The nonnegativity of the CR Paneitz operator plays a crucial role in three-dimensional CR geometry. In this paper, we prove this nonnegativity for embeddable CR manifolds. This result gives an affirmative solution to the CR Yamabe problem for embeddable CR manifolds. We also show the existence of a contact form with zero CR $Q$-curvature and generalize the total $Q$-prime curvature to embeddable CR manifolds with no pseudo-Einstein contact forms. Furthermore, we discuss the logarithmic singularity of the Szegő kernel.

The aim of this article is to provide an analogue of the Ball–Rivoal theorem for $p$-adic $L$-values of Dirichlet characters. More precisely, we prove, for a Dirichlet character $\chi$ and a number field $K$, the formula ${dim}_K(K+{\sum }_{i=2}^{s+1}{L}_p(i,\chi {\omega }^{1-i}\left)K\right)\ge \frac{(1-ϵ)log\left(s\right)}{2[K:\mathbb{Q}](1+log2)}$. As a by-product, we establish an asymptotic linear independence result for the values of the $p$-adic Hurwitz zeta function.

For a semisimple Lie group $G$ satisfying the equal-rank condition, the most basic family of unitary irreducible representations is the discrete series found by Harish-Chandra. In our work here we study some of the branching laws for discrete series when restricted to a subgroup $H$ of the same type by combining classical results with recent work of Kobayashi; in particular, we prove discrete decomposability under Harish-Chandra’s condition of cusp form on the reproducing kernel. We show a relation between discrete decomposability and representing certain intertwining operators in terms of differential operators.

In this paper, we prove that Bernoulli percolation on bounded degree graphs with isoperimetric dimension $d>4$ undergoes a nontrivial phase transition (in the sense that $p_c<1$). As a corollary, we obtain that the critical point of Bernoulli percolation on infinite quasitransitive graphs (in particular, Cayley graphs) with superlinear growth is strictly smaller than $1$, thus answering a conjecture of Benjamini and Schramm. The proof relies on a new technique based on expressing certain functionals of the Gaussian free field (GFF) in terms of connectivity probabilities for a percolation model in a random environment. Then we integrate out the randomness in the edge-parameters using a multiscale decomposition of the GFF. We believe that a similar strategy could lead to proofs of the existence of a phase transition for various other models.

We determine the minimum positive entropy of complex Enriques surface automorphisms. This together with McMullen’s work completes the determination of the minimum positive entropy of complex surface automorphisms in each class of the Enriques–Kodaira classification of complex surfaces.