For each d, we construct $\mathrm{CAT}(0)$ cube complexes on which Cremona groups of rank d act by isometries. From these actions we deduce new and old group-theoretical and dynamical results about Cremona groups. In particular, we study the dynamical behavior of the irreducible components of exceptional loci. This leads to proofs of regularization theorems, such as the regularization of groups with property FW. We also find new constraints on the degree growth for non-pseudo-regularizable birational transformations, and we show that the centralizer of certain birational transformations is small.

The number of real roots has been a central subject in the theory of random polynomials and random functions since the fundamental papers of Littlewood, Offord, and Kac in the 1940s. The main task here is to determine the limiting distribution of this random variable. In 1974, Maslova famously proved a central limit theorem (CLT) for the number of real roots of Kac polynomials. It has remained the only limiting theorem available for the number of real roots for more than four decades. In this paper, using a new approach, we derive a general CLT for the number of real roots of a large class of random polynomials with coefficients growing polynomially. Our result both generalizes and strengthens Maslova’s theorem.

We discover a rigidity phenomenon within the volume-preserving partially hyperbolic diffeomorphisms with 1-dimensional center. In particular, for smooth ergodic perturbations of certain algebraic systems—including the discretized geodesic flows over hyperbolic manifolds and certain toral automorphisms with simple spectrum and exactly one eigenvalue on the unit circle—the smooth centralizer is either virtually ${\mathbb{Z}}^{\ell }$ or contains a smooth flow.

At the heart of this work are two very different rigidity phenomena. The first was discovered by Avila, Viana, and the second author: for a class of volume-preserving partially hyperbolic systems including those studied here, the disintegration of volume along the center foliation is equivalent either to Lebesgue or atomic. The second phenomenon, described by the first and third authors, is the rigidity associated to several commuting partially hyperbolic diffeomorphisms with very different hyperbolic behavior transverse to a common center foliation.

We employ a variety of techniques, among them a novel geometric approach to building new partially hyperbolic elements in hyperbolic Weyl chambers using Pesin theory and leafwise conjugacy, measure rigidity via thermodynamic formalism for circle extensions of Anosov diffeomorphisms, partially hyperbolic Livšic theory, and nonstationary normal forms.

We prove an abstract compactness theorem for a family of generalized Seiberg–Witten equations in dimension 3. This result recovers Taubes’s compactness theorem for stable flat $\mathbf{P}{\mathrm{SL}}_2(\mathbf{C})$-connections as well as the compactness theorem for Seiberg–Witten equations with multiple spinors by Haydys and Walpuski. Furthermore, this result implies a compactness theorem for the ${\mathrm{ADHM}}_{1,2}$ Seiberg–Witten equation, which partially verifies a conjecture by Doan and Walpuski.