The aim of this paper is to construct (i) infinitely many families of nonrational $\mathbb{Q}$-Fano varieties of arbitrary dimension $\ge 4$ with at most quotient singularities, and (ii) twelve families of nonrational $\mathbb{Q}$-Fano threefolds with at most terminal singularities among which two are new and the remaining ten give an alternate proof of nonrationality to known examples. These are constructed as weighted hypersurfaces with the reduction mod $p$ method introduced by Kollár [10].

We propose a new approach to the value distribution theory of entire holomorphic curves. We define \emph{packing density} of Brody curves, and show that it has various non-trivial properties. The packing density of Brody curves can be considered as an infinite dimensional version of characteristic number, and it has an application to Gromov's mean dimension theory.

Generalizing the concepts of Stanley-Reisner and affine monoid algebras, one can associate to a rational pointed fan $\Sigma$ in $\mathbb{R}^{d}$ the $\mathbb{Z}^{d}$-graded toric face ring $K[\Sigma]$. Assuming that $K[\Sigma]$ is Cohen-Macaulay, the main result of this paper is to characterize the situation when its canonical module is isomorphic to a $\mathbb{Z}^{d}$-graded ideal of $K[\Sigma]$. From this result several algebraic and combinatorial consequences are deduced. As an application, we give a relation between the cleanness of $K[\Sigma]$ and the shellability of $\Sigma$.

For a finite set $S$ of primes of a number field $K$ and for $\sigma_{1}, \dots, \sigma_{e} \in \operatorname{Gal}(K)$ we denote the field of totally $S$-adic numbers by $K_{{\rm tot}, S}$ and the fixed field of $\sigma_{1}, \dots, \sigma_{e}$ in $K_{{\rm tot}, S}$ by $K_{{\rm tot}, S}({\boldsymbol\sigma})$. We prove that for almost all ${\boldsymbol\sigma} \in \operatorname{Gal}(K)^{e}$ the absolute Galois group of $K_{{\rm tot}, S}({\boldsymbol\sigma})$ is the free product of ${\hat F}_{e}$ and a free product of local factors over $S$.

Given a smooth minimal surface $F : \Omega \rightarrow \mathbb{R}^{3}$ defined on a simply connected region $\Omega$ in the complex plane $\mathbb{C}$, there is a regular SG circle pattern $Q_{\Omega}^{\epsilon}$. By the Weierstrass representation of $F$ and the existence theorem of SG circle patterns, there exists an associated SG circle pattern $P_{\Omega}^{\epsilon}$ in $\mathbb{C}$ with the combinatoric of $Q_{\Omega}^{\epsilon}$. Based on the relationship between the circle pattern $P_{\Omega}^{\epsilon}$ and the corresponding discrete minimal surface $F^{\epsilon} : V_{\Omega}^{\epsilon} \rightarrow \mathbb{R}^{3}$ defined on the vertex set $V_{\Omega}^{\epsilon}$ of the graph of $Q_{\Omega}^{\epsilon}$, we show that there exists a family of discrete minimal surface $\Gamma^{\epsilon} : V_{\Omega}^{\epsilon} \rightarrow \mathbb{R}^{3}$, which converges in $C^{\infty}(\Omega)$ to the minimal surface $F : \Omega \rightarrow \mathbb{R}^{3}$ as $\epsilon \rightarrow 0$.

We study the canonical basis for the negative part $\mathbf{U}^{-}$ of the quantum generalized Kac-Moody algebra associated to a symmetric Borcherds-Cartan matrix. The algebras $\mathbf{U}^{-}$ associated to two different matrices satisfying certain conditions may coincide (6.3). We show that the canonical bases coincide provided that the algebras $\mathbf{U}^{-}$ coincide (Theorem 6.3.5). We also answer partially a question by Lusztig in [L3] (Theorem 7.1.1).