There exist a variety of tools for studying multiple scale behaviors in nonlinear systems, and the method of choice for a particular problem often depends on whether the investigator is an analyst, an applied mathematician or a physicist. An important goal is to effectively merge the various approaches, and develop new ways of thinking about multi-scale problems. To this end, this project focuses on investigating five interdisciplinary problems, using ideas from functional analysis, geometry, topology, matched asymptotics and scaling arguments, in conjunction with numerical simulations. These problems are (1) Generalized crumpling: Combining functional analytic methods with geometric/topological ideas to study singularities and microstructure; (2) Blowup in Parabolic PDEs: Applying Morse theory to the dynamics of blowup solutions; (3) Dynamics of microstructure: Studying global bifurcations involving the change of microstructure in a model system; (4) Pattern formation and non-equilibrium phase transitions: Investigating multiple scale behaviors in nonlinear systems in the presence of noise; and (5) Topological transitions in fluid interfaces: Using tools from PDE to investigate topological transitions in 2 fluid systems.
Many real world systems are interesting precisely because they exhibit different behaviors on different scales. This is certainly true for living organisms, geological and geophysical systems, technologically important composite materials and even social structures and hierarchies. Thus researchers across many disciplines grapple with the following two questions, which are the essence of multiple scale analysis: (1) How does the large scale (macroscopic) behavior emerge out of the collective behavior of the small scale (microscopic) units?, and (2) What are the rules governing the large scale behavior, and how does this influence the behavior of the small scale units? The research component of this project studies these questions in a mathematical setting through problems that arise in material science and in physics. The overall goal is to meld together a variety of techniques to develop tools that can successfully handle complex real-world multiple scale problems. This is combined with an integrated approach to pedagogy, that features a strong involvement in undergraduate and graduate research, development of research opportunities for groups that are under-represented in mathematics and the physical sciences, curriculum development both at the graduate and the undergraduate level, and development of materials for scientific outreach to the general public.
Date: December 17, 2001
