Variational Methods for Materials and Imaging Sciences

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The objectives of this project are the identification and pursuit of emerging areas of applied analysis, motivated by contemporary issues in imaging and materials science at the core of advances in high-end technology and of national scientific importance. The two main topics of the project are(1) the mathematical study of modern semiconductors and nano structures, of pivotal importance in microelectric and optoelectronic technologies, such as reflective or anti-reflective coatings for optics, the fabrication of layers of insulators and semiconductors for integrated circuits, quantum well lasers, and(2) the analytical investigation of image segmentation and inpainting and recolorization for color images, fundamental to the advance of computer vision, medical imaging, film restoration, and scanning probe microscopy. Postdocs and graduate students are trained in the course of the project. Common features of the projects include the treatment of energies that involve terms of different dimensionality. These often exhibit a large range of length and time scales, higher order derivatives, and discontinuous underlying fields. Such features prevent the use of well understood functional analytic frameworks, they escape traditional mathematical theories, and they require state-of-the-art techniques, creative ideas, and the introduction of innovative mathematical tools. The investigator and her collaborators use new and recently developed methods and a deep articulation of ideas in the calculus of variations, geometric measure theory, and nonlinear partial differential equations, to address problems that include in topic (1) epitaxy and the formation of quantum dots, the onset and propagation of dislocations, homogenization of composite materials, and in topic (2) signal denoising and detexturing, dejittering, inpainting, and recolorization. These topics offer new opportunities for the integration of applied analysis in research and in the education of advanced graduate students and postdoctoral fellows, thus allowing for the training of a new generation of applied analysts at the forefront of contemporary mathematics as it interfaces with materials and imaging sciences.