Random Differential Topology

This manuscript collects three independent works: arXiv:1902.03805, arXiv:1906.04444, with Antonio Lerario and arXiv:2103.10853, together with some additional results, observations, examples and comments, some of which were taken up in the subsequent work arxiv:2010.14553 (with Antonio Lerario). The topic discussed in this thesis are at the crossroad of Differential Topology and Random Geometry. The common thread of these works is the study of topological and geometric properties of random smooth maps. The first chapter contains the motivations and the main results of the thesis. In particular it describes how these works are related as parts of a general method to study topological properties of smooth random maps. In the second chapter a general framework to deal with issues of differential geometric and topological nature regarding smooth Gaussian Random Fields is developed. The main results in this context are: a characterization of the convergence in law in terms of the covariance functions and a probabilistic version of Thom's jet transversality theorem. The third chapter is devoted to a generalization of the famous Kac-Rice formula. The formula presented here calculates the expected number of points at which a smooth random map meets a given (deterministic) submanifold of the codomain, whereas the standard formula deals with preimages of a point. In the fourth chapter all the previous methods are applied to Kostlan random polynomials. The fifth and last chapter of the thesis is devoted to present and prove an original theorem of Differential Topology. It says that the Betti numbers of the solution of a system of regular equations cannot decrease under a $\mathcal{C}^0$-small perturbation of the equations.