We consider Reeb dynamics on the $3$-sphere associated to a tight contact form. Our main result gives necessary and sufficient conditions for a periodic Reeb orbit to bound a disk-like global section for the Reeb flow, when the contact form is assumed to be nondegenerate.

Let $f$ traverse a sequence of classical holomorphic newforms of fixed weight and increasing square-free level $q\to \infty$. We prove that the pushforward of the mass of $f$ to the modular curve of level $1$ equidistributes with respect to the Poincaré measure.

Our result answers affirmatively the square-free level case of a conjecture spelled out in 2002 by Kowalski, Michel, and Vanderkam in the spirit of a conjecture that Rudnick and Sarnak made in 1994.

Our proof follows the strategy of Holowinsky and Soundararajan, who showed in 2008 that newforms of level $1$ and large weight have equidistributed mass. The new ingredients required to treat forms of fixed weight and large level are an adaptation of Holowinsky’s reduction of the problem to one of bounding shifted sums of Fourier coefficients, a refinement of his bounds for shifted sums, an evaluation of the $p$-adic integral needed to extend Watson’s formula to the case of three newforms of not necessarily equal square-free levels, and some additional technical work in the problematic case that the level has many small prime factors.

We use motivic fundamental groups to show that $S$-integral points on a unirational variety over a totally real number field whose fundamental group is nonabelian enough in a certain sense can be covered by zero loci of finitely many nonzero $p$-adic analytic functions. In particular, in the $1$-dimensional case we obtain a motivic proof of finiteness of $S$-integral points of punctured projective line over totally real number fields, which gives as a special case a motivic proof of Siegel’s theorem over $\mathbb{Q}$ and totally real quadratic number fields.

Let $A$ be a self-adjoint operator acting over a space $X$ endowed with a partition. We give lower bounds on the energy of a mixed state $\rho$ from its distribution in the partition and the spectral density of $A$. These bounds improve with the refinement of the partition, and generalize inequalities by Li and Yau and by Lieb and Thirring for the Laplacian in ${\mathbb{R}}^n$. They imply an uncertainty principle, giving a lower bound on the sum of the spatial entropy of $\rho$, as measured from $X$, and some spectral entropy, with respect to its energy distribution. On ${\mathbb{R}}^n$, this yields lower bounds on the sum of the entropy of the densities of $\rho$ and its Fourier transform. A general log-Sobolev inequality is also shown. It holds on mixed states, without Markovian or positivity assumption on $A$.

In this work we obtain a new criterion to establish ergodicity and nonuniform hyperbolicity of smooth measures of diffeomorphisms of closed connected Riemannian manifolds. This method allows us to give a more accurate description of certain ergodic components. The use of this criterion in combination with topological devices such as blenders lets us obtain global ergodicity and abundance of nonzero Lyapunov exponents in some contexts.

In the partial hyperbolicity context, we obtain that stably ergodic diffeomorphisms are $C^1$-dense among volume-preserving partially hyperbolic diffeomorphisms with 2-dimensional center bundle. This is motivated by a well-known conjecture of Pugh and Shub.