The purpose of this paper is twofold. First we give a survey on the recent developments of curve counting invariants on Calabi–Yau 3-folds, for example, Gromov–Witten theory, Donaldson–Thomas theory, and Pandharipande–Thomas theory. Next we focus on the proof of the rationality conjecture of the generating series of PT invariants and discuss its conjectural Gopakumar–Vafa form.

In this paper, we first recall and apply the fundamental techniques of constructing bad Noetherian local domains, due to C. Rotthaus, T. Ogoma, R. C. Heitmann, and M. Brodmann and C. Rotthaus, to show several basic examples:

(1) a three-dimensional Nagata normal local domain, which is a complete intersection, whose regular locus is not open;

(2) a three-dimensional Henselian Nagata normal local domain, which is not catenary.

Next we present a unified version of Brodmann and Rotthaus’s and Ogoma’s methods in order to obtain a particular local domain $A$ with a specified prime element $x$ such that the local domain $A/xA$ is the bad Noetherian local domain given above:

(3) a three-dimensional unmixed local domain $A$ that has $xA=\mathfrak{p}\in Spec\left(A\right)$ such that $A/\mathfrak{p}$ is not unmixed.

Finally we follow Ogoma’s construction of factorial local domains whose completions are designated complete local domains. Then, we gather some examples of bad factorial local domains.

In this paper, we first derive the CR analogue of the almost Schur lemma on a pseudo-Hermitian $(2n+1)$-manifold $(M,J,\theta )$ for $n\ge 2$. Second, we study a sufficient condition for the existence of a pseudo-Einstein contact form when the CR structure of $M$ has vanishing first Chern class which is related to the J. M. Lee conjecture.

In this paper, we review several recent results concerning well-posedness of the one-dimensional, cubic nonlinear Schrödinger equation (NLS) on the real line $\mathbb{R}$ and on the circle $\mathbb{T}$ for solutions below the $L^2$-threshold. We point out common results for NLS on $\mathbb{R}$ and the so-called Wick-ordered NLS (WNLS) on $\mathbb{T}$, suggesting that WNLS may be an appropriate model for the study of solutions below $L^2\left(\mathbb{T}\right)$. In particular, in contrast with a recent result of Molinet, who proved that the solution map for the periodic cubic NLS equation is not weakly continuous from $L^2\left(\mathbb{T}\right)$ to the space of distributions, we show that this is not the case for WNLS.

Let $f$ be a cusp form on ${GL}_2$ over an imaginary quadratic field $F$ of class number 1, and let $p$ be an odd prime which satisfies some mild conditions. Then we show the existence of a finite-order Hecke character $\phi$ of $F_{\mathbf{A}}^{\times }$ such that the algebraic part of the special value of $L$-functions of $f\otimes \phi$ at $s=1$ is a $p$-adic unit. This is an analogous result to the result of A. Ash and G. Stevens for ${GL}_2$ over the field of rationals obtained in [AS].

Saito–Kurokawa liftings from Jacobi forms of degree one of weight $k$ of level $N$ to Siegel modular forms of degree two of weight $k$ of level $N$ with or without character are explicitly given by describing their Fourier expansions. Their $L$-functions are also given as well as the action of Hecke operators including some operators at bad primes. Also, a practical way of constructing Jacobi forms of given level is explained. This is explicitly executed for Jacobi forms of level up to $5$, and their explicit structure theorems are given.

This paper addresses Gabor families in $l^2\left({\mathbb{Z}}^d\right)$. The discrete Gabor families have interested many researchers due to their good potential for digital signal processing. Gabor analysis in $l^2\left({\mathbb{Z}}^d\right)$ is more complicated than that in $l^2\left(\mathbb{Z}\right)$ since the geometry of the lattices generated by time-frequency translation matrices can be quite complex in this case. In this paper, we characterize window functions such that they correspond to complete Gabor families (Gabor frames) in $l^2\left({\mathbb{Z}}^d\right)$; obtain a necessary and sufficient condition on time-frequency translation for the existence of complete Gabor families (Gabor frames, Gabor Riesz bases) in $l^2\left({\mathbb{Z}}^d\right)$; characterize duals with Gabor structure for Gabor frames; derive an explicit expression of the canonical dual for a Gabor frame; and prove its norm minimality among all Gabor duals.