The probability mass function $1/j(j+1)$ for $j\geq 1$ belongs to the domain of attraction of a completely asymmetric Cauchy distribution.

The purpose of the talk is to review some of applications of this simple observation to limit theorems related to the destruction of random recursive trees.

Specifically, a random recursive tree of size $n+1$ is a tree chosen uniformly at random amongst the $n!$ trees on the set of vertices $\{0,1, 2, ...,  n\}$ such that the sequence of vertices along any segment starting from the root $0$ increases. One destroys this tree by removing its edges one after the other in a uniform random order. It was first observed by Iksanov and M\"ohle that the central limit theorems for the random walk with step distribution given above explains the fluctuations of the number of cuts needed to isolate the root. We shall discuss further results in the same vein.

We consider classes of diffeomorphisms of Euclidean space with partial asymptotic expansions at infinity; the remainder term lies in a weighted Sobolev space whose properties at infinity fit with the desired application. We show that two such classes of asymptotic diffeomorphisms form topological groups under composition. As such, they can be used in the study of fluid dynamics according to the method of V. Arnold. Specific applications have been obtained for the Camassa-Holm equation and the Euler equations.

I shall discuss an entropy functional defined for convex bodies and its related analysis in the study of the large time asymptotics of the Gauss curvature flow. This is a joint work with Pengfei Guan at McGill.