In this talk, I will introduce briefly a moving mesh method based on harmonic mapping. As a rare character, the unique existence of the harmonic mapping is the basic motivation for us to develop this method. The method is implemented in finite element and an iterative procedure is adopted to avoid mesh tangling caused by numerical factors. Our method can move the mesh interior the domain and mesh on the boundary in coupling for both 2D and 3D problems. The moving mesh module can be a black box added on the whole solver to the PDE under consideration that it is very convenient for coding - no modifications to the solver of the PDE are required. The inter-mesh mesh updation is implemented by a linear convection equation instead of generally adopted interpolation methods, thus the method can be easily to applied to problem as incompressible Navier-Stokes equation where the divergence free interpolation can be a big problem, and problem as conservation laws where the conservative interpolation is not trival to be implemented. Numerical results including viscos Burgers equation, reaction-diffusion equation, incompressible Navier-Stokes equation and its coupling with level set method will be shown.

Probability of a gap in the spectrum of a matrix from the
Gaussian Unitary Ensemble is given by a Fredholm determinant. Its asymptotics when the gap becomes large is an interesting problem related to Painleve equations, random permutations, etc. These asymptotics for the gap in the bulk of the spectrum were conjectured by Dyson. The proof was given over the years by Widom, Deift, Its , Zhou, and the speaker. In particular, the proof of the multiplicative constant in the asymptotics
was the last difficulty recently resolved. I will explain the method of determining this constant and the rest of the asymptotics (applicable also to other important Fredholm, and also Hankel, and Toeplitz determinants where the corresponding constant is not yet determined). The method uses the Riemann-Hilbert approach. This part of the talk will be based on the works of Deift, Its, Zhou, and the speaker.

We will discuss two approaches for modeling cancer growth. One approach is to employ a deterministic space-independent model that includes separate components to represent specific and non-specific immune function. The other approach employs a spatially dependent hybrid cellular-automata (HCA) model whose rules are driven by cellular metabolic function. For the determinstic model, numerical simulations of mixed chemo-immuno therapy and vaccine therapy using both mouse and human parameters are presented. We illustrate situations for which neither chemotherapy nor immunotherapy alone are sufficient to control tumor growth, but in combination the therapies are able to eliminate the entire tumor burden. The HCA model is in its initial stages of development, and preliminary results will be presented.