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.
Joint Mathematics Meetings, San Diego CA, January 2018
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
Bi-college colloquium (Bryn Mawr and Haverford Colleges), April 2018
Latinos in the Mathematical Sciences, IPAM at UCLA, March 2018
On a lower bound for volume of k-free hyperbolic 3-manifolds,
with P. Shalen.
PAPERS AND PREPRINTS:
The geometry of k-free hyperbolic 3-manifolds, with
P. Shalen. Submitted for publication. Preprint here, arXiv:1802.08350.
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. Submitted for publication. Preprint
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 here, arXiv:1801.09530.
Hyperbolic 3-manifolds with k-free fundamental
group. Topology and its Applications,
173(1):142--156, 2014. Published version here.
Women's History Month. Notices of the American Mathematical
Society, 65(3):268--269, 2018. Published version here.