CRII: III: Effective Geometry of Urban Travel Patterns

城市化的速度和规模给可持续城市的发展带来了巨大挑战。全球一半的人口已经生活在城市，到2050年，预计全球三分之二的人口将生活在城市地区。该项目旨在利用新颖的数学和计算工具解决城市规划中的开放性挑战，如交通拥堵和可达性。具体来说，该项目将黎曼几何的数学与道路网络中城市旅行的大规模计算机模拟相结合。这项工作将产生新的算法来提取城市出行的几何特征和规则，从而促进城市可持续发展的城市交通解决方案的设计。此外，这项研究将支持STEM研究人员的培训，以及开源软件的开发，这将使其他研究人员更容易获得这一新的研究领域。该项目的技术目标分为两个重点。第一个重点是开发理论和算法工具来计算黎曼度量，使网络上的流量是该度量的距离最小化测地线。我们可以想象，在一张地图上，所有的道路都有相同的速度，而不是固定的比例（例如，地图上的每一英寸对应1英里）。因此，地图上的距离代表的是旅行时间，而不是实际距离。其主要思想是通过向网络中的每个节点附加一个矢量来表示网络流，该矢量表示从该节点到某个固定目标节点的流的方向和速度。研究人员将开发策略，从这些向量场中推断出度量张量，并显示几何特征（如曲率和体积）与原始网络的拓扑特征之间的联系。第二个重点是分析世界各地城市的真实道路地图和旅行时间数据，并使用该框架根据城市的几何形状及其随时间的变化对城市进行分类。这项工作将展示几何工具如何用于解决城市交通系统中的开放性挑战，如交通拥堵和可达性。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The speed and scale of urbanization brings tremendous challenges in developing sustainable cities. Half of the global population already lives in cities, and by 2050 two-thirds of the world’s people are expected to live in urban areas. This project aims to address open challenges in urban planning, such as traffic congestions and accessibility, using novel mathematical and computational tools. Specifically, the project combines the mathematics of Riemannian geometry with large-scale computer simulations of urban travel in road networks. The work will result in new algorithms to extract the geometrical features and rules of urban travel, which facilitate the design of urban mobility solutions for the sustainable development of cities. Furthermore, this research will support the training of STEM researchers, and the development of open-source software that will make this new line of research more accessible to other researchers. The technical aims of the project are divided into two thrusts. The first thrust develops theoretical and algorithmic tools to compute a Riemanian metric such that flows on the network are distance-minimizing geodesics for that metric. One can imagine a road map where instead of having a fixed scale (for example, every inch on the map corresponds to 1 mile), all roads have the same speed. Thus, distances on the map represent travel time rather than actual distances. The main idea is to represent network flows by vector fields attaching a vector to every node in the network, which shows the direction and speed of flow from that node to some fixed target node. The researchers will develop strategies to infer a metric tensor from these vector fields and show connections between geometric features (such as curvature and volume) and topological features of the original network. The second thrust will focus on analyzing real road maps and travel time data in cities around the world, and use the framework to classify cities based on their geometry and how it varies with time. The work will demonstrate how geometrical tools can be used to address open challenges in urban transportation systems such as traffic congestions and accessibility.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.