Random Algebraic Topology

Purdue University, West Lafayette IN

Investigators

Abstract

This project involves two areas of mathematics, probability and algebraic topology. The significant interaction between these areas has recently begun with "applied" algebraic topology, finding a wide range of applications in sensor networks, bioinformatics, computational chemistry, manifold learning, linguistics, and cosmology. This is due to the fact that many of the data set handled by applied topologists are based on a random sample from a larger population or contain different types of errors. Failure to capture the stochastic nature of topological data can result in a serious misunderstanding of the topology of the system. For example, the layer of extraneous topologies triggered by heavy-tailed errors, as a phenomena analogous to that of audio crackle in temporal signal analysis, will make it very hard to recover an underlying topology of the system. Another example includes topology on non-Euclidean spaces, especially those negatively curved. One good motivation for this study lies in its tree-like structure as a result of negative curvature. By virtue of this structure, the resulting non-Euclidean model can potentially be employed for analyses of various complex networks in physics, computer science, and cosmology. Combining topological concepts with the notion of randomness, this projects aims to properly understand random topology of the system. This project addresses research questions on random algebraic topology. Despite its recent development, most of the studies on random algebraic topology do not pay sufficient attentions to how heaviness or lightness of the tail of probability distributions affects the topology. One of the main focuses is on topological crackle - the layered structure of topological elements, typically caused by a heavy-tailed distribution. Although topological crackle has been recognized, at least empirically, in the statistical manifold learning, this project investigates it thoroughly from a probabilistic viewpoint. In light of the involvement of heavy-tailed distributions, extreme value theory (EVT) will be selected as a main tool for this study. Relying on EVT, the PI aims to establish a theoretical foundation for a graphical descriptor of the birth and death of topological cycles, called persistence diagram. One of the main focuses is on the random configuration of points in the persistence diagram by regarding them as a random closed set and discussing their asymptotics in terms of the Fell topology. As the second main topic, the project attempts to investigate random topology on the negatively curved non-Euclidean space. The PI mainly deals with a theoretical question: How do the parameters of the hyperbolic space, including its negative curvature, characterize the random topology? As expected, new phenomena will be uncovered that are never seen in the usual Euclidean setting. 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.