CAREER: Singularities and Microstructure - Multiple Scale Analysis for Nonlinear Partial Differential Equations (PDE), Geometric Problems, and the Physical Sciences

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DMS Award AbstractAward #: 0135078PI: Venkataramani, Shankar Institution: University of ChicagoProgram: Applied MathematicsProgram Manager: Catherine MavriplisTitle: CAREER: Singularities and Microstructure - Multiple Scale Analysis for Nonlinear PDE, Geometric Problems, and the Physical SciencesThere exist a variety of tools for studying multiple scale behaviorsin nonlinear systems, and the method of choice for a particularproblem often depends on whether the investigator is an analyst, anapplied mathematician or a physicist. An important goal is toeffectively merge the various approaches, and develop new ways ofthinking about multi-scale problems. To this end, this project focuseson investigating five interdisciplinary problems, using ideas fromfunctional analysis, geometry, topology, matched asymptotics andscaling arguments, in conjunction with numerical simulations. Theseproblems are (1) Generalized crumpling: Combining functional analyticmethods with geometric/topological ideas to study singularities andmicrostructure; (2) Blowup in Parabolic PDEs: Applying Morse theory tothe dynamics of blowup solutions; (3) Dynamics of microstructure:Studying global bifurcations involving the change of microstructure ina model system; (4) Pattern formation and non-equilibrium phasetransitions: Investigating multiple scale behaviors in nonlinearsystems in the presence of noise; and (5) Topological transitions influid interfaces: Using tools from PDE to investigate topologicaltransitions in 2 fluid systems.Many real world systems are interesting precisely because they exhibitdifferent behaviors on different scales. This is certainly true forliving organisms, geological and geophysical systems, technologicallyimportant composite materials and even social structures andhierarchies. Thus researchers across many disciplines grapple with thefollowing two questions, which are the essence of multiple scaleanalysis: (1) How does the large scale (macroscopic) behavior emergeout 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? Theresearch component of this project studies these questions in amathematical setting through problems that arise in material scienceand in physics. The overall goal is to meld together a variety oftechniques to develop tools that can successfully handle complexreal-world multiple scale problems. This is combined with anintegrated approach to pedagogy, that features a strong involvement inundergraduate and graduate research, development of researchopportunities for groups that are under-represented in mathematics andthe physical sciences, curriculum development both at the graduate andthe undergraduate level, and development of materials for scientificoutreach to the general public.Date: December 17, 2001