By using quasiconformal flows, we establish that exponentials of logarithmic potentials of measures of small mass are comparable to Jacobians of quasiconformal homeomorphisms of ${\mathbb{R}}^n$, $n\ge 2$. As an application, we obtain the fact that certain complete conformal deformations of an even-dimensional Euclidean space ${\mathbb{R}}^n$ with small total Paneitz or $Q$-curvature are bi-Lipschitz equivalent to standard ${\mathbb{R}}^n$

First-return and first-hitting times, local times, and first-intersection times are studied for planar finite-horizon Lorentz processes with a periodic configuration of scatterers. Their asymptotic behavior is analogous to the asymptotic behavior of the same quantities for the two-dimensional simple symmetric random walk (see classical results of Darling and Kac [DK] and Erdős and Taylor [ET]. Moreover, asymptotical distributions for phases in first hittings and in first intersections of Lorentz processes are also proved. The results are also extended to the quasi-one-dimensional model of the linear Lorentz process

A possible evolution of a compact hypersurface in ${\mathbb{R}}^{n+1}$ by mean curvature past singularities is defined via the level set flow. In the case where the initial hypersurface has positive mean curvature, we show that the Brakke flow associated to the level set flow is actually a Brakke flow with equality. As a consequence, we obtain the fact that no mass drop can occur along such a flow. A further application of the techniques used above is to give a new variational formulation for mean curvature flow of mean convex hypersurfaces

In this article, a positive answer is given to the following question posed by Hayman [35, page 326]: if a polyharmonic entire function of order $k$ vanishes on $k$ distinct ellipsoids in the Euclidean space ${\mathbb{R}}^n$, then it vanishes everywhere. Moreover, a characterization of ellipsoids is given in terms of an extension property of solutions of entire data functions for the Dirichlet problem, answering a question of Khavinson and Shapiro [39, page 460]. These results are consequences from a more general result in the context of direct sum decompositions (Fischer decompositions) of polynomials or functions in the algebra $A(B_R)$ of all real-analytic functions defined on the ball $B_R$ of radius $R$ and center zero whose Taylor series of homogeneous polynomials converges compactly in $B_R$. The main result states that for a given elliptic polynomial $P$ of degree $2k$ and for sufficiently large radius $R>0$, the following decomposition holds: for each function $f\in A(B_R)$, there exist unique $q,r\in A(B_R)$ such that $f=\mathrm{Pq}+r$ and ${\Delta }^{k}r=0$. Another application of this result is the existence of polynomial solutions of the polyharmonic equation ${\Delta }^{k}u=0$ for polynomial data on certain classes of algebraic hypersurfaces

We prove sharp limit theorems on random walks on graphs with values in finite groups. We then apply these results (together with some elementary algebraic geometry, number theory, and representation theory) to finite quotients of lattices in semisimple Lie groups (specifically, $\mathrm{SL}(n,\mathbb{Z})$ and $\mathrm{Sp}(2n,\mathbb{Z})$) to show that a random element in one of these lattices has irreducible characteristic polynomials (over $\mathbb{Z}$). The term random can be defined in at least two ways: first, in terms of height; second, in terms of word length in terms of a generating set. We show the result using both definitions.

We use these results to show that a random (in terms of word length) element of the mapping class group of a surface is pseudo-Anosov and that a random free group automorphism is irreducible with irreducible powers (or fully irreducible*)