Research

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.

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:

  • Phylogenetics
  • Discrete Morse Theory, Vector Fields, and Materials Science

    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:

  • 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

    UPCOMING TALKS:

  • Bi-college colloquium (Bryn Mawr and Haverford Colleges), April 2018
  • Latinos in the Mathematical Sciences, IPAM at UCLA, March 2018


    IN PREPARATION:

  • 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 here, arXiv:1708.02626.

  • 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.

    OTHER MEDIA:

  • Women's History Month. Notices of the American Mathematical Society, 65(3):268--269, 2018. Published version here.

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