Applications of Frames to Problems in Mathematics and Engineering II

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Casazza1307685 The investigator and his students work on four fundamental problems concerning applications of Hilbert space frames. These problems all require developing completely new techniques for frame design. (1) Fusion frames have a host of applications in distributed processing, wireless sensor networks, and other areas. The investigator and his students construct fusion frames designed specifically for applications. (2) It is now known that the 1959 Kadison-Singer Problem is equivalent to fundamental unsolved problems in a dozen areas of research in pure mathematics, applied mathematics, and engineering. The recent surprising positive solution to the problem raises a number of fundamental questions about the implementation of this result to all these areas of research. Advances here affect a dozen important research areas in a fundamental way. (3) The investigator and his students derive new classes of Grassmannian frames. These frames have broad application to communication theory, graph theory, and quantum physics -- especially quantum information theory, quantum state tomography, quantum cryptography, and foundational issues in quantum mechanics. (4) The investigator attacks the famous Paulsen Problem in frame theory, which has important implications for operator theory and other areas of pure mathematics. The investigator constructs and designs fusion frames for a host of applications to problems in engineering. One emphasis is on designing wireless sensor networks. which are an emerging technology with the potential to detect chemical, biological, radiological, and nuclear weapons in transport. Another application is to distributed processing where data systems are too large for central processing. This overload of data now appears in engineering, physics, astronomy, medicine, biology, and industrial applications that demand real-time processing on as ever-increasing amounts of data. Frames also have application to a number of important problems in pure mathematics.