Suppose that $G$ is a simple adjoint reductive group over $\mathbf{Q}$, with an exceptional Dynkin type and with $G\left(\mathbf{R}\right)$ quaternionic (in the sense of Gross and Wallach). Then there is a notion of modular forms for $G$, anchored on the so-called quaternionic discrete series representations of $G\left(\mathbf{R}\right)$. The purpose of this paper is to give an explicit form of the Fourier expansion of modular forms on $G$, along the unipotent radical $N$ of the Heisenberg parabolic $P$ of $G$.

Catanese has recently asked if there exists an element of the absolute Galois group $\sigma \in Gal\left(\overline{\mathbb{Q}}\right)$ for which there is a Kodaira fibration $f:S\to B$ defined over a number field such that the universal covers of $S$ and its Galois conjugate surface $S^{\sigma }$ are not isomorphic. The main result of this article is that every element $\sigma \ne \mathrm{Id}$ has this property.

For $0<p<+\infty$, we prove a global $W^{2,p}$-estimate for potentials of optimal transport maps between convex domains in the plane. Among the tools developed for that purpose are obliqueness in general convex domains and estimates for the growth of eccentricity of sections of the potentials.

We provide the first tight bounds on the lower tail probability of the one-point distribution of the Kardar–Parisi–Zhang (KPZ) equation with narrow wedge initial data. Our bounds hold for all sufficiently large times $T$ and demonstrates a crossover between superexponential decay with exponent $\frac{5}{2}$ (and leading prefactor $\frac{4}{15\pi }{T}^{1/3}$) for tail depth greater than $T^{2/3}$, and exponent $3$ (with leading prefactor at least $\frac{1}{12}$) for tail depth less than $T^{2/3}$.