CAREER: Tripodal Geometry and Applications

三脚架空间是一种度量空间，给定任意三个点，它们之间的测地线可以选择相交于一个公共点，称为中点。这种度量有许多自然的例子，包括树和树的乘积，平面上的出租车度量或系统发育树的空间。我计划发展这个三脚架几何的概念，并展示更多允许三脚架度量的空间例子，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留一些三脚架特征。作用于三脚架空间上的群具有一些低复杂度的特征，例如在欧几里得空间中可以很好地嵌入。群体出现在许多科学领域，因为它们编码物理、生物或其他系统的对称性，因此研究它们很重要。我们不可能对所有的小组都说些有用的东西，所以我们把他们分成班级，每个班级分别学习。我对由几何物体对称而产生的群类很感兴趣，比如三脚架空间。对这些物体几何形状的了解通常可以转化为群体的代数信息。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.