Here we give a brief survey on recent mathematical works concerning the Faddeev and Skyrme models.One of the most facinating phenomena descibed by these models are the knoted topological soliton solutions which are fundamentally different from many other well-known feild theory models such as instantons and monopoles in the Yang-Mills or the general gauge field theory,bubbles in the nonlinear sigma models or ferromagnetisms and vortices in superconductors and superfluids.In this lecture we shall illustrate some key features of these models that lead to the exisitence of stable knoted solitons and to discuss some possible implications in other problems.
The importance of the conserved quantities were well recoganized in the physical sciences. In this lecture, through several examples, we shall illustrate the fundamental roles played by such conserved quatities in the multiscale analysis. One therefore has to put such quatities into serious considerations also in both modelings and computations.
The Mathematical Biosciences Institute (MBI) was established at The Ohio State University in 2002, with funding from the NSF. The MBI brings mathematicians and statisticians together with bio-scientists from all over the country and the world in order to work on significant problems in biology and medicine. In this talk, I shall give examples where mathematics does contribute to the solution of important problems in the biosciences. (i.e. tumor growth). I shall also briefly describe new mathematical problems, which arise from biological models.
