The Topology of Random Manifolds

We will investigate zero sets of random functions, starting in the univariate case with a simple question motivated by the fundamental theorem of algebra, How many zeros of a random polynomial are real? . After discussing the answer provided by M. Kac, we then consider the case of a random manifold determined by the zero set of a random function in several variables. We will focus on a question motivated by seeking a broad point of view on Hilbert\u27s sixteenth problem: What is the topology of a random real algebraic manifold? . Recent progress addresses the average number of connected components of a random manifold, the embedding of components in the ambient space, and higher-dimensional Betti numbers. We survey these results and give an overview of related problems representing several new avenues of research