduate research projects co-advised with my colleague, Dr. Ruth
Davidson, on topics including:

Joshua Hunt, a student of Henry Wilton at Cambridge, has a paper on arxiv here on a counterexample to a conjecture of mine which relates ranks of joins and intersections of two rank-m (m ≥ 3) subgroups of a free group, which stemmed from results in the rank-2 case as shown by Kent and Louder-McReynolds, and by me in the rank-3 case.

RECENT TALKS:

UPCOMING

Joint Mathematics Meetings, San Diego CA, January 2018

PAPERS AND IN PREPARATION: