In previous works, we showed that the internal diffusion-limited aggregation (DLA) cluster on ${\mathbb{Z}}^d$ with $t$ particles is almost surely spherical up to a maximal error of $O(logt)$ if $d=2$ and $O\left(\sqrt{logt}\right)$ if $d\ge 3$. This paper addresses average error: in a certain sense, the average deviation of internal DLA from its mean shape is of constant order when $d=2$ and of order $r^{1-d/2}$ (for a radius $r$ cluster) in general. Appropriately normalized, the fluctuations (taken over time and space) scale to a variant of the Gaussian free field.

Kaimanovich and Masur showed that a random walk on the mapping class group for an initial distribution whose support generates a nonelementary subgroup when projected into Teichmüller space converges almost surely to a point in the space $\mathcal{PMF}$ of projective measured foliations on the surface. This defines a harmonic measure on $\mathcal{PMF}$. Here, we show that when the initial distribution has finite support, the corresponding harmonic measure is singular with respect to the natural Lebesgue measure class on $\mathcal{PMF}$.

We consider the nonlinear Schrödinger equation $i{\partial }_{t}u+\Delta u+u|u{|}^{p-1}=0$ in dimension $N\ge 2$ and in the mass supercritical and energy subcritical range $1+\frac{4}{N}<p<min\hspace{0.17em}\{\frac{N+2}{N-2},5\}$. For initial data $u_0\in H^1$ with radial symmetry, we prove a universal upper bound on the blow-up speed. We then prove that this bound is sharp and attained on a family of collapsing ring blow-up solutions first formally predicted in Fibich et al.

We prove an analogue of the classical Ax–Lindemann theorem in the context of compact Shimura varieties. Our work is motivated by Pila’s strategy for proving the André–Oort conjecture unconditionally.