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

AF：媒介：协作研究：同调中的最优性 - 算法和应用

• 批准号: 1064416
• 负责人: Tamal Dey
• 金额: $ 35.29万
• 依托单位: Ohio State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-08-01 至 2016-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1064416&HistoricalAwards=false
• 关键词: AF; Medium; Collaborative; Research; Optimality

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.

科学和工程中的许多应用都遇到了识别和处理几何形状或数据的拓扑上有趣的特征的问题。对于某些度量（测量），这种特征通常需要最佳。 ISRERED认为，来自代数拓扑的同源组在这些计算中起着不可能的作用。尽管对同源组的结构性研究的研究在数学方面具有丰富的历史，但它们与几何形状结合的计算并没有那么精心研究。首席研究人员（PIS）提议彻底研究这些基础问题，以及他们的科学和工程学的界面问题。智能优点：最佳问题的有效解决方案无性问题计算需要数学和算法，也需要算法。 PI带上这些必需的专业知识。 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从蛋白质分子中的隧道到大型巨星的空隙，科学和工程应用。这些问题的解决方案可以帮助制造啤酒机器，设计新药以及对物体的快速建模。该项目的教育影响在于数学和计算机科学之间的协同作用，这是动机的byreal应用程序。通过该项目开发的课程说明，互联网分布和软件系统将使科学社区能够研究几何，拓扑，拓扑，andalgorithms中的具有挑战性的问题。该项目支持的研究生将开发数学和理论计算机科学领域的技能，并为强大，高效和用户友好的软件提供重要的信息。