CAREER: Persistence Theories for Computational Topology

Dartmouth College, Hanover NH

Investigators

Abstract

Proposal: CCF- 0845716 Title: CAREER: Persistence Theories for Computational Topology PI: Zomorodian, Afra Institution: Dartmouth College ABSTRACT Data is often in the form of a set of points, sampled from some underlying original space. For example, a computer representation of a three-dimensional object is usually constructed from a set of points sampled from its surface using a 3D laser scanner. Another example is the set of snapshots from the simulated trajectory of a folding protein, a four-dimensional point set. Given any set of points, topological analysis tries to recover how the original space was connected. As such, topological analysis is a significant first step in understanding data, discovering its global structure. Due to its fundamental nature, topological analysis has applications in diverse disciplines, such as shape description, terrain analysis, sensor networks, and dynamical systems. This research focuses on the theoretical and algorithmic development of persistence theories. Intuitively, persistence is the technique of identifying features by analyzing geometric histories of data. The key notion is that features persist through history while noise is short-lived. Currently, there exists a persistence theory only for one class of data: the theory of persistent homology for static point cloud data. This aim of this research is to develop persistence theories for other major classes of data that arise in the practice of computational topology, such as parameterized data, dynamic data, and data in diagrams.