Research

See my papers on arXiv. To avoid spamming the arXiv, the "preprint" version here is usually the most up-to-date. An ** is attached to the bracket [...] containing the links if the discrepancies are more than typos and minor changes.

• Effective equidistribution of intersection points in hyperbolic manifolds (joint with Tina Torkaman).
  [preprint]
  we prove effective equidistribution of transverse intersection points between totally geodesic submanifolds of complementary dimensions in an arithmetic hyperbolic manifold with respect to the hyperbolic volume measure, as the volume of the submanifolds tends to infinity.

• Disk patterns, quasi-duality and the uniform bounded diameter conjecture (joint with Yusheng Luo).
  Submitted. [preprint | arxiv]
  We show that the diameter of the image of the skinning map on the deformation space of an acylindrical reflection group is bounded by a constant depending only on the topological complexity of the components of its boundary, answering a conjecture of Minsky in the reflection group setting. This result can be interpreted as a uniform rigidity theorem for disk patterns. Our method also establishes a connection between the diameter of the skinning image and certain discrete extremal width on the Coxeter graph of the reflection group.

• Circle packings, renormalizations and subdivision rules (joint with Yusheng Luo).
  Submitted. [preprint | arxiv]** The preprint has been reorganized, and includes a revised introduction with some new corollaries.
  In this paper, we use iterations of skinning maps on Teichmüller spaces to study circle packings. This allows us to develop a renormalization theory for circle packings whose nerves satisfy certain subdivision rules. We characterize when the skinning map has bounded image. Under the corresponding condition, we prove that the renormalization operator is uniformly contracting. This allows us to give complete answers for the existence and moduli problems for such circle packings. The exponential contraction has many consequences. In particular, we prove that homeomorphisms between any two such circle packings are asymptotically conformal.

• Isoperiodic foliation of the stratum H(1,1,-2) (joint with Gianluca Faraco and Guillaume Tahar).
  Submitted. [preprint | arxiv]
  In this paper, we give a complete description of the isoperiodic leaves in stratum H(1,1,-2) of meromorphic 1-forms with two simple zeroes and a pole of order two on an elliptic curve. Topologically, each leaf is a Loch Ness monster.  Some features of the leaves include essential singularities, a wall-and-chamber structure, and a ramified cover of infinite degree over the flat torus defined by the lattice of absolute periods.

• On quasiconformal non-equivalence of gasket Julia sets and limit sets (joint with Yusheng Luo).
  Ergodic Theory Dynam. Systems, to appear [preprint | arxiv | journal]
  This paper studies quasiconformal non-equivalence of Julia sets and limit sets. We proved that any Julia set is quasiconformally different from the Apollonian gasket. We also proved that any Julia set of a quadratic rational map is quasiconformally different from the gasket limit set of a geometrically finite Kleinian group.

• Geodesic planes in a geometrically finite end and the halo of a measured lamination (joint with Tina Torkaman).
  Adv. Math. 456 (2024), article no. 109902 [preprint | arxiv | journal]
  In this paper, we investigate two related problems: (I) The growth rate of intersection of geodesic rays with a given measured geodesic lamination on a hyperbolic surface of finite area; and (II) The topological behavior of geodesic planes outside the convex core of a geometrically finite manifold. Our main results establish the existence of exotic rays for the first problem, and exotic roofs for the second. The former have the unexpected property that they have finite intersection number with the lamination, even though they meet the lamination infinitely often; and the latter are geodesic planes disjoint from the convex core, yet cannot be separated from it by a support plane of the convex core boundary. These two objects are related: we show the existence of exotic rays is necessary for the existence of exotic roofs.

• Elementary planes in the Apollonian orbifold.
  Trans. Amer. Math. Soc. 376 (2023), pp. 453 -- 506 [preprint | arxiv | journal]
  In this paper, we give a complete list of elementary planes in the Apollonian orbifold M_A, and show that (1) the area of an elementary plane in M_A is uniformly bounded above; (2) the union of all elementary planes is closed. The existence of these elementary planes leads to the following failure of equidistribution: there exists a sequence of closed geodesic planes in M_A limiting only on a finite union of closed geodesic planes.

• Train tracks, entropy, and the halo of a measured lamination (joint with Tina Torkaman).
  [preprint | arxiv]
  This paper is subsumed by "The halo of a measured..." above. Only Problem (I) is studied here, but there are alternative constructions included in this version that might be of interest.

• Existence of an exotic plane in an acylindrical 3-manifold.
  Math. Res. Lett. 30 (2023), pp. 611 -- 631 [preprint | arxiv | journal]
  As an application of the paper below, we show that there exists an acylindrical 3-manifold and a geodesic plane that is closed in the convex core of the manifold but not closed in the whole manifold.

• Construction of acylindrical hyperbolic 3-manifolds with quasifuchsian boundary.
  Exp. Math. 31 (2022), pp. 883 -- 896 [preprint | journal]
  In this paper, we construct a one (real) dimensional deformation of an acylindrical hyperbolic 3-manifold. A shorter version focusing only on the construction is here.

• Geodesic planes in hyperbolic 3-manifolds, PhD thesis, with figures of high (~45mb) / low (~6mb) resolution.

From undergrad:

• Generalized factorials on the image set of polynomials with integer coefficients.
  [final report]
  Directed Studies in Mathematics at HKU, under the supervision of Prof. Kai Man Tsang.

• Smale's mean value conjecture for finite Blaschke products (joint with T. W. Ng).
  J. Anal. 24 (2016), pp. 331 -- 345. [preprint | arxiv | journal]
  Summer research fellowship at Faculty of Science, HKU, under the supervision of Prof. Tuen Wai Ng.

• Colloquium, William & Mary, Sept 2025
  Uniform quasi-duality of moduli on graphs

• Fields-CNAM Nonlinear Days 2025: Structures and Processes: Dynamical, Geometric, and Random, Fields Institute, Aug 2025
  Geometry and dynamics of infinite circle packings [videos]

• Dynamics Seminar, Cornell University, May 2025
  Effective equidistribution of intersection points in hyperbolic manifolds

• Dynamical Systems Seminar, Stony Brook University, April 2025
  Effective equidistribution of intersection points in hyperbolic manifolds

• Group Actions and Dynamics Seminar, University of Wisconsin Madison, Mar 2025
  Disk patterns, discrete modulus and the uniform bounded diameter conjecture

• Dynamics Seminar, University of Chicago, Feb 2025
  Circle packings and Kleinian reflection groups [slides]

• Topology Seminar,  Shanghai Center for Mathematical Sciences, Jan 2025
  Disk patterns, discrete modulus and the uniform bounded diameter conjecture

• Dynamical Systems Seminar, Stony Brook University, Oct 2024
  Disk patterns, discrete modulus and the uniform bounded diameter conjecture

• Exchange Seminar, Seoul National University, July 2024
  Existence and rigidity of infinite circle packings

• Computational Aspects of Thin Groups, IMS, National University of Singapore, Jun 2024
  (Lightning talk) Combinatorial rigidity of circle packing limit sets

• Dynamical Systems and Fractal Geometry Conference, University of North Texas, May 2024
  Rigidity of infinite circle packings

• Early Career Math Colloquim, University of Oklahoma, April 2024
  Existence and rigidity of infinite circle packings

• Complex Analysis and Dynamics Seminar, CUNY-Graduate Center, Oct 2023
  Quasiconformal nonequivalence of gasket Julia sets and limit sets

• Dynamics, Rigidity and Arithmetic in Hyperbolic Geometry, ICERM, May 2023
  Minicourse on geodesic planes in hyperbolic 3-manifolds of infinite volume
  Videos: Lecture 1, Lecture 2.

• Geometry & Topology Seminar, University of Milano-Bicocca, March 2023
  Geodesic planes in quasifuchsian manifolds and geodesic rays for measured laminations

• Spring Topology and Dynamics Conference, Geometric Group Theory Special Sessions, Rhodes College, March 2023 (Virtual)
  Contact graphs of Kleinian and Julia gaskets

• Postdoc seminar, Stony Brook University, Oct 2022
  The pressure metric on quasifuchsian space

• Dynamical Systems Seminar, Stony Brook University, Sept 2022
  Geodesic planes in quasifuchsian manifolds and geodesic rays for measured laminations

• Lightning talks, Geometry, Arithmetic, and Groups (a conference in honor of A. Reid), Jun 2022
  Geodesic planes in hyperbolic 3-manifolds of infinite volume

• Geometry-Topology Seminar, Oregon State University, Jan 2022
  Elementary planes in the Apollonian orbifold

• MPIM Oberseminar, MPIM Bonn, Dec 2021
  Measured laminations, train tracks, and geodesic planes

• Oberseminar Differentialgeometrie, MPIM Bonn, Nov 2021
  Geodesic planes in hyperbolic 3-manifolds of inifinte volume

• Geometric group theory and topology seminar, Tufts University, Feb 2021
  Geodesic planes in hyperbolic 3-manifolds of infinite volume

• Group action and dynamics seminar, Yale University, Oct 2020
  Geodesic planes in acylindrical and quasifuchsian 3-manifolds

• Informal geometry and dynamics seminar, Harvard University, Oct 2019
  Elementary surfaces in the Apollonian manifold

• Group action and dynamics seminar, Yale University, Oct 2018
  Construction of an acylindrical hyperbolic 3-manifold with quasifuchsian boundary

• Geometry seminar, HKU, June 2018
  Construction of an acylindrical hyperbolic 3-manifold with quasifuchsian boundary

• Informal geometry and dynamics seminar, Harvard University, Apr 2018
  Andreev's theorem on hyperbolic polyhedra and Kleinian reflection groups

• Undergraduate Summer Research Colloquium, HKU, Oct 2014
  Smale's mean value conjecture for finite Blaschke products