Applications of frames to problems in mathematics and engineering

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CasazzaDMS-1008183 The investigator works on several fundamental questionsconcerning applications of Hilbert space frames. Much of thework concerns "fusion frames," which provide a natural frameworkfor performing hierarchical data processing. Recent advances inhardware technology have enabled the economic production anddeployment of a large number of low-cost components, which incombination enable reliable and efficient operation. Across manydisciplines there is a fundamental shift from centralizedinformation processing to distributed or network-wide informationprocessing. Data communication is shifting from point-to-pointcommunication to packet transport over wide area networks wherenetwork management is distributed and the reliability of individuallinks is less critical. Radar imaging is moving away from singleplatforms to multiple platforms that cooperate to achieve betterperformance. Wireless sensor networks are emerging as a newtechnology that potentially enables cost-effective and reliablesurveillance. All these applications involve a large number of datastreams, which need to be integrated at a central processor. Fusionframes are a recent development designed precisely for theseapplications. The investigator develops concrete constructions offusion frames and the algorithms needed for their implementation. Working directly with engineers guarantees that advances here haveImmediate implementation. The techniques are also applied toproduce Grassmannian frames for quantum physics -- especiallyQuantum information theory, quantum state tomography, and quantumcryptography. Finally, frame theory has shown that the 1959Kadison-Singer Problem is equivalent to fundamental unsolved problemsin a dozen areas of research in pure mathematics, applied mathematics,and engineering. This project develops the needed frames for resolvingwhat is now one of the most significant open problems in mathematics. We have now entered a new era where the Defense Department,Homeland Security, and industry must process amounts of data thatoverwhelm even current computer capacities. Especially fordefense, it is critical to be able to process these data in realtime, which is becoming more and more difficult. The problemscaused by this "information overload" are escalating rapidlyevery year and need to be addressed immediately and quickly. Therefore, it is of vital national interest to automate as muchof the information fusion process as possible. But at this time,in the area of information fusion the fundamental mathematicsnecessary for such an automated system is far behind the currentneeds. The investigator and his students develop the mathematicsnecessary for tackling these critical needs. This includesmethods for storing and processing information in parallel --i.e. dividing it into multiple subsystems for simpler processingand then "fusing" the results to get the desired outcomes. Italso includes developing distributed sensing networks that cankeep an area under surveillance while taking into account severephysical limitations of the systems such as low communicationbandwidth, limited signal processing power, limited battery life,or the topography of the surveillance area. Such an automatedsystem must be able to combine a large number of sources of quitedisparate information while leaving almost no room for error. The investigator and his students address these needs as well astry to anticipate the next level of difficulties that will arise.