Fractal-based methods in analysis

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Based on the work of recent years, "fractal-based methods" refers not just to the theory and tools of fractals themselves, but also to the driving philosophy behind this mathematics. For example, in my recent collaborative work in inverse problems for ordinary and partial differential equations, no fractal appears, yet the mathematical approach is clearly inspired by successful approaches in fractal imaging. The work in this proposal extends from this fractal-based core to include broad elements from analysis, dynamical systems, mathematical modeling, numerical analysis, and other topics in mathematics and the specific application domain research.The goals of the proposed research program are to develop fractal-based frameworks for analyzing direct and inverse problems in(A) physics and engineering, including- porous media and perforated domains, building upon some preliminary collaborative work that I have published in the past year. Typically, differential equations on perforated domains are treated with homogenization theory in which heterogeneous material is replaced by a fictitious homogenous medium; my recent work rigorously justifies that this sort of scale shift can also be done for related inverse problems. For example, one can recover the variable thermal diffusivity of a porous medium by using observational data to solve the corresponding inverse problem on the medium with no holes- PDE problems with fractal-like boundary data(B) environmental science, with a focus on sustainability, including- pollution-driven population dynamics, perhaps with economic impact- ecosystem modeling- an inverse problem solution framework for coupled systems of reaction-diffusion equations with delay(s)In both problems, a key point of interest is sustainability, particularly in the presence of undesirable elements (some tolerable pollution level) or possibly large perturbations (harvest). (C) biomedical science, including continuing work on tumor detection, modeling, and analysis, and the development of an image-driven inverse problem solution framework(D) new imaging frameworks based on very recent ideas in fractal-based analysis, including- star-shaped set inversion systems- iterated multifunction systemsThe proposed research program combines rigorous theoretical elements (analysis, fractal geometry, differential and integral equations) with application-driven matters (modeling, signal/image processing) and practical issues (programming, algorithm design, numerical analysis, approximations). Students working in the program will receive both deep and broad training across this spectrum. Some initial small pieces of the research program proposal have been developed by current/recent students, with subsequent pieces to be established by future students. Other parts of the proposal reflect the next step of very recent work in current collaborations.