Persistence, Combinatorial Morse Functions and Shape Spaces and their Applications

本提案涉及探索机器人、地理信息系统和生物应用中出现的计算问题的新方法。这些方法涉及数学和计算机科学交叉的技术，以及在开发解决各种问题的新算法中应用数学观点的技术。主要目标是利用莫尔斯理论、同调理论、几何群论和组合学的技术来研究计算机科学中的形状、配置、运动规划和结构问题。应用包括计算机图形学、科学数据的可视化、分子对接问题的计算分析和机器人技术。一个关键因素是，这项工作将在一个跨学科团队中完成，该团队包括杜克大学、北卡罗来纳大学和斯坦福大学的计算机科学、生物化学和化学研究人员。该项目的研究将对促进我们对许多重要领域的理解具有重要意义。所开发的技术将为分析来自x射线晶体学、脑部扫描、卫星图像和海洋温度等数据集的噪声提供新的方法。机器人工作将增强我们创造在危险或远程环境中执行任务的自动化设备的能力。生物学工作的长期目标是开发更好的计算工具，帮助生物医学研究人员找到可以相互作用的合适分子

This proposal concerns the exploration of new approaches to computational problems that arise in robotics, geographic information systems, and in biological applications. These approaches involve techniques that lie at the intersection of mathematics and computer science and the application of mathematical perspectives in the development on new algorithms to solve a variety of problems. The primary goal is to use techniques from Morse theory, homology theory, geometric group theory and combinatorics to study problems of shape, configuration, motion planning, and structure in computer science. Applications include computer graphics, visualization of scientific data, computational analysis of molecular docking problems and robotics. A key element is that this work will be done within an interdisciplinary team that includes researchers in computer science, biochemistry and chemistry at Duke, UNC and Stanford.The research in this project will be important to advance our understanding in a number of important areas. Techniques developed will give new approaches to analyzing noise in data sets such as those from x-ray Crystallography, brain scans, satellite images, and ocean temperature. The robotics work will enhance our ability to create automated devices that perform tasks in dangerous or remote environments. And the biology work has as its long term goal the development of better computational tools to help biomedical researchers find appropriate molecules that can interact with