An analogue of the well-known $$ \frac{3}{{16}} $$ lower bound for the first eigenvalue of the Laplacian for a congruence hyperbolic surface is proven for a congruence tower associated with any non-elementary subgroup$L$of SL(2,$Z$). The proof in the case that the Hausdorff of the limit set of$L$is bigger than $$ \frac{1}{2} $$ is based on a general result which allows one to transfer such bounds from a combinatorial version to this archimedian setting. In the case that delta is less than $$ \frac{1}{2} $$ we formulate and prove a somewhat weaker version of this phenomenon in terms of poles of the corresponding dynamical zeta function. These “spectral gaps” are then applied to sieving problems on orbits of such groups.

We study the averaged behavior of interfaces moving with oscillatory normal velocity that is periodic in time and stationary ergodic in space. This problem can be interpreted as a homogenization problem of a Hamilton-Jacobi equation with a positively homogeneous and time dependent Hamiltonian. In an earlier work, we studied the setting when the environment is periodic in space and random in time, and established homogenization results through the averaging of reachable sets. In the present setting, we show that the minimal travel time between two spatial points, which now depends also on a starting time, has a deterministic averaging limit that is independent of the starting time. This averaged travel time function is then shown to determine the effective behavior of the moving interfaces.

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We consider the Schrödinger equation for the harmonic oscillator$i$$∂$_{$t$}$u$=$Hu$, where$H$=−Δ+|$x$|^{2}, with initial data in the Hermite–Sobolev space$H$^{−$s$/2}$L$^{2}(ℝ^{$n$}). We obtain smoothing and maximal estimates and apply these to perturbations of the equation and almost everywhere convergence problems.

For the d-dimensional incompressible Euler equation, the standard energy method gives local wellposedness for initial velocity in Sobolev space with regularity above some critical threshold. The borderline case was a folklore open problem. In this paper we consider the physical dimension d=2 and show that if we perturb any given smooth initial data in critical norm, then the corresponding solution can have infinite critical norm instantaneously at t>0. In a companion paper we settle the 3D and more general cases. The constructed solutions are unique and even infinitely-smooth (locally) in some cases. To prove these results we introduce a new strategy: large Lagrangian deformation induces critical norm inflation. As an application we also settle several closely related open problems.