In this paper, we consider the universality of the local eigenvalue statistics of random matrices. Our main result shows that these statistics are determined by the first four moments of the distribution of the entries. As a consequence, we derive the universality of eigenvalue gap distribution and$k$-point correlation, and many other statistics (under some mild assumptions) for both Wigner Hermitian matrices and Wigner real symmetric matrices.

Let ($M$,$ω$) be a connected, symplectic 4-manifold. A$semitoric integrable system$on ($M$,$ω$) essentially consists of a pair of independent, real-valued, smooth functions$J$and$H$on$M$, for which$J$generates a Hamiltonian circle action under which$H$is invariant. In this paper we give a general method to construct, starting from a collection of five ingredients, a symplectic 4-manifold equipped a semitoric integrable system. Then we show that every semitoric integrable system on a symplectic 4-manifold is obtained in this fashion. In conjunction with the uniqueness theorem proved recently by the authors, this gives a classification of semitoric integrable systems on 4-manifolds, in terms of five invariants.

Given a fixed$p$≠2, we prove a simple and effective characterization of all radial multipliers of $ \mathcal{F}{L^p}\left( {{\mathbb{R}^d}} \right) $ , provided that the dimension$d$is sufficiently large. The method also yields new$L$^{$q$}space-time regularity results for solutions of the wave equation in high dimensions.

We construct a hereditarily indecomposable Banach space with dual space isomorphic to$ℓ$_{1}. Every bounded linear operator on this space is expressible as λ$I$+$K$, with λ a scalar and$K$compact.

We develop the affine sieve in the context of orbits of congruence subgroups of semisimple groups acting linearly on affine space. In particular, we give effective bounds for the saturation numbers for points on such orbits at which the value of a given polynomial has few prime factors. In many cases these bounds are of the same quality as what is known in the classical case of a polynomial in one variable where the orbit consists of the integers. When the orbit is the set of integral matrices of a fixed determinant, we obtain a sharp result for the saturation number, and thus establish the Zariski density of matrices all of whose entries are prime numbers. Among the key tools used are explicit approximations to the generalized Ramanujan conjectures for such groups, and sharp and uniform counting of points on such orbits when ordered by various norms.