We propose virtual element methods for solving one- or multi-domain elliptic interface problems using interface-fitted meshes. The main challenge is to design an efficient and robust mesh that can capture certain properties while preserving arbitrary complex geometries of the interface. Moreover, it is a tricky problem in classical finite element methods when the domain is decomposed into tetrahedra due to the existence of slivers in three dimensions. In our mesh generation, every element in three dimensions could be any polyhedron instead of tetrahedron. Then we apply virtual element methods for solving elliptic interface problems with solution and flux jump conditions. The purpose of using virtual element methods rather than classical finite element methods is that every element could be a different shape. We use multi-grid solvers to solve the discrete system. Lastly, numerical examples demonstrating the theoretical results for linear elements are shown.

Sums of two Cantor sets arise naturally in homoclinic bifurcations, Markov and Lagrange dynamical spectra, and the spectrum of the square Fibonacci Hamiltonian. In the 1970s Palis conjectured that for generic pairs of regular Cantor sets either the sum has zero Lebesgue measure or else it contains an interval. This problem is known to be extremely difficult, and is still open for affine Cantor sets. In this talk, we will discuss the history of sums of two Cantor sets, and also introduce my recent results about sums of two homogeneous Cantor sets. 

In this talk, I will introduce a novel low dimensional manifold model for image processing problem.This model is based on the observation that for many natural images, the patch manifold usually has low dimension structure. Then, we use the dimension of the patch manifold as a regularization to recover the original image. Using some formula in differential geometry, this problem is reduced to solve Laplace-Beltrami equation on a manifold. The Laplace-Beltrami equation is solved by the point integral method. Numerical tests show that this method gives very good results in image inpainting, denoising and super-resolution problem. This is joint work with Stanley Osher and Wei Zhu.

We will compute the mantle, the intersection of all grounds, of the minimal class size mouse with a Woodin cardinal and a Strong cardinal.