CAREER: Tripodal Geometry and Applications

Ohio State University Research Foundation -Do Not Use, Columbus OH

Investigators

Abstract

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.