We consider Fourier integral operators with symbols in modulation spaces and non-smooth phase functions whose second orders of derivatives belong to certain types of modulation space. We prove continuity and Schatten–von Neumann properties of such operators when acting on$L$^{2}.

We are studying the Diophantine exponent μ_{$n$,$l$}defined for integers 1≤$l$<$n$and a vector α∈ℝ^{$n$}by letting $$\mu_{n,l}=\sup\{\mu\geq0: 0 < \Vert\underline{x}\cdot\alpha\Vert<H(\underline{x})^{-\mu}\ \text{for infinitely many}\ \underline{x}\in\mathcal{C}_{n,l}\cap\mathbb{Z}^n\},$$ where $\cdot$ is the scalar product, $\|\cdot\|$ denotes the distance to the nearest integer and $\mathcal{C}_{n,l}$ is the generalised cone consisting of all vectors with the height attained among the first$l$coordinates. We show that the exponent takes all values in the interval [$l$+1,∞), with the value$n$attained for almost all α. We calculate the Hausdorff dimension of the set of vectors α with μ_{$n$,$l$}(α)=μ for μ≥$n$. Finally, letting$w$_{$n$}denote the exponent obtained by removing the restrictions on $\underline{x}$ , we show that there are vectors α for which the gaps in the increasing sequence μ_{$n$,1}(α)≤...≤μ_{$n$,$n$-1}(α)≤$w$_{$n$}(α) can be chosen to be arbitrary.

A recent approach of D. S. Lubinsky yields universality in random matrix theory and fine zero spacing of orthogonal polynomials under very mild hypothesis on the weight function, provided the support of the generating measure μ is [-1,1]. This paper provides a method with which analogous results can be proven on general compact subsets of$R$. Both universality and fine zero spacing involves the equilibrium measure of the support of μ. The method is based on taking polynomial inverse images, by which results can be transferred from [-1,1] to a system of intervals, and then to general sets.

We compute local Gromov–Witten invariants of cubic surfaces at all genera. We use a deformation a of cubic surface to a nef toric surface and the deformation invariance of Gromov–Witten invariants.

Stein’s higher Riesz transforms are translation invariant operators on$L$^{2}($R$^{$n$}) built from multipliers whose restrictions to the unit sphere are eigenfunctions of the Laplace–Beltrami operators. In this article, generalizing Stein’s higher Riesz transforms, we construct a family of translation invariant operators by using discrete series representations for hyperboloids associated to the indefinite quadratic form of signature ($p$,$q$). We prove that these operators extend to$L$^{$r$}-bounded operators for 1<$r$<∞ if the parameter of the discrete series representations is generic.