My formal research statement can be viewed here.
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.
Discrete Morse Theory, Vector Fields, and Materials Science
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.
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.
Institute for Advanced Study, Women and Math Program, May 2017
JL Doob Colloquium at University of Illinois at Urbana-Champaign, May 2017
Women in Mathematics Symposium at the University of Chicago, February 2017
Joint Mathematics Meetings, San Diego CA, January 2018
PAPERS AND IN PREPARATION:
Nerves, $k$-free groups, and Quantitative Mostow Rigidity, with
P. Shalen. Preprint now available here.
A combinatorial method for connecting BHV spaces representing different
numbers of taxa}, with J. Bi, R. Davidson, M. Delcourt, C.Monical,
J. Sanchez, Y. Ren, and S. Zha. preprint at
http://arxiv.org/abs/1708.02626. Submitted for publication. Preprint available here.
Image-based data analysis via discrete Morse theory and
persistent homology, with R. Davidson, C. Du, A. Manawa, N. Rasekh,
C. Szul, T. Wibowo. To be submitted. Preprint now available here.
Hyperbolic 3-manifolds with k-free fundamental
group. Topology and its Applications, 173(1):142--156.