AF: Medium: Collaborative Research: Optimality in Homology - Algorithms and Applications

在科学和工程中的许多应用都遇到了在几何或数据的数字表示中识别和处理拓扑感兴趣的特征的问题。这些特性通常需要在某些度量(度量)方面是最优的。人们认识到，来自代数拓扑的同调群在这些计算中起着至关重要的作用。虽然同调群的结构性质的研究在数学上已经有了丰富的历史，但它们与几何相结合的计算却没有得到很好的研究。首席研究人员(PI)建议深入研究这些基本问题及其与科学和工程中的实际问题的联系。智力优势：同调计算中的最优化问题的有效解决需要数学和算法的发展。私人投资经理带来了这些所需的专业知识。除了数学和理论计算机科学研究的协同效应外，科学和工程中与各种应用的密切联系将在计算机图形学、计算机视觉、传感器网络、计算机辅助设计等计算领域和生物、物理、化学等科学领域之间发挥协同作用。同调基团方面的优化在从蛋白质分子的隧道到大型机器的空洞等许多科学和工程应用中具有重要的意义。这些问题的解决可以帮助制造更好的机器，设计新药，以及快速建立定制对象的模型。这个项目的教育影响是数学和计算机科学之间的协同作用，受到实际应用的推动。通过该项目开发的课程笔记、互联网分布和软件系统将使科学界能够研究几何学、拓扑学和算法中具有挑战性的问题。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，并编写健壮、高效和用户友好的软件。

Many applications in science and engineering encounter the problem ofidentifying and processing topologically interesting features in thedigital representation of a geometry or data. Such features often needto be optimal with respect to some metric (measurement). It isrecognized that homology groups from algebraic topology play anessential role in these computations. Although the study of structuralproperties of the homology groups has a rich history in mathematics,their computations in combination with geometry are not that wellstudied. The principal investigators (PIs) propose to study thesefundamental questions thoroughly, along with their connections topractical problems from science and engineering.Intellectual merit: Efficient solutions of the optimality questions inhomology computations require both mathematical and algorithmicdevelopments. The PIs bring aboard these required expertise. Apartfrom the synergistic effect of the proposed study on mathematics andtheoretical computer science, the close ties with various applicationsin science and engineering will play a synergistic role betweencomputational fields such as computer graphics, computer vision,sensor networks, computer aided design, and scientific fields such asbiology, physics, chemistry, and others.Optimization of aspects of homology groups providesimportant insights in many scientific and engineering applicationsranging from tunnels in protein molecules to voids in largemachines. Solutions of such problems can aid in the manufacturing ofbetter machines, designing of new drugs, and rapid modeling ofcustomized objects. The educational impact of this project is in alarge synergy between mathematics and computer science motivated byreal applications. Course notes, internet distributions, and softwaresystems developed through the project will enable the scientificcommunity to study challenging problems in geometry, topology, andalgorithms. Graduate students supported by the project will developskills in mathematics and theoretical computer science and also inwriting robust, efficient, and user-friendly software.