The Anderson operator is a perburbation of the Laplace-Beltrami operator by a space white noise potential. I will explain how to get a short self-contained functional analysis construction of the operator and how a sharp description of its heat kernel leads to useful quantitative estimates on its eigenvalues and eigenfunctions. One can associate to Anderson operator a (doubly) random field called the Anderson Gaussian free field. The law of its (random) partition function turns out to characterize the law of the spectrum of the operator. The square of the Anderson Gaussian free field turns out to be related to a probability measure on paths built from the operator, called the polymer measure.

Tensor decomposition express a tensor as a linear combination of elementary tensors. They have applications in chemometrics, computer science, machine learning, psychometrics, and signal processing. Their uniqueness properties render them suitable for data analysis tasks in which the elementary tensors are the quantities of interest. However, in applications, the idealized mathematical model is corrupted by measurement errors. For a robust interpretation of the data, it is therefore imperative to quantify how sensitive these elementary tensors are to perturbations of the whole tensor. I will give an overview of recent results on the condition number of tensor decompositions, established with my collaborators C. Beltran, P. Breiding, and N. Dewaele.