We prove that the unit cube $B_{\infty }^n$ is a strict local minimizer for the Mahler volume product ${\mathrm{vol}}_n(K){\mathrm{vol}}_n(K^{*})$ in the class of origin-symmetric convex bodies endowed with the Banach-Mazur distance.

The local Langlands correspondence can be used as a tool for making verifiable predictions about irreducible complex representations of $p$-adic groups and their Langlands parameters, which are homomorphisms from the local Weil-Deligne group to the $L$-group. In this article, we refine a conjecture of Hiraga, Ichino, and Ikeda which relates the formal degree of a discrete series representation to the value of the local gamma factor of its parameter. We attach a rational function in $x$ with rational coefficients to each discrete parameter, which specializes at $x=q$, the cardinality of the residue field, to the quotient of this local gamma factor by the gamma factor of the Steinberg parameter. The order of this rational function at $x=0$ is also an important invariant of the parameter—it leads to a conjectural inequality for the Swan conductor of a discrete parameter acting on the adjoint representation of the $L$-group. We verify this conjecture in many cases. When we impose equality, we obtain a prediction for the existence of simple wild parameters and simple supercuspidal representations, both of which are found and described in this article.

We show that certain lamplighter groups that are quasi-isometric to each other are not bilipschitz equivalent. This gives a positive answer to a question of de la Harpe.

We analyze Loewner traces driven by functions asymptotic to $\kappa \sqrt{1-t}$. We prove a stability result when $\kappa \ne 4$, and we show that $\kappa =4$ can lead to nonlocally connected hulls. As a consequence, we obtain a driving term $\lambda (t)$ so that the hulls driven by $\kappa \lambda (t)$ are generated by a continuous curve for all $\kappa >0$ with $\kappa \ne 4$, but not when $\kappa =4$, so that the space of driving terms with continuous traces is not convex. As a byproduct, we obtain an explicit construction of the traces driven by $\kappa \sqrt{1-t}$ and a conceptual proof of the corresponding results of Kager, Nienhuis, and Kadanoff.

We show that the group of bounded automatic automorphisms of a rooted tree is amenable, which implies amenability of numerous classes of groups generated by finite automata. The proof is based on reducing the problem to showing amenability just of a certain explicit family of groups (mother groups) which is done by analyzing the asymptotic properties of random walks on these groups.

It was proved by Devaney and Krych, by McMullen, and by Karpińska that, for $0<\lambda <1/e$, the Julia set of $\lambda e^z$ is an uncountable union of pairwise disjoint simple curves tending to infinity, and the Hausdorff dimension of this set is $2$, but the set of curves without endpoints has Hausdorff dimension $1$. We show that these results have $3$-dimensional analogues when the exponential function is replaced by a quasi-regular self-map of ${\mathbb{R}}^3$ introduced by Zorich.