We study CMV matrices (discrete one-dimensional Dirac-type operators) with random decaying coefficients. Under mild assumptions, we identify the local eigenvalue statistics in the natural scaling limit. For rapidly decreasing coefficients, the eigenvalues have rigid spacing (like the numerals on a clock); in the case of slow decrease, the eigenvalues are distributed according to a Poisson process. For a certain critical rate of decay, we obtain the $\beta$-ensembles of random matrix theory. The temperature ${\beta }^{-1}$ appears as the square of the coupling constant

We study the average size of shifted convolution summation terms related to the problem of quantum unique ergodicity (QUE) on ${\mathrm{SL}}_2(\mathbb{Z})\backslash \mathbb{H}$. Establishing an upper-bound sieve method for handling such sums, we achieve an unconditional result that suggests that the average size of the summation terms should be sufficient in application to quantum unique ergodicity. In other words, cancellations among the summation terms, although welcomed, may not be required. Furthermore, the sieve method may be applied to shifted sums of other multiplicative functions with similar results under suitable conditions

The main result of this article is that every closed Hamiltonian $S^1$-manifold is uniruled, (i.e., it has a nonzero Gromov-Witten invariant, one of whose constraints is a point). The proof uses the Seidel representation of ${\pi }_1$ of the Hamiltonian group in the small quantum homology of $M$ as well as the blow-up technique recently introduced by Hu, Li, and Ruan [15, Th. 5.15]. It applies more generally to manifolds that have a loop of Hamiltonian symplectomorphisms with a nondegenerate fixed maximum. Some consequences for Hofer geometry are explored. An appendix discusses the structure of the quantum homology ring of uniruled manifolds

Igusa varieties are smooth varieties over ${\stackrel{̲}{\mathbb{F}}}_p$ which are higher-dimensional analogues of Igusa curves. They were introduced by Harris and Taylor [HT] to study the bad reduction of some PEL Shimura varieties and were generalized by Mantovan [M1], [M2]. The present article gives a group-theoretic formula for the traces of certain operators on the cohomology of Igusa varieties, suitable for applications via comparison with the Arthur-Selberg trace formula. Our formula generalizes the results of [HT, Chap. V, Prop. 4.8] to the case of any PEL Shimura varieties of types (A) and (C) and puts it in a more natural framework, in the spirit of [K7]