In this talk I will define Gaussian random fields on manifolds and study their topological and geometric properties.
The first main result is a probabilistic version of Thom transversality theorem - a necessary lubricant for the generic argument in Differential Topology (it is a more general version of Sard's theorem) - that allows to change the word "generic" to "almost every" in almost every situations.
Secondly, I will address the problem of describing the asymptotic behaviour of a sequence of Gaussian Random Fields. The prototypical example is a sequences of random (Kostlan) polynomials having {degree $\to\infty$}.
I will present a general method to deal with the limit probability of differential geometric events (for instance: the probability of having a critical point inside a certain domain; the probability that a level set is diffeomorphic to some fixed closed manifold; etc.) and apply it to the case of Kostlan polynomials to obtain a generalization of the results by Gayet and Welschinger. Time permitting, I will discuss how to study the expected topology (Betti numbers) of singular sets.
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