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 field 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.
subsection{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 and (ii) structure and function of proteins.

(i) Fluid-structure interaction
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I Basic fluid mechanics: Stokes flows and Navier-Stokes flows
I Computation of Stokes flows and Navier-Stokes flows
I Basic elasticity
I Allan-Cahn and Cahn-Hilliard models for interfaces
I Structure of interfaces
I Geometry and evolution of interfaces: level set method
I Diffusion interface methods
I Fluid-structure interaction
I Surfacants
I Complex fluids
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(ii) Structure and function of proteins.
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I Basic electrostatics
I Numerical method for Poisson equation
I Poisson-Boltzmann model for protein in solution
I Elliptic interface problems
I Computation of Poisson-Boltzmann equation
I Poisson-Nerst-Planck models
I Layer structure
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