Sorbonne Université

In this talk I will give an overview of Seba's billiard as a popular model in the field of Quantum Chaos. Consider a rectangular billiard with a Dirac mass placed in its interior. Whereas this mass has essentially no effect on the classical dynamics, it does have an effect on the quantum dynamics, because quantum wave packets experience diffraction at the point obstacle. Numerical investigations of this model by Petr Seba suggested that the spectrum and the eigenfunctions of the Seba billiard resemble the spectra and eigenfunctions of billiards which are classically chaotic.

I will give an introduction to this model and discuss recent results on quantum ergodicity, superscars and the validity of Berry's random wave conjecture. This talk is based on joint work with Par Kurlberg and Zeev Rudnick.

Several works (by Kontsevich, Soibelman, Berkovich, Nicaise, Boucksom, Jonsson...) have shown that the limit behavior of a one-parameter family $(X_t)$ of complex algebraic varieties can often be described using the associated Berkovich t-adic analytic space $X^b$. In a work in progress with E. Hrushovski and F. Loeser, we provide a new instance of this general phenomenon. Suppose we are given for every t an  $(n,n)$-form $ω_t$ on $X_t$ (for n= dim X). Then under some assumptions on the formula that describes $ω_t$, the family $(ω_t)$ has a "limit" ω, which is a real valued  (n,n)-form in the sense of Chambert-Loir and myself on the Berkovich space $X^b$, and the integral of $ω_t$ on $X_t$ tends to the integral of ω on $X^b$. 
In this talk I will first make some reminders about Berkovich spaces and (n,n)-forms in this setting, and then discuss the above result. 
In fact, as I will explain, it is more convenient to formulate it with  $(X_t)$ seen as a single algebraic variety over a non-standard model *C of C and (ω_t) as a (n,n) differential form on this variety. The field *C also carries a t-adic real valuation which makes it a model of ACVF (and enables to do Berkovich geometry on it), and our proof uses repeatedly RCF and ACVF theories.