Research

My research is primarily concerned with geometric structures on surfaces and 3-manifolds. In particular, my interests include :

• hyperbolic geometry; identities on hyperbolic manifolds
• Teichmüller theory and Kleinian groups
• computational approaches to hyperbolic 3-manifolds
• convex real projective structures; Hitchin representations
• low-dimensional topology; mapping class groups
• circle packings on complex projective structures
• knowledge graph emebddings and machime learning in geometric spaces

My recent and active projects include volumes of link complements in Seifert fibered spaces (with with Tommaso Cremaschi) and José Andrés Rodíguez Migueles)), circle packings of projective structures (with Jean-Marc Schlenker), random geodesic triangles on surfaces (with Olivier Glorieux), and deformation spaces of infinite-type hyperbolic 3-manfiolds (with Tommaso Cremaschi). My works also incldues several projects classifying large families of hyperbolic 3-manifolds with Rob Meyerhoff, David Gabai, Nathaniel Thurston, and Robert Haraway.

Below, you can find a list of my publication and preprints.

In preparation

• Bounds on exceptional Dehn surgery with David Gabai and Marc Lackenby.
• On canonical knot complements in projective bundles with Tommaso Cremaschi and Jacob Intrater.
• Circle packing complex projective tori
• Quasi-conformal deformation space of infinite type hyperbolic 3-manifolds with Tommaso Cremaschi.

Preprint

• Hyperbolic 3-manifolds of low cusp volume
  with David Gabai, Robert Haraway, Rob Meyerhoff, and Nathaniel Thurston.
• Volumes and filling collections of multicurves
  with Tommaso Cremaschi and José Andrés Rodíguez Migueles. (submitted | arXiv)
• Properness for circle packings and Delaunay circle patterns on complex projective structures
  with Jean-Marc Schlenker. (submitted | arXiv)
• Improved bounds for average bending on the convex core of a Kleinian group. (pdf)

Publications

• The Two Eyes Lemma: a linking problem for table-top necklaces
  with David Gabai and Rob Meyerhoff. (to appear in Graphs and Combinatorics | arXiv)
• Random triangles on surfaces
  with Olivier Glorieux. (to appear in Rocky Mountain Journal of Mathematics | arXiv)
• Basmajian’s identity for higher Teichmüller-Thurston theory
  with Nicholas G. Vlamis.
  Journal of Toplogy, Volume 10, Issue 3, September 2017 (published | arXiv)
• The Bridgeman-Kahn identity for hyperbolic manifolds with cusped boundary
  with Nicholas G. Vlamis.
  Geometriae Dedicata, 2017 (published | arXiv)
• A new bound for Sullivan’s Theorem for simply connected domains
  with Martin Bridgeman and Richard D. Canary.
  Proceedings of the London Mathematical Society, January 2016 (published | arXiv)

Notes, slides, and not for publication

• Seminar talk on Properness for circle packings and Delaunay circle patterns on complex projective structures at Rutgers Newark. (pdf)
• Conference talk on Hyperbolic 3-manifolds of low cusp volume at NUS. (pdf)
• Informal talk about hyperbolic geomtery at the University of Luxembourg. (pdf)
• Convex hulls in hyperbolic 3-space and generalized orthospectral identities. (pdf)
  PhD thesis under Martin Bridgeman defended in spring 2016.
• Curves, Surfaces, and Hyperbolic 3-Manifolds. (pdf)
  Notes on a minicourse by Yair Minsky at the GEAR Junior Retreat 2014.
• Poster for A new bound for Sullivan’s Theorem for simply connected domains. (pdf)
• Informal notes on the cohomology of compact Lie groups. (pdf)
• Persistence and Computation of the Cup Product. (pdf)
  Undergraduate honors thesis under Mikael Vejdemo-Johansson.