Let $B_{1,t}^{H},\ldots ,B_{d,t}^{H}$ be $d$ independent fractional Brownian motions with Hurst parameter $H\in (0, 1)$. Denote $X_{t}=(B_{1,t}^{H},\ldots ,B_{d,t}^{H})$ and let $\delta$ be the Dirac delta function. It is shown that when $H< \mathrm{min}(3/(2d), 2/(d+2))$, the (renormalized) self-intersection local time of fractional Brownian motion, $\int _{0}^{T}\int _{0}^{t}\delta (X_{t}-X_{s})dsdt-\mathbb{E}\int _{0}^{T}\int _{0}^{t}\delta (X_{t}-X_{s})dsdt$, is in $D_{1,2}$, where $D_{1,2}$ is the Meyer-Watanabe test functional space, i.e. the $L^{2}$ space of “differentiable” functionals, whose precise meaning is given in Section 2.

A theorem of Hardy asserts that a function on the real line and its Fourier transform cannot both be very small. We generalize Hardy’s theorem for the Heckman-Opdam transform associated with hypergeometric functions.

We compute the intersection pairings on the configuration spaces of points in $\mathbf{CP}^{1}$ with arbitrary weight from an algebro-geometric point of view. We also derive their generating function, which was originally proved by Takakura based on symplectic technique and representation theory.

There are a number of characterizations of convex subregions $\Omega$ of the complex plane $\mathbb{C}$ in terms of the density $\lambda_{\Omega}(w)$ of the hyperbolic metric $\lambda_{\Omega}(w)|dw|$ for $\Omega$. We derive analogous characterizations for spherically convex regions $\Omega$ on the Riemann sphere $\mathbb{P}$ in terms of the spherical density $\mu_{\Omega}(w) = (1+|w|^{2})\lambda_{\Omega}(w)$ of the hyperbolic metric. A proper subregion $\Omega$ of $\mathbb{P}$ is spherically convex if for all pairs $A$, $B$ of points in $\Omega$ the spherical geodesic (the shorter arc of the great circle) joining $A$ and $B$ lies in $\Omega$. As a limiting case of our results we obtain known characterizations of convex regions in $\mathbb{C}$.

We compute the torsion part of the Picard group of a numerical Godeaux surface Y which was constructed by Stagnaro [7] as a double cover of $\mathbb{P}^{2}$ branching along a curve of degree 10. We show that this surface is a classical Godeaux surface whose universal cover is the Fermat quintic in $\mathbb{P}^{3}$ (cf. [1, p.170]).

We formulate the modularity conjecture for rigid Calabi-Yau threefolds defined over the field $\mathbf{Q}$ of rational numbers. We establish the modularity for the rigid Calabi-Yau threefold arising from the root lattice $A_{3}$. Our proof is based on geometric analysis.

Let $C$ be a compact orientable hyperbolic 3-cone-manifold with cone-type singularity along simple closed geodesics $\Sigma$. Let $\{ C_{i}\}_{i=1}^{\infty}$ be a sequence consisting of deformations of $C$ and $\Sigma _{i}$ be the singular set of $C_{i}$ so that the cone angles along $\Sigma _{i}$ all are less than $2\pi$. In this paper, we will show that, if tubular neighborhoods of the singular sets $\Sigma _{i}$ can be taken to be uniformly thick, then there is a subsequence $\{ C_{i_{k}}\}_{k=1}^{\infty}$ which converges strongly to a hyperbolic 3-cone-manifold $C_{*}$ homeomorphic to $C$.