Ph.D Thesis

Ph.D StudentBobrowski Omer
SubjectAlgebraic Topology of Random Fields and Complexes
DepartmentDepartment of Electrical and Computer Engineering
Supervisor PROFESSOR EMERITUS Robert Adler
Full Thesis textFull thesis text - English Version


Algebraic topology studies the topological spaces using algebraic machinery. One of its main aims lies in the fact that assigning algebraic structures (e.g. homology groups) to topological spaces can be used to classify them into classes of “similar” (e.g. homotopy equivalent) spaces, to study their properties and to study the behavior of mappings between them. The field of ‘Applied Algebraic Topology’ focuses on applying algebraic topology methods to study features of surfaces and functions arising in engineering scenarios. This field has generated considerable interest over the past few years. However, despite the fact that many of the problems in this area involve random data collection, its probabilistic foundations are still at a very preliminary stage. The main goal of this thesis is to explore such problems, and to supply at least some of them with rigorous probabilistic statements. We focus on two different probabilistic setups that generate intricate topological spaces which we are interested in studying using the methods of algebraic topology.

Random fields are stochastic processes defined over parameter spaces of dimension greater than one. For example, consider a noisy image as a random field on [0,1]2. As the domain of the process is of dimension greater than one, the graph of the process is typically a (random) manifold, rather than a simple one-dimensional line. Thus, many intriguing probabilistic questions on the geometrical and topological structure of the image arise.

A simplicial complex is a collection of vertices, edges, triangles, and simplexes of higher dimension, following a few basic rules, so one can think of it as a generalization of a graph. A geometric complex is a simplicial complex, where in order to decide whether to include a k-dimensional simplex or not, we need to verify whether its k vertices satisfy a certain geometrical property. Choosing the vertices of a geometric complex at random yields a random topological space with many interesting features.

In the first part of the thesis we study the persistent homology of Gaussian random fields, and compute its expected Euler characteristic. The results we present also have surprising and interesting consequences related to the critical points of Gaussian fields.  In the second part we focus on the limiting behavior of the Betti numbers of random geometric complexes, as the number of vertices goes to infinity. We study different ways to construct a geometric complex, each resulting in a completely different structure.