In this article, we combine complex-analytic and arithmetic tools to study the preperiodic points of $1$-dimensional complex dynamical systems. We show that for any fixed $a,b\in \mathbb{C}$ and any integer $d\ge 2$, the set of $c\in \mathbb{C}$ for which both $a$ and $b$ are preperiodic for $z^d+c$ is infinite if and only if $a^d=b^d$. This provides an affirmative answer to a question of Zannier, which itself arose from questions of Masser concerning simultaneous torsion sections on families of elliptic curves. Using similar techniques, we prove that if rational functions $\varphi ,\psi \in \mathbb{C}(z)$ have infinitely many preperiodic points in common, then all of their preperiodic points coincide (and, in particular, the maps must have the same Julia set). This generalizes a theorem of Mimar, who established the same result assuming that $\varphi$ and $\psi$ are defined over $\overline{\mathbb{Q}}$. The main arithmetic ingredient in the proofs is an adelic equidistribution theorem for preperiodic points over number fields and function fields, with nonarchimedean Berkovich spaces playing an essential role.

In this article, we consider a fairly general potential in the plane and the corresponding Boltzmann-Gibbs distribution of eigenvalues of random normal matrices. As the order of the matrices tends to infinity, the eigenvalues condensate on a certain compact subset of the plane—the “droplet.” We prove that fluctuations of linear statistics of eigenvalues of random normal matrices converge on compact subsets of the interior of the droplet to a Gaussian field, and we discuss various ramifications of this result.

In this article, we use the Lopez-Ros deformation to show that any embedded genus $1$ helicoid (or “genus-one helicoid”) must be symmetric with respect to rotation by $180°$ around a normal line. This partially answers a conjecture of Bobenko. We also show that this symmetry holds for an embedded genus $k$ helicoid $\Sigma$, provided that the underlying conformal structure of $\Sigma$ is hyperelliptic.

$W$-algebras of finite type are certain finitely generated associative algebras closely related to the universal enveloping algebras of semisimple Lie algebras. In this article, we prove a conjecture of Premet that gives an almost complete classification of finite-dimensional irreducible modules for $W$-algebras. A key ingredient in our proof is a relationship between Harish-Chandra bimodules and bimodules over $W$-algebras that is also of independent interest.

We give a new explicit construction of $n\times N$ matrices satisfying the Restricted Isometry Property (RIP). Namely, for some $ϵ>0$, large $N$, and any $n$ satisfying $N^{1-ϵ}\le n\le N$, we construct RIP matrices of order $k\ge n^{1/2+ϵ}$ and constant $\delta =n^{-ϵ}$. This overcomes the natural barrier $k=O(n^{1/2})$ for proofs based on small coherence, which are used in all previous explicit constructions of RIP matrices. Key ingredients in our proof are new estimates for sumsets in product sets and for exponential sums with the products of sets possessing special additive structure. We also give a construction of sets of $n$ complex numbers whose kth moments are uniformly small for $1\le k\le N$ (Turán's power sum problem), which improves upon known explicit constructions when $(\log N{)}^{1+o(1)}\le n\le (\log N{)}^{4+o(1)}$. This latter construction produces elementary explicit examples of $n\times N$ matrices that satisfy the RIP and whose columns constitute a new spherical code; for those problems the parameters closely match those of existing constructions in the range $(\log N{)}^{1+o(1)}\le n\le (\log N{)}^{5/2+o(1)}$.