The topic discussed in this thesis are at the crossroad of Differential Topology and Random Geometry.
In the first 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 second chapter is devoted to a generalization of the famous Kac-Rice formula. The formula presented here calculates the expected value of the number of points at which a smooth random map meets a given (deterministic) submanifold of the range. Such formula holds true under a series of natural hypotheses, particularly easy to verify in the gaussian case.
In the third chapter all the previous methods are applied to Kostlan random polynomials. The framework develped in the previous chapters allows to give simpler and more general proofs of many known results in this context. In particular, for certain types of singular loci of Kostlan polynomials of degree d on the m-sphere, the asymptotic behaviour of the expected Betti numbers is proven to be of order d^(m/2), while their maximal behaviour is of order d^m, in accordance with the so called "square root law" principle.
The last chapter aims to present a deterministic original result of Differential Topology saying that the Betti numbers of the solution of a system of regular equations cannot decrease under a C^0-small perturbation of the equations. Coupled with approximation theorems, it is used to prove one of the results of chapter 2 and to obtain an analogue of Milnor-Thom bound for the Betti numbers of smooth compact hypersurfaces of R^n.
