• Teichmüller theory via random simple closed curves (joint work with Curtis T. McMullen), Its writing is in Preparation

We introduce a canonical geodesic current K_X associated to each point X in the Teichmüller space Tg of genus g hyperbolic surfaces. This current is defined as the Thurston-average of measured laminations in the unit ball, normalized so that its self-intersection number equals one. We show that the map sends X to K_X defines a proper embedding of Tg into the space of geodesic currents. Moreover, this embedding yields a compactification of Teichmüller space by projective measured laminations PML. This compactification agrees with Thurston's compactification on a dense subset of PML but differs at infinitely many points. 

• Effective equidistribution of intersection points in hyperbolic manifolds, (joint work with Yongquan Zhang), arXiv

In this project, we establish effective equidistribution of transverse intersection points between totally geodesic submanifolds of complementary dimensions in a finite-volume hyperbolic manifold with respect to the hyperbolic volume measure, as the volume of the submanifolds tends to infinity.

• Geodesic planes in a geometrically finite end and the halo of a measured lamination (Joint work with Yongquan Zhang), Advances in Mathematics (accepted 2024) [ journal | arXiv]

Recent works have shed light on the topological behavior of geodesic planes in the convex core of a geometrically finite hyperbolic 3-manifold M of infinite volume. In this project, we focus on the remaining case of geodesic planes outside the convex core of M, giving a complete classification of their closures in M.

• Intersection points of the closed geodesics on finite volume hyperbolic surfaces, IMRN (accepted 2025) [journal | arXiv]

Let X be a complete hyperbolic surface of finite area. In this project, we establish that the intersection points of closed geodesics with length  <T are equidistributed on X as T approaches infinity.

*When X has cusps, we need to show that there is no escape of mass to the cusps, and the challenge is that the intersection number has different behavior when there are cusps. When X is compact, the intersection number is bounded by a multiple of the product of lengths; however, in the presence of cusps, as illustrated in the picture, the intersection number can be exponential. 

• Intersection number, length, and systole on compact hyperbolic surfaces, Geometriae Dedicata (accepted 2025).[journal | arXiv]

The interaction strength I(X) of a compact hyperbolic surface X is the best upper bound for the intersection number of two closed geodesics divided by the product of their lengths. Let M_g be the moduli space of compact hyperbolic surfaces of genus g and sys(X) the length of a shortest closed geodesic on X in M_g. In this project, we determine the asymptotic behavior of I(X) as X approaches infinity in M_g, in terms of sys(X). We also determine the approximate behavior of the minimum of I(X) over M_g, as g goes to infinity. 

*We can see that I(X) is approximately the ratio between the intersection number and the product of the lengths of the curves  in the picture.