We study noncommutative maximal inequalities and ergodic theorems for group actions on von Neumann algebras. Consider a locally compact group $G$ of polynomial growth with a symmetric compact subset $V$. Let $\alpha$ be a continuous action of $G$ on a von Neumann algebra $\mathcal{M}$ by trace-preserving automorphisms. We then show that the operators defined by $$A_{n}x=\frac{1}{m\left(V^n\right)}{\int }_{V^n}{\alpha }_{g}xdm\left(g\right),x\in L_p\left(\mathcal{M}\right),n\in \mathbb{N},1\le p\le \infty ,$$ are of weak type $(1,1)$ and of strong type $(p,p)$ for $1<p<\infty$. Consequently, the sequence $(A_{n}x{)}_{n\ge 1}$ converges almost uniformly for $x\in L_p\left(\mathcal{M}\right)$ for $1\le p<\infty$. Also, we establish the noncommutative maximal and individual ergodic theorems associated with more general doubling conditions, and we prove the corresponding results for general actions on one fixed noncommutative $L_p$-space which are beyond the class of Dunford–Schwartz operators considered previously by Junge and Xu. As key ingredients, we also obtain the Hardy–Littlewood maximal inequality on metric spaces with doubling measures in the operator-valued setting. After the groundbreaking work of Junge and Xu on the noncommutative Dunford–Schwartz maximal ergodic inequalities, this is the first time that more general maximal inequalities are proved beyond Junge and Xu’s setting. Our approach is based on quantum probabilistic methods as well as random-walk theory.

In the following, we prove the geometric Bogomolov conjecture over a function field of characteristic $0$.

According to the Langlands functoriality conjecture, broadened to the setting of spherical varieties (of which reductive groups are special cases), a map between $L$-groups of spherical varieties should give rise to a functorial transfer of their local and automorphic spectra. The “beyond endoscopy” proposal predicts that this transfer will be realized as a comparison between limiting forms of the (relative) trace formulas of these spaces. In this paper, we establish the local transfer for the identity map between $L$-groups, for spherical affine homogeneous spaces $X=H\backslash G$ whose dual group is ${SL}_2$ or ${PGL}_2$ (with $G$ and $H$ split). More precisely, we construct a transfer operator between orbital integrals for the $(X\times X)/G$-relative trace formula, and orbital integrals for the Kuznetsov formula of ${PGL}_2$ or ${SL}_2$. Besides the $L$-group, another invariant attached to $X$ is a certain $L$-value, and the space of test measures for the Kuznetsov formula is enlarged to accommodate the given $L$-value. The transfer operator is given explicitly in terms of Fourier convolutions, making it suitable for a global comparison of trace formulas by the Poisson summation formula, hence for a uniform proof, in rank $1$, of the relations between periods of automorphic forms and special values of $L$-functions.

We prove that the minimal diameter of a closed orientable hyperbolic surface of genus $g$ is asymptotic to $logg$ as $g\to \infty$. The proof relies on a random construction, which we analyze using lattice-point counting theory and the exploration of random trivalent graphs.