Geometric and Combinatorial Viewpoints in Complex and Harmonic Analysis

Michigan State University, East Lansing MI

Investigators

Abstract

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). With regards to the intellectual content of the project, 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. More specifically, the PI will pursue (1) Sharp distortion examples for K-QC maps. (2) Sharp sufficient conditions for removability of sets under bounded quasiregular (QR) maps (related to sharp QC distortion theorems.) (3) Buffon needle probability (Favard length) for Cantor sets (4) Questions involving uniformly rectifiable sets and harmonic measure. (5) Problems on distance sets relating Fourier analysis and geometric combinatorics. 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 K-QC mappings (mappings sending an infinitesimal circle/ball to an infinitesimal ellipse/ellipsoid with eccentricity controlled by K), 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.) With regards to contextualizing the proposed research within a broader mathematical and scientific framework, note that the mathematical objects involved have found abundant applications in other disciplines, so the problems proposed will advance knowledge in those areas and hence impact other areas of mathematics, science, or engineering. More specifically, fractals (geometric measure theory) 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. Quasiconformal 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. In terms of human resource development, 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. Research results will be disseminated via participation in US and international meetings, which will facilitate collaborations with both established and young researchers.