Problems from materials science provide a rich source of mathematical and computational challenges. These problems often have a multiscale character; for example, the atomic structure of a material influences its macroscopic properties. Purely continuum models also raise multiscale issues - involving for example the formation and evolution of patterns as a consequence of loading, phase transformation, or coarsening.
These problems are driving new developments in numerical analysis, the calculus of variations, and nonlinear partial differential equations. The developments are typically problem-driven, complementing work being done in the materials community. For example, numerical methods combining atomic-scale and continuum phenomena differ widely in their stability, efficiency, and accuracy. At a more macroscopic level, analysis of models for coarsening or phase transformation provides fresh understanding and sometimes reveals unexpected phenomena.
The interdisciplinary character of this area can be a barrier to entry. The goal of this summer school is to overcome that barrier. There will be four tutorials (5 hours lecture and 2 hours recitation each) by leading experts in the field, on four main themes of the program:
The origins and consequences of material microstructure
Kaushik Bhattacharya (California Institute of Technology)
Self-similarity in mean-field models for coarsening and coagulation
Barbara Niethammer (University of Oxford)
Atomistic-to-continuum coupling methods for crystalline solids
Christoph Ortner (University of Warwick)
Pattern formation and scaling laws in materials science
Felix Otto (Max-Planck-Institut für Mathematik in den Naturwissenschaften)
These tutorials will be complemented by additional lectures by other specialists:
Joseph Conlon (University of Michigan)
Cyrill Muratov (New Jersey Institute of Technology)
Harold Park (Boston University)
Robert Pego (Carnegie Mellon University)
Anja Schlömerkemper (University of Wurzburg)
Christian Seis (University of Toronto)
Lev Truskinovsky (Ecole Polytechnique)
Brian Van Koten (University of Minnesota)
While participants at all levels are welcome, this event is intended particularly for PhD students and young scientists. The tutorials will assume familiarity with partial differential equations, and some exposure to continuum mechanics and numerical analysis will also be helpful. However participants are not expected to have prior experience with the research topics under discussion.