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

科学和工程中的许多应用都遇到了在几何图形或数据的数字表示中识别和处理拓扑上有趣的特征的问题。这些特征通常需要在某些度量（度量）方面是最优的。人们认识到，代数拓扑的同调群在这些计算中起着至关重要的作用。虽然对同调群的结构性质的研究在数学上有着丰富的历史，但它们与几何相结合的计算并没有得到很好的研究。主要研究者（pi）建议深入研究这些基本问题，以及它们与科学和工程实际问题的联系。智力优势：同调计算中最优性问题的有效解决需要数学和算法的发展。pi带来了这些所需的专业知识。除了拟议的数学和理论计算机科学研究的协同效应外，与科学和工程中的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与生物学，物理，化学等科学领域之间发挥协同作用。同源群的优化在许多科学和工程应用中提供了重要的见解，从蛋白质分子的隧道到大型机器的空隙。这些问题的解决有助于制造更好的机器，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巨大协同作用，这些协同作用是由实际应用驱动的。通过该项目开发的课程笔记、网络分发和软件系统将使科学界能够研究几何、拓扑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技能，以及编写健壮、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.