We discuss localization and local eigenvalue statistics for Schr\"odinger operators with random point interactions on $R^d$, for $d=1,2,3$. The results rely on probabilistic estimates, such as the Wegner and Minami estimate, for the eigenvalues of the Schr\"odinger operator restricted to cubes. The special structure of the point interactions facilitates the proofs of these eigenvalue correlation estimates.
One of the main results is that the local eigenvalue statistics is given by a Poisson point process in the localization regime, one of the first examples of Poisson eigenvalue statistics for multi-dimensional random Schr\"odinger operators in the continuum. This is joint work with M.\ Krishna and W.\ Kirsch.
Abstract: Renormalization provides a powerful tool to approach universality and
rigidity phenomena in dynamical systems. In this talk, I will discuss
recent results on renormalization and rigidity theory of circle
diffeomorphisms (maps) with a break (a single point where the derivative
has a jump discontinuity) and their relation with generalized interval
exchange transformations introduced by Marmi, Moussa and Yoccoz. In a
joint work with K.Khanin, we proved that renormalizations of any two
sufficiently smooth circle maps with a break, with the same irrational
rotation number and the same size of the break, approach each other
exponentially fast. For almost all (but not all) irrational rotation
numbers, this statement implies rigidity of these maps: any two
sufficiently smooth such maps, with the same irrational rotation number
(in a set of full Lebesgue measure) and the same size of the break, are
$C^1$-smoothly conjugate to each other. These results can be viewed as
an extension of Herman's theory on the linearization of circle
Uniform laws of large numbers provide theoretical foundations for statistical learning theory. This talk will focus on quantitative uniform laws of large numbers for random matrices. A range of illustrations will be given in high dimensional geometry and data science.