CAREER: Tripodal Geometry and Applications

三脚架空间是公制空间，对于给定的任意三个点，它们之间存在一条测地线，它们相交于一个公共点，称为中点。这种类型的度量有许多自然的例子，包括树及其乘积、平面上的出租车度量或系统发育树的空间。我计划发展这种三脚架几何的概念，并展示更多允许三脚架度量的空间的例子，例如SL(4，R)。几乎接受三脚架度量的空间允许更多的例子保持一些三脚架的特征。作用在三脚架空间上的群具有可嵌入欧几里得空间等低复杂度的特点，由于群编码了物理、生物或其他系统的对称性，所以群在许多科学领域都有出现，因此对群的研究具有重要意义。不可能说出关于所有小组的任何有用的东西，所以我们把他们分成班级，分别研究每个班级。我对几何对象的对称性所产生的一类群感兴趣，例如三脚架空间。理解这些对象的几何知识通常可以转化为关于群的代数信息。我计划开发代数和分析的工具来研究三脚架几何，特别强调理解只有在大范围内才变得明显的几何性质。

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.