Genetic instability is a major cause for abnormal cell
replication and carcinogenesis. But the mutant cells that replicate abnormally are also weaker and die at a more rapid rate. Hence, genetic instability is a two-edge sword in inducing cancer. The determination of the best time varying cell mutation rate for the fastest time to cancer can be formulated as a nonlinear optimal control problem. As generally the case for nonlinear optimal control problems, there is no general sure fire method for the solution of our problem. The talk will show how the unique solution of the problem can be obtained by ad hoc elementary analyses of the relevant boundary value problem for a systems of nonlinear differential equations. The
method of solution illustrates how important problems in application can be solved by elementary use of classical analysis.

Nonlinear Diffusions exhibit a variety of interesting and
sometimes unexpected behaviors. I shall
give a brief overview of this broad research area and emphasize some
applications.

In the numerical simulation of many practical problems in physics and
engineering, it is always an active research topic to efficiently and effectively
solve a set of partial differential equations (PDEs), which represents the
mathematical model of practical problems concerned. This talk is on the study of
advanced numerical methods for partial differential equations that arise from
scientific and engineering applications. The theme of research is on the
development, application and analysis of multilevel adaptive finite element methods.