AF: Small: Analyzing Spaces and Scalar Fields via Point Clouds

扫描技术的最新进展和科学与工程的总体进展使得对显式或隐式呈现的几何图形进行采样变得容易。在某些情况下，采样允许为数据配备在采样域上定义的标量函数的值。目标是处理产生的点云数据（PCD），以近似原始空间的不同几何和拓扑结构和/或在其上定义的函数。如何实现这一目标是本奖项的主要议题。从PCD中提取几何和拓扑结构首先要了解原始采样空间上的这些结构。在某些情况下，需要调整现有的数学概念，以适应算法开发中固有的离散化。在其他方面，需要开发新的分析技术和新的算法工具。pi更希望后者能够驯服维度和规模的诅咒。两位pi将他们在计算几何和计算拓扑方面的专业知识带到董事会来执行该项目。数据分析中的一个核心问题是设计从大量高维数据中表示和检索信息的计算方法。各种科学、工程和社会研究，如产品开发、医学、经济学、气候、疾病控制等领域的研究，产生的数据可能是对位于高维的隐藏参数空间进行采样。他们将受益于本项目开发的技术。由于这些问题的本质，这个项目将加强数学、理论计算机科学和数据分析等领域之间的协同作用。除了标准的研究论坛之外，项目的结果将通过课程笔记、教程和网页传播，以达到更广泛的受众。该项目支持的研究生将发展数学和理论计算机科学的技能，尤其是几何和拓扑，以及编写健壮、高效和用户友好的软件。

Recent advances in scanning technology and general progress in science and engineering have made it easy to sample a geometry presented explicitly or implicitly. In some cases, sampling allows the data to be equipped with values of a scalar function defined on the sampled domain. The goal is to process the resulting point cloud data (PCD) for approximating different geometric and topological structures of the original space and/or functions defined on them. How can this be achieved is the main agenda of this award.The extraction of geometric and topological structures from a PCD begins with the understanding of these structures on the original sampled space. In some cases one needs to adapt existing mathematical concepts to fit the discretization inherent in algorithmic development. In others new analysis techniques and new algorithmic tools need to be developed. PIs expect the more of the latter to tame the curse of dimensionality and scale. The two PIs bring their expertise in computational geometry and computational topology to the board to conduct the project. A central problem in data analysis is to design computational methods for representing and retrieving information from massive and high-dimensional data. Various scientific, engineering, and social studies such as the ones in product development, medicine, economics, climate, disease control produce data that presumably sample a hidden parameter space sitting in a high dimension. They will benefit from the techniques developed in this project. By the very nature of the problems, this project will enhance a synergy between fields such as mathematics, theoretical computer science, and data analysis.Other than standard research forums, results from the project will be disseminated through course notes, tutorials, and web-pages to reach wider audience. Graduate students supported by the project will develop skills in mathematics and theoretical computer science, most notably in geometry and topology and also in writing robust, efficient and user-friendly software.