CAREER: Tripodal Geometry and Applications

职业：三足几何及其应用

• 批准号: 0644613
• 负责人: Indira Chatterji
• 金额: $ 40万
• 依托单位: Ohio State University Research Foundation -DO NOT USE
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-15 至 2011-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0644613&HistoricalAwards=false
• 关键词: CAREER; Tripodal; Geometry; Applications

Tripodal spaces are metric spaces for which given any three points, there exists a choice of geodesics between them that intersect in a common point, called a median point. This type of metric has many natural examples including trees and products of those, taxi-cab metrics on the plane or the space of phylogenetic trees.I plan on developing this notion of tripodal geometry and show many more examples of spaces admitting tripodal metrics, examples such as SL(4,R). Spaces that almost admit a tripodal metric allow for many more examples keeping some of the tripodal features. Groups acting on tripodal spaces have some low complexity features such as being nicely embeddable in Euclidean spaces.Groups arise in many scientific fields because they encode symmetries of physical, biological or other systems, therefore it is important to study them. It is impossible to say anything useful about all groups, so we divide them into classes and study each class separately. I am interested in classes of groups that arise as symmetries of geometric objects, such as tripodal spaces.Understanding something about the geometry of these objects often can be translated into algebraic information about the group. I plan to develop tools from algebra and analysis to study tripodal geometry, with a particular emphasis on understanding geometric properties which only become apparent on a large scale.

三脚架空间是度量空间，给定任何三个点，它们之间存在一个相交于公共点（称为中点）的测地线选择。这种度量有许多自然的例子，包括树和产品，出租车度量在平面上或空间的系统发育树。我计划发展这个概念的三脚架几何和显示更多的例子空间接纳三脚架度量，例如SL（4，R）。几乎允许三脚架度量的空间允许更多的例子保持一些三脚架特征。作用在三脚架空间上的群具有一些低复杂性的特征，例如可以很好地嵌入到欧氏空间中。群出现在许多科学领域，因为它们编码了物理，生物或其他系统的对称性，因此研究它们很重要。不可能对所有的群体都说些有用的话，所以我们把他们分成几类，分别研究每一类。我感兴趣的是几何对象（如三足空间）的对称性所产生的群的类别，理解这些对象的几何通常可以转化为关于群的代数信息。我计划从代数和分析中开发工具来研究三脚架几何，特别强调理解只有在大规模上才变得明显的几何特性。