Title: Homological Percolation: The Formation of Giant k-Cycles
Speaker: Omer Bobrowski
Speaker Info: Technion
The field of percolation studies the formation of “giant” connected components in various types of random media. In this talk, we will discuss a higher dimensional analogue of this phenomenon, where instead of connected components, we are looking at k-dimensional cycles — topological objects that describe structures in various dimensions. For example, a 0-cycles is a connected component, a 1-cycle is a loop that surrounds a hole, and a 2-cycle is a surface that encloses a cavity. We focus on a continuum percolation model, where the underlying point process is generated on a compact manifold M. Among all the k-cycles formed at random, we consider as ``giant" those that correspond to one of the k-cycles of M. Similarly to the classical study in percolation theory, our goal is to analyze the phase transitions describing the emergence of these giant cycles. We will also present an unexpected (heuristic) connection to the Euler characteristic.Date: Tuesday, June 09, 2020
This is joint work with Primoz Skraba from Queen Mary University of London.