We will explain the equivalences between p-adic Galois representations and various types of $(\varphi,\Gamma)$-modules.
The condition number of a matrix is at the heart of numerical linear algebra. In the 1940s von-Neumann and Goldstine, motivated by the problem of inverting, posed the following question:
(1) What is the condition number of a random matrix ?
During the years, this question was raised again and again, by various researchers (Smale, Demmel etc). About ten years ago, motivated by "Smoothed Analysis", Spielman and Teng raised a more general question:
(2) What is the condition number of a randomly perturbed matrix ?