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

在科学和工程中的许多应用遇到的问题ofidentifying和处理拓扑有趣的功能在thedigraphy的几何或数据。这些特征通常需要在某些度量（测量）方面是最佳的。人们认识到，从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有着丰富的历史，但它们的计算与几何的结合却没有得到很好的研究。主要研究者（PI）建议彻底研究这些基本问题，沿着它们与科学和工程实际问题的联系。智力价值：在同调计算中有效解决最优性问题需要数学和算法的发展。PI带来了这些所需的专业知识。除了数学和理论计算机科学的协同效应之外，与科学和工程中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学、物理学、化学等科学领域之间发挥协同作用。更广泛的影响：从蛋白质分子中的隧道到大型机器中的空隙，同源基团的优化在许多科学和工程应用中提供了重要的见解。这些问题的解决方案可以帮助制造更好的机器，设计新的药物，并快速建模的定制对象。这个项目的教育影响是数学和计算机科学之间的巨大协同作用，其动机是实际应用。课程笔记、网络发行版和通过该项目开发的软件系统将使科学界能够研究几何学、拓扑学和算法方面的挑战性问题。该项目支持的研究生将发展数学和理论计算机科学方面的技能，并编写强大，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.Broader impacts: 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.