Let $S$ be a compact complex surface with ordinary singularities. We denote by $\Theta_S$ the sheaf of germs of holomorphic tangent vector fields on $S$. In this paper we shall give a description of the cohomology $H^1(S, \Theta_S)$, which is called the infinitesimal locally trivial deformation space of $S$, using a 2-cubic hyper-resolution of $S$ in the sense of F. Guillén, V. Navarro Aznar et al. ([1]). As a by-product, we shall show that the natural homomorphisim $H^1(S, \Theta_S)\rightarrow H^1(X, \Theta_X(-\log D_X))$ is injective under some condition, where $X$ is the (non-singular) normal model of $S$, $D_X$ the inverse image of the double curve $D_S$ of $S$ by the normalization map $f\colon X\rightarrow S$, and $\Theta_X(-\log D_X)$ the sheaf of germs of logarithmic tangent vector fields along $D_X$ on $X$. Note that the cohomology $H^1(X, \Theta_X(-\log D_X))$ is nothing but the infinitesimal locally trivial deformation space of a pair $(X, D_X)$.

Suppose that $M$ is a compact, orientable three-manifold such that each piece of the canonical decomposition along embedded spheres, discs and tori admits one of the eight geometric structures of three-manifolds in the sense of Thurston. Let $G$ be a subgroup of $\pi_1(M)$. If $G$ has property $T$ in the sense of Kazhdan, then $G$ is finite.

We show that there are no stochastically complete minimal submanifolds immersed in Cartan-Hadamard manifolds which lie in a non-degenerate cone type domain.

For a holomorphic self-map $\varphi$ of the unit disk of the complex plane, the compactness of the composition operator $C_{\varphi}(f) = f\circ \varphi$ on the Hardy spaces is known to be equivalent to the various function theoretic conditions on $\varphi$, such as Shapiro's Nevanlinna counting function condition, MacCluer's Carleson measure condition, Sarason condition and Yanagihara-Nakamura condition, etc. A direct function-theoretic proof of Shapiro's condition and Sarason's condition was recently given by Cima and Matheson. We give another direct function-theoretic proof of the equivalence of these conditions by use of Stanton's integral formula.

We find the uniformizing equation, governing the developing map, of a complex hyperbolic structure on the (4-dimensional) moduli space of marked cubic surfaces. Our equation is invariant under the action of the Weyl group of type $E_6$.

We exibit some simplest nontrivial relations among the values of the $\mathfrak{sl}_3$ weight systems which seem to generate a large portion of the kernel of the weight system.

Let $f>1$ be given. Whereas a simple formula for the mean value of $|L(1, \chi)|^2$ for odd Dirichlet characters modulo $f$ is known, we explain why there is no hope of ever finding a simple formula for the mean value of $|L(1, \chi)|^2$ for primitive odd Dirichlet characters modulo $f$.