Conditioned on the Riemann hypothesis, we show that the conjecture that the zeros of the Riemann zeta function resemble the eigenvalues of a random matrix is logically equivalent to a statement about the distribution of primes.

This generalizes well-known work that the pair correlation conjecture is equivalent to a statement about the variance of prime counts in short intervals and complements the work of Farmer, Gonek, Lee, and Lester, who have considered similar questions conditioned on additional hypotheses, which are not required here. As a byproduct of this argument, conditioned on the Riemann hypothesis, we derive upper bounds for all moments of the logarithmic derivative of the Riemann zeta function.

We also discuss a conjecture for the covariance in short intervals of counts of almost-primes, weighted by the higher-order von Mangoldt function, and show the GUE Conjecture implies a weighted version of this conjecture. The covariance is surprisingly simple to write down.

We improve on the result of Hacon and Witaszek by showing that the MMP for semistable fourfolds in mixed characteristic terminates in several new situations. In particular, we show the validity of the MMP for strictly semistable fourfolds over excellent Dedekind schemes globally when the residue fields are perfect and have characteristics $\mathit{p}>5$.

The aim of this paper is a construction of new explicit annihilators of the minus part of the ideal class group of an imaginary Abelian number field M, that is, annihilators that are outside of the Stickelberger ideal, their usual source. This construction works for quite a large class of Abelian fields M; a sufficient condition to get a new annihilator is that there is an odd prime $\ell \mid [\mathit{M}:\mathbb{Q}]$, unramified in $\mathit{M}/\mathbb{Q}$, and two primes q and ${\mathit{q}}^{\prime }$ ramifying in $\mathit{M}/\mathbb{Q}$, having their decomposition groups ${\mathit{D}}_{\mathit{q}}\subseteq {\mathit{D}}_{{\mathit{q}}^{\prime }}$ cyclic of ℓ-power order.

Work of numerous authors has shown that any smooth, orientable, closed 4-manifold may be described as a loop of Morse functions on a surface, a loop in the cut complex, a loop in the pants complex, or as a multisection. In this paper, we prove a corresponding uniqueness theorem for each of these descriptions so that, for example, any two loops of Morse functions on a surface yielding diffeomorphic 4-manifolds are related by a given set of moves.

We study relations between the property of being log abundant for lc pairs and the termination of log MMP with scaling. We prove that any log MMP with scaling of an ample divisor starting with a projective dlt pair contains only finitely many log abundant dlt pairs. We also discuss log MMP for slc pairs.

We answer two questions about the topology of end spaces of infinite type surfaces and the action of the mapping class group that have appeared in the literature. First, we give examples of infinite type surfaces with end spaces that are not self-similar, but a unique maximal type of end, either a singleton or a Cantor set. Secondly, we use an argument of Tsankov to show that the “local complexity” relation ≼ on end types gives an equivalence relation that agrees with the notion of being locally homeomorphic.