Advances in random topology

Random topology refers to the study of topological properties of randomly generated spaces.This area of research dates back to the seminal result by Erdős and Rényi on the connectivity of random graphs (Erdős and Rényi 1959).Over the past two decades this field has gained increasing attention, driven by two primary developments.The first is the elegant generalization of the Erdős-Rényi result, by Linial and Meshulam (2006), from connectivity in random graphs to homological-connectivity in random simplicial complexes.This result has facilitated a new area of study in combinatorial topology.The second development is the rise of Topological Data Analysis (TDA).This field focuses on utilizing methods from algebraic topology in the analysis of data and networks, based on their global shape.As data is random, probabilistic analysis is essential for developing solid statistical tests.In this line of research, the focus is on objects of geometric nature (i.e., geometric complexes, random fields).Though these two lines of research have evolved largely in parallel, they share numerous concepts and methods.In this special issue, we tried to gather topics that cover the wide spectrum of studies that one might consider as "random topology".The types of models and phenomena that fall under this category are quite varied.The most fruitful line of study has been the topology of random simplicial complexes.The result in Linial and Meshulam (2006) has sparked a whole line of research on combinatorial models.There is a variety of models including the random k-complex, clique complex, multi-parameter complex, Rado complex, and more.There are also various phenomena studied, including phase transitions for homology and the fundamental group, spectral properties, central limit theorems, and more.The study of random geometric complexes was initiated by Kahle (2011), extending the notion of random geometric graphs (Penrose 2003).Here, the most well studied models are the Čech and the Vietoris-Rips complexes, which are also the ones often used in TDA.The range of phenomena studied in the geometric