This course is a problem-oriented course. It is designed for those students who are interested
in the interaction of mathematics and science,and plan to do research in this direction. Applied
mathematics is a huge eld and is growing prosperously. In the International Congress of Industrial
and Applied Mathematics 2011 at Vancouver, there are more than 3000 talks covering mathematical
applications raging from material science, biological science, data science to social science.
In this course, I shall focus only on one current research topics, the interface problems. I will
start from basic background, following literature study and computation experiment, then provide
some small research projects.
During this problem-oriented study, you will learn three basic tools in applied mathematics,
namely, mathematical modeling, applied analysis, and computations.

1.1 Interface Problems
Interface problems are those physical problems whose solutions are composed of several components
separated by interfaces. These interfaces could be material interfaces, phase boundaries,
flame fronts,
physical boundaries, etc. Sometimes, they may dynamically move. The interface problems appear
in fluid mechanics, solid mechanics, electrodynamics, material science, biochemistry, etc. They are
of fundamental in science and engineering. In this topics, we shall focus on two subtopics: (i)
fluid-structure interaction, (ii) structure and function of proteins, and (iii) Bose-Einstein condensates.
(i) Fluid-structure interaction
1. Basic fluid mechanics: Stokes flows and Navier-Stokes flows
2. Computation of Stokes flows and Navier-Stokes flows
3. Basic elasticity
4. Allan-Cahn and Cahn-Hilliard models for interfaces
5. Structure of interfaces
6. Geometry and evolution of interfaces: level set method
7. Diffusion interface methods
8. Fluid-structure interaction
9. Surfacants
10. Complex fluids

(ii) Structure and function of proteins.
1. Basic electrostatics
2. Numerical method for Poisson equation
3. Poisson-Boltzmann model for protein in solution
4. Elliptic interface problems
5. Computation of Poisson-Boltzmann equation
6. Poisson-Nerst-Planck models
7. Layer structure

(iii) Bose-Einstein condensates
1. Spin and spin-spin interaction
2. Mean field theory and Gross-Pitaevisky equation
3. Ground states
4. Vortex states