CAREER: Weighted Inequalities and their Applications to Quasiconformal Maps

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The aim of this proposal is the study of interactions between quasiconformal (QC) mappings, geometric analysis (in particular uniform rectifiability), Fourier analysis, and geometric combinatorics. Five problems will be addressed. The unifying method to be employed is an underlying geometric-combinatorial vision which often manifests itself through multiscale analysis (i.e., the analysis of a problem on different scales.) This method will be applied in the contexts of so called "K-QC mappings" (mappings sending an infinitesimal circle/ball to an infinitesimal ellipse/ellipsoid), geometric measure theory (GMT, which analyzes sets and measures on them -these are generalizations of length, area and volume-), harmonic analysis (decomposing a signal into elementary pieces of wavelike character), and potential theory (study of Coulombic potential and related topics.) The mathematical objects involved have found abundant applications in other disciplines, so the problems proposed will advance knowledge in those areas. Fractals (GMT) appear naturally in electrodeposition and Diffusion Limited Aggregation. The internal structure of lungs has a high fractal dimension (to capture more oxygen.) Fourier analysis is often applied in signal and image processing. QC maps are solutions to problems in non-linear elasticity, and have found applications in string theory. Uniform rectifiability appears in minimizers of the Mumford-Shah functional (originally used for image segmentation.) Geometric combinatorics is used for fair division and voting problems in the social sciences, and for phylogenetic trees models in biology. Distance sets are used in industry to study the dimensionality of data sets. The PI will continue preparing students for the Putnam Competition, participating in the Math Club, and mentoring graduate students informally in the context of graduate courses. Fractals and geometric combinatorics are excellent areas for promoting teaching and training of undergraduates and postdocs. The basic notions of multiscale analysis, dimension, combinatorics, etc. are deep enough to convey some flavor of research yet can be successfully explained in an elementary way.