The analysis of structures that span scales pose challenges to mathematics that have led in recent years to new ideas and theories. The investigator in this work focuses on issues associated with the stability of permanent structures that have contrasting structure at the different scales. Their presence in a given system is the result of a balance between nonlinearity and a spatio-physical effect such as dispersion (e.g., for a solitary wave in optical communications), diffusion (e.g., propagating waves in biological media) or convection (e.g., shocks in materials or gases). The issue of stability requires a deep analysis that can reveal more intricate features in the inner workings of a given system. The work of this project will address four issues: (1) the determination of the stability for waves that are held together on different scales by different spatio-physical effects; (2) The assessment of the stability of patterns in multi-dimensions that have substructures on different scales; (3) The effects of stability at the macro-scale on controlling variations on a smaller scale; and (4) For complex structures, the best way to measure the manifestations of instability that influence a range of scales.
Critical questions in a physical or biological model often concern the stability of a permanent structure, such as a nonlinear wave or pattern. The work of this project aims at developing techniques for assessing the stability of structures that cross physical scales. A variety of information may be available at the different scales and the goal is to develop approaches for patching together these contrasting pieces of information to answer the relevant physical questions. Problems in advanced technology and communications routinely exploit the optical properties of materials. The work described in this proposal will supply techniques for key structures of existing and promised technologies, such as high bit-rate communications and Bose-Einstein condensates. At a more general level, the techniques to be developed promise a new viewpoint on stability determinations for higher-dimensional structures and patterns. Another dimension is the training of graduate students and postdocs in interdisciplinary mathematics. This work will both further that effort and systematize some of the key activities in the training of students for interdisciplinary work. In particular, a focused effort will be aimed at teaching problem discernment while working in a team with domain scientists. The involvement of under-represented minorities will also be a goal and it is anticipated that this effort will bring a number of minorities and women to a competitive level in modern interdisciplinary mathematics.
