I will discuss the notion of Random Map and present the space $\mathcal{G}^{r}(M,\mathbb{R}^{k})$ of $\mathcal{C}^r$ Gaussian Random Fields on a smooth manifold $M$. For example a polynomial with gaussian coefficients $p(u)=\sum_{\beta\le d}\xi_\beta u^\beta$ is a gaussian random field belonging to the space $\mathcal{G}^{\infty}(\mathbb{R},\mathbb{R})$.

This kind of random maps are determined by a deterministic object called *covariance function*.

A problem which arises frequently in the field of stochastic geometry is to understand the asymptotic behaviour of $\mathbb{P}\{X_d\in U\}$, as $d\to +\infty$, for a given sequence $X_d\in \mathcal{G}^{r}(M,\mathbb{R}^{k})$ and an open subset $U\subset \mathcal{C}^{r}(M,\mathbb{R}^{k})$. For example, $U$ can be the subset of $\mathcal{C}^{1}(S^1,\mathbb{R}^{3})$ of all embeddings that represent the trefoil knot or the set of all vector fields in $\mathcal{C}^{\infty} \left( \mathbb{R}^2,\mathbb{R}^2 \right)$ with exactly $576$ saddle points inside a fixed ball. In the case of a compact manifold $M^m$, one can consider $U$ to be the subset of $\mathcal{C}^{\infty}(M,\mathbb{R})$ of all functions whose zero set is diffeomorphic to $S^{m-1}$.

I will describe a strategy in two steps to study the behaviour of $\mathbb{P}\{X_d\in U\}>0$ as $d\to \infty$, by looking at the corresponding covariance functions. I will present also a probabilistic version of Thom's transversality theorem, with which it is possible to give a good criteria for the existence of the limit.

*This is a joint work with Antonio Lerario.*