The past decade has witnessed the rapid development of mathematical and computational researches in materials science. Applied mathematics has made much contribution to the understanding of some of the fundamental principles and basic mechanisms of complex material processes that are often characterized by singularities, fluctuation, and multi-scale. The advance in materials science and nanotechnology today, however, has posed new challenges and opportunities to applied and computational mathematics.

This course will review some of the mathematical and computational aspects of the materials science. It will cover a variety of interesting topics, some of which have been much studied and some are still relatively new. They include:

1. Weak convergence methods for variational problems modeling crystalline microstructure;
2. Nonlinear evolutionary differential equations modeling the dynamics of internal layers, phase separation, and coarsening;
3. Continuum models for epitaxial growth of thin films;
4. The level set method for the simulation of interface motion;
5. The heterogeneous multi-scale methods for material simulations.

1. A.-L. Barabási and H. E. Stanley, Fractal concepts in surface growth, Cambridge Univ. Press, 1995.
2. K. Bhattacharya, Theory of martensitic microstructure and the shape-memory effect, 2003.
3. R. E. Caflisch, Mathematical problems in materials science, Unpublished lecture notes, UCLA, 1999.
4. W. E and B. Engquist, The heterogeneous multi-scale methods, Comm. Math. Sci., 1:87-133, 2003.
5. P. Fife, Dynamics of internal layers and diffusive interfaces, SIAM, 1988.
6. B. Li, Weak convergence methods for variational problems modeling crystalline microstructure, Unpublished lecture notes, UCLA, 1998.
7. S. Müller, Variational models for microstructure and phase transitions, Lecture notes, Max-Planck Inst. for Math. in Sci., Leipzig, 1998.
8. S. Osher and R. Fedkiw, Level set methods and dynamic implicit surfaces, Springer, 2003.
9. A. Pimpinelli and J. Villain, Physics of crystal growth, Cambridge Univ. Press, 1998.
10. J. A. Sethian, Level set methods and fast marching methods, 1999.

This course is designed for graduate students who are interested in applied partial differential equations, nonlinear dynamics, scientific computing, and numerical analysis. The undergraduate course on partial differential equations or equivalent will be a prerequisite. But no knowledge on materials science is assumed. The material presented in the course will be essentially self-closed. Along the course, reference material and lecture notes will be distributed, and possible research topics will be discussed.

For more information, please contact the instructor.