We prove dynamical upper bounds for discrete one-dimensional Schrödinger operators in terms of various spacing properties of the eigenvalues of finite-volume approximations. We demonstrate the applicability of our approach by a study of the Fibonacci Hamiltonian.

We prove bounds of the form ${\displaystyle \sum _{e\in I\cap {\sigma }_d(H)}\mathrm{dist}(e,{\sigma }_e(H){)}^{1/2}\le L^1-\mathrm{norm} \mathrm{of} a \mathrm{perturbation},}$ where $I$ is a gap. Included are gaps in continuum one-dimensional periodic Schrödinger operators and finite gap Jacobi matrices, where we get a generalized Nevai conjecture about an $L^1$-condition implying a Szegő condition. One key is a general new form of the Birman-Schwinger bound in gaps.

We prove that the family of functionals $(I_{\delta })$ defined by ${\displaystyle I_{\delta }(g)=\underset{|g(x)-g(y)|>\delta }{\underset{{\mathbb{R}}^N\times {\mathbb{R}}^N}{\int \int }}\frac{{\delta }^p}{|x-y{|}^{N+p}}\mathrm{dx}\mathrm{dy},\hspace{1em}\forall g\in L^p({\mathbb{R}}^N),}$ for $p\ge 1$ and $\delta >0$, $\Gamma$-converges in $L^p({\mathbb{R}}^N)$, as $\delta$ goes to zero, when $p\ge 1$. Hereafter $\left| \right|$ denotes the Euclidean norm of ${\mathbb{R}}^N$. We also introduce a characterization for bounded variation (BV) functions which has some advantages in comparison with the classic one based on the notion of essential variation on almost every line.

We develop a mixed-characteristic version of the Mori-Mukai technique for producing rational curves on K$3$ surfaces. We reduce modulo $p$, produce rational curves on the resulting K$3$ surface over a finite field, and lift to characteristic zero. As an application, we prove that all complex K$3$ surfaces with Picard group generated by a class of degree two have an infinite number of rational curves.

We first consider the Monge problem in a convex bounded subset of ${\mathbb{R}}^d$. The cost is given by a general norm, and we prove the existence of an optimal transport map under the classical assumption that the first marginal is absolutely continuous with respect to the Lebesgue measure. In the final part of the paper we show how to extend this existence result to a general open subset of ${\mathbb{R}}^d$.