Applications of frames to problems in mathematics and engineering

CasazzaDMS-1008183 研究了Hilbert空间框架应用中的几个基本问题。 大部分的工作涉及“融合框架”，它提供了一个自然的框架进行分层数据处理。 硬件技术的最新进展使大量低成本组件的经济生产和部署成为可能，这些组件结合在一起可以实现可靠和高效的操作。 在许多学科中，从集中式信息处理到分布式或网络范围的信息处理有了根本性的转变。 数据通信正在从点对点通信转变为广域网上的分组传输，广域网的网络管理是分布式的，各个链路的可靠性不那么重要。 雷达成像正从单一平台向多平台发展，这些平台相互协作以获得更好的性能。 无线传感器网络是一种新兴的技术，它有可能实现成本效益和可靠的监测。 所有这些应用都涉及大量的数据流，需要在中央处理器上进行集成。 融合帧是最近的发展，专门为这些应用而设计。 调查员开发的具体建设offunctionalframes和它们的实施所需的算法。 直接与工程师合作，保证了这里的进步立即实施。 这些技术也被应用于产生量子物理学的格拉斯曼框架，特别是量子信息理论，量子态层析成像和量子密码学。 最后，框架理论表明，1959年的Kadison-Singer问题等价于纯数学、应用数学和工程学中十几个研究领域的基本未解决问题。 这个项目开发了解决现在数学中最重要的开放问题之一所需的框架。 我们现在已经进入了一个新时代，国防部、国土安全部和工业界必须处理大量的数据，这些数据甚至超过了当前计算机的能力。 特别是对于国防来说，能够实时处理这些数据是至关重要的，这变得越来越困难。 这种“信息过载”所造成的问题每年都在迅速升级，需要立即迅速加以解决。因此，它是至关重要的国家利益，自动化尽可能多的信息融合过程。 但目前，在信息融合领域，这种自动化系统所需的基础理论远远落后于当前的需求。 研究人员和他的学生开发了解决这些关键需求所必需的方法。 这包括并行存储和处理信息的方法--也就是说，将信息分成多个子系统进行简单的处理，然后将结果“融合”以获得预期的结果。 它还包括开发分布式传感网络，可以保持一个区域的监视，同时考虑到系统的严重物理限制，如低通信带宽，有限的信号处理能力，有限的电池寿命，或监视区域的地形。 这样一个自动化系统必须能够联合收割机将大量完全不同的信息源结合起来，同时几乎不留任何出错的余地。研究者和他的学生解决这些需求，并努力预测将出现的下一个层次的困难。

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.