Kalton Program, Ribe Program, and Metric Invariants

Geometric approachs often provide an effective framework to study a great variety of problems, ranging from modeling the interactions of elementary particles to practical problems such as computer vision. Networks such as social networks or telecommunication networks can naturally be seen as a geometric object by considering the number of edges of the shortest path connecting two nodes in the network as a quantity measuring their proximity. A graph equipped with its shortest path distance is an example of what mathematicians call a metric space. This extremely useful abstract concept generalizes the classical notion of distance and plays a pivotal role in mathematical models for optimization problems in networks. Understanding whether a graph, which is a nonlinear object, can be faithfully represented in a linear space allows one to leverage a wealth of geometric tools to gain insight. This project will also provide training to graduate students and junior researchers. In this project, the Principal Investigator will use various curvature-like inequalities to measure the distortion of a geometric structure when it is mapped into curved space using non-standard probabilistic framework. The study of curvature-like inequalities and metric embeddings is strongly connected to a central aspect of the Ribe program. The Kalton program consists in the discovery of metric invariants that capture the geometry of graphs and characterize local and asymptotic properties of Banach spaces. On a conceptual level, the project helps to explain how problems in the Kalton program can be seen as limits of problems in the Ribe program. At the same time, the new probabilistic intuition should provide new approaches to attack long-standing open problems in the Kalton program. The project concentrates on new metrics invariants capturing the geometry of countably branching diamond graphs and trees and to formulate an asymptotic Enflo problem that is central to the study of the geometry of countably branching Hamming graphs. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.