Coarse geometry and cohomology of large data sets

数字经济是建立在数据基础上的。当前的趋势是开发智能的、客户主导的、交互式的、实时的系统，这就要求能够有效、快速地处理和解释大量的数据，并具有与要求相对应的准确度。最近的两个重要事态发展很好地强调了这一点。由IBM支持的智慧地球计划设想了“仪表化的互连数据系统”，其中物理环境的主要元素配备了不断交换信息的传感器。其次，英国政府新的透明度驱动将使公众可以获得大量数据集，创造“构建创新应用程序的机会，这将带来显著的经济效益”。在数据分析中需要合成的几何方法，因为涉及的集合的大尺寸和高维度。该建议将扩展最近的重要理论结果，以创建一套几何和拓扑工具的数据分析，特别强调灵活性，效率，并密切配合潜在的实际应用。这是一个理想的和非常令人兴奋的时间来启动一个项目的性质，其结果很可能有直接和重要的后果，从上述举措和许多其他可能的应用程序的角度来看。该提案的一个中心主题是研究各种尺度的大型数据集的几何特性，这对应于不同程度的“清晰度”，数据集被视为。例如，在搜索大量的数字照片中包含人的照片时，需要与试图识别特定人时不同的分辨率。这一建议为发展纯数学方法提供了一个非常令人兴奋的机会，使它们能够直接应用于重要，困难和及时的实际问题。拟议的工作是冒险的，跨学科的，并汇集了纯数学家和应用数学家，专家或，计算机科学，统计学和能源系统。长期实际应用的潜力将在更广泛的智能飞机计划范围内的两个特定应用领域进行测试。该项目的一个主要目标是开发几何和cohomological工具的尺度依赖粗几何，特别强调应用有限度量空间，更具体地说，数据集。我们将非常重视可以开发成有效的数据分析工具的方法，研究将通过从理论到更实际的应用中产生的具体问题来了解。我们将测试的理论思想和结果两个重要的情况下：一，数据集所产生的英国政府的开放数据倡议。其次，在智能电网的背景下，我们将考虑由大量传感器生成的数据，这些传感器监测电网性能的各个方面，目的是提供供需之间的准确匹配。

The digital economy is founded on data. The current trend to develop intelligent, customer-led, interactive, real-time systems requires the ability to handle and interpret vast amounts of data efficiently, quickly, and with a degree of accuracy corresponding to the requirements. This point is underlined very well by two important recent developments. The Smarter Planet initiative, supported by the IBM, envisages 'instrumented, interconnected data systems' where main elements of the physical environment are equipped with sensors constantly exchanging information. Secondly, the new transparency drive of the UK government will make huge data sets available to the public, creating 'an opportunity to build innovative applications which will bring significant economic benefit'. The need for synthetic geometric methods in data analysis arises because of the large size and of high dimensionality of the sets involved. This proposal will extend recent important theoretic results to create a set of geometric and topological tools for data analysis, placing special emphasis on flexibility, efficiency, and on close alignment with potential practical applications. This is an ideal and a very exciting time to launch a project of this nature, and its results are very likely to have direct and important consequences from the point of view of initiatives mentioned above and many other possible applications. A central theme of the proposal is the study of geometric properties of large data sets at various scales, which corresponds to varying degree of 'sharpness' with which a data set is viewed. For example, in searching large numbers of digital photographs for those that contain pictures of of people one requires a different resolution than when trying to identify a specific person. This proposal offers a very exciting opportunity for developing pure mathematical methods to the point where they can be directly applied to important, difficult and timely practical problems. The proposed work is adventurous, interdisciplinary, and brings together pure and applied mathematicians, experts in OR, computer science, statistics, and energy systems. Potential for long-term practical applications will be tested in two specific areas of applications within the context of the wider Smarter Plane initiative. A main objective of the project is to develop geometric and cohomological tools of scale-dependent coarse geometry with special emphasis on applications to finite metric spaces and more specifically, to data sets. We will place strong emphasis on methods that can be developed into efficient tools for data analysis, and the research will be informed by specific problems arising from applications which range from the theoretical to the more practical. We will test the theoretical ideas and results two important cases: one, data sets arising from the UK Government's Open data initiative. Secondly, within the context of smart grids, we will consider data generated by large number of sensors monitoring various aspects of the performance of a power grid with the objective to provide an accurate matching between supply and demand.