We initiate the study of $\left(1+2\right)$-dimensional wave maps on a curved space time in the low-regularity setting. Our main result asserts that in this context the wave maps equation is locally well-posed at almost critical regularity.

As a key part of the proof of this result, we generalize the classical optimal bilinear $L^2$ estimates for the wave equation to variable coefficients by means of wave packet decompositions and characteristic energy estimates. This allows us to iterate in a curved $X^{s,b}$ space.

This paper is devoted to the derivation and mathematical analysis of a wave-structure interaction problem which can be reduced to a transmission problem for a Boussinesq system. Initial boundary value problems and transmission problems in dimension $d=1$ for $2\times 2$ hyperbolic systems are well understood. However, for many applications, and especially for the description of surface water waves, dispersive perturbations of hyperbolic systems must be considered. We consider here a configuration where the motion of the waves is governed by a Boussinesq system (a dispersive perturbation of the hyperbolic nonlinear shallow water equations), and in the presence of a fixed partially immersed obstacle. We shall insist on the differences and similarities with respect to the standard hyperbolic case, and focus our attention on a new phenomenon, namely, the apparition of a dispersive boundary layer. In order to obtain existence and uniform bounds on the solutions over the relevant time scale, a control of this dispersive boundary layer and of the oscillations in time it generates is necessary. This analysis leads to a new notion of compatibility condition that is shown to coincide with the standard hyperbolic compatibility conditions when the dispersive parameter is set to zero. To the authors’ knowledge, this is the first time that these phenomena (likely to play a central role in the analysis of initial boundary value problems for dispersive perturbations of hyperbolic systems) are exhibited.

We prove global-in-time Strichartz-type estimates for the Schrödinger equation on manifoldsof the form ${ℝ}^n\times {\mathbb{𝕋}}^d$, where ${\mathbb{𝕋}}^d$ is a $d$-dimensional torus. Our results generalize and improve a global space-time estimate for the Schrödinger equation on $ℝ\times {\mathbb{𝕋}}^2$ due to Z. Hani and B. Pausader. As a consequence we prove global existence and scattering in $H^{1∕2}$ for small initial data for the quintic NLS on $ℝ\times \mathbb{𝕋}$ and the cubic NLS on ${ℝ}^2\times \mathbb{𝕋}$.

We consider large Hermitian matrices whose entries are defined by evaluating the exponential function along orbits of the skew-shift $\left(\begin{array}{c}j\\ 2\end{array}\right)\omega +jy+x\mathrm{mod}1$ for irrational $\omega$. We prove that the eigenvalue distribution of these matrices converges to the corresponding distribution from random matrix theory on the global scale, namely, the Wigner semicircle law for square matrices and the Marchenko–Pastur law for rectangular matrices. The results evidence the quasirandom nature of the skew-shift dynamics which was observed in other contexts by Bourgain, Goldstein and Schlag and Rudnick, Sarnak and Zaharescu.

We revisit Matui and Sato’s notion of property (SI) for ${\text{C}}^{\ast }$-algebras and ${\text{C}}^{\ast }$-dynamics. More specifically, we generalize the known framework to the case of ${\text{C}}^{\ast }$-algebras with possibly unbounded traces. The novelty of this approach lies in the equivariant context, where none of the previous work allows one to (directly) apply such methods to actions of amenable groups on highly nonunital ${\text{C}}^{\ast }$-algebras, in particular to establish equivariant Jiang–Su stability. Our main result is an extension of an observation by Sato: for any countable amenable group $\mathrm{\Gamma }$ and any nonelementary separable simple nuclear $C^{\ast }$-algebra $A$ with strict comparison, every $\mathrm{\Gamma }$-action on $A$ has equivariant property (SI). A more general statement involving relative property (SI) for inclusions into ultraproducts is proved as well. As a consequence we show that if $A$ also has finitely many rays of extremal traces, then every $\mathrm{\Gamma }$-action on $A$ is equivariantly Jiang–Su stable. We moreover provide applications of the main result to the context of strongly outer actions, such as a generalization of Nawata’s classification of strongly outer automorphisms on the (stabilized) Razak–Jacelon algebra.

We initiate the study of multilayer potential operators associated with any given homogeneous constant-coefficient higher-order elliptic system $L$ in an open set $\mathrm{\Omega }\subseteq {ℝ}^n$ satisfying additional assumptions of a geometric measure theoretic nature. We develop a Calderón–Zygmund-type theory for this brand of singular integral operators acting on Whitney arrays, starting with the case when $\mathrm{\Omega }$ is merely of locally finite perimeter and then progressively strengthening the hypotheses by ultimately assuming that $\mathrm{\Omega }$ is a uniformly rectifiable domain (which is the optimal setting where singular integral operators of principal value type are known to be bounded on Lebesgue spaces), and conclude by indicating how this body of results is significant in the context of boundary value problems for the higher-order system $L$ in such a domain $\mathrm{\Omega }$.

We first establish short-time existence and a Shi-type estimate of second Ricci flow on complete noncompact Hermitian manifolds. As an application, we use the second Ricci flow to discuss the existence of a Kähler–Ricci metric on complete noncompact Hermitian manifolds.