Let D be a disk in R n and f ∈ C r+2 (D, R k ). We deal with the problem of the algebraic approximation of the set j r f −1 (W ) consisting of the set of points in the disk D where the r-th jet extension of f meets a given semialgebraic set W ⊂ J r (D, R k ). We call such sets type-W singularities; examples of sets arising in this way are the zero set of f , or the set of its critical points.Under some transversality conditions, we prove that f can be approximated with a polynomial map p : D → R k such that the corresponding singularity is diffeomorphic to the original one, and such that the degree of this polynomial map can be controlled by the C r+2 data of f . More precisely, denoting by ∆W ⊂ C r+1 (D, R k ) the set of maps whose r-th jet extension is not transverse to W , we show that there exists a polyomial p such that:

Motivated by numerous questions in random geometry, given a smooth manifold M , we approach a systematic study of the differential topology of Gaussian Random Fields (GRF) X : M → R k , i.e. random variables with values in C ∞ (M, R k ) inducing on it a Gaussian measure. We endow the set of GRFs with the narrow topology and we prove a results relating the convergence in the Whitney C ∞ topology of the covariance structure of X and the random variable X ∈ C ∞ (M, R k ) itself. When dealing with a convergent family {X d } d∈N of GRFs, these results allow to compute the limit probabilities of a family of events in terms of the probability distribution of the limit GRF.We complement this study by proving an important technical tools: an infinite dimensional, probabilistic version of Thom Transversality Theorem, which ensures that, under some conditions, a GRF is almost surely transversal to any given submanifold of the jet space.

We prove a generalized Kac-Rice formula that, in a well defined regular setting, computes the expected cardinality of the preimage of a submanifold via a random map, by expressing it as the integral of a density. Our proof starts from scratch and although it follows the guidelines of the standard proofs of Kac-Rice formula, it contains some new ideas coming from the point of view of measure theory. Generalizing further, we extend this formula to any other type of counting measure, such as the intersection degree.We discuss in depth the specialization to smooth Gaussian random sections of a vector bundle. Here, the formula computes the expected number of points where the section meets a given submanifold of the total space, it holds under natural nondegeneracy conditions and can be simplified by using appropriate connections. Moreover, we point out a class of submanifolds, that we call sub-Gaussian, for which the formula is locally finite and depends continuously with respect to the covariance of the first jet. In particular, this applies to any notion of singularity of sections that can be defined as the set of points where the jet prolongation meets a given semialgebraic submanifold of the jet space.Various examples of applications and special cases are discussed. In particular, we report a new proof of the Poincaré kinematic formula for homogeneous spaces and we observe how the formula simplifies for isotropic Gaussian fields on the sphere. Contents 0.1. Overview 0.2. Structure of the paper 0.3. Aknowledgements

Spin (spherical) random fields are very important in many physical applications, in particular they play a key role in Cosmology, especially in connection with the analysis of the Cosmic Microwave Background radiation. These objects can be viewed as random sections of the s-th complex tensor power of the tangent bundle of the 2-sphere. In this paper, we discuss how to characterize their expected geometry and topology. In particular, we investigate the asymptotic behaviour, under scaling assumptions, of general classes of geometric and topological functionals including Lipschitz-Killing Curvatures and Betti numbers for (properly defined) excursion sets; we cover both the cases of fixed and diverging spin parameters s. In the special case of monochromatic fields (i.e., spin random eigenfunctions) our results are particularly explicit; we show how their asymptotic behaviour is non-universal and we can obtain in particular complex versions of Berry's random waves and of Bargmann-Fock's models as subcases of a new generalized model, depending on the rate of divergence of the spin parameter s.

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