In this talk, we will consider the homogenization of $p$-Laplacian with
obstacles in perforated domain, where the holes are periodically
distributed and have random size. And we also assume that the $p$-capacity
of each hole is stationary ergodic.

In this talk, we will describe some sharp geometric
inequalities on the Heisenberg group and CR spheres, which includes
the best constants and extremal functions for the Moser-Trudinger
inequalities on Heisenberg group, CR spheres, and Adams' high order
Moser's inequality. We will also discuss some recent work on sharp
Moser's inequalities on unbounded domains in CR setting.

A survey of automatic continuity in Banach algebras with a focus on the case of a derivation on a Banach Jordan triple. (Joint work with Antonio Peralta)

Cheng-Yau's local gradient estimate for harmonic functions is of fundamental importance in geometric
analysis. I will discuss recent work on local gradient estimate for p-harmonic functions on Riemannian
manifolds. This is a joint work with Lei Zhang at Univ. of Florida.

In this talk, I will discuss the quadruple junction solutions in
the entire three dimensional space to a vector-valued Allen-Cahn equation
which models multiple phase separation.

In the joint work with X. Huang, a monotonicity property has been detected for a CR embedding from a Levi non-degenerate hypersurface into another one with the same signature. Roughly speaking, the CR embedding decreases the Chern-Moser-Weyl curvature along the null space of the Levi-form. The criterion allows us to construct many algebraic Levi non-degenerate hypersurfaces non-embeddable into hyperquadrics of the same signature.

A new characterization for Carleson measures in terms of integration on a non-tangential cone is established. Applications on the bounedness of area operator and Volterra operator on Hardy spaces are discussed.

In the talk, I give some my recent results on
the regularity for the solution of $p$-Laplacian
on Heisenberg group.

Motivated by the infinite horizon discounted problem, we study the convergence of solutions of the Hamilton Jacobi equation when the discount vanishes. If the Aubry set consists in a
finite number of hyperbolic critical points, we give an explicit expression for the limit.