Given the standard Gaussian measure $\gamma$ on the countable product of lines ${\mathbb{R}}^{\infty }$ and a probability measure $g\cdot \gamma$ absolutely continuous with respect to $\gamma$, we consider the optimal transportation $T\left(x\right)=x+\nabla \phi \left(x\right)$ of $g\cdot \gamma$ to $\gamma$. Assume that the function $|\nabla g{|}^2/g$ is $\gamma$-integrable. We prove that the function $\phi$ is regular in a certain Sobolev-type sense and satisfies the classical change of variables formula $g={det}_2(\mathrm{I}+D^2\phi )exp(\mathcal{L}\phi -\frac{1}{2}|\nabla \phi {|}^2)$. We also establish sufficient conditions for the existence of third-order derivatives of $\phi$.

We classify all Gieseker semistable sheaves on the complex projective plane that have dimension $1$ and multiplicity $6$. We decompose their moduli spaces into strata which occur naturally as quotients modulo actions of certain algebraic groups. In most cases we give concrete geometric descriptions of the strata.

In the case of Krein’s strings with spectral functions of polynomial growth a necessary and sufficient condition for the Krein’s correspondence to be continuous is given.

In this paper, we give an extension of the classical story of the elliptic modular function to the Hilbert modular case for $\mathbb{Q}\left(\sqrt{5}\right)$. We construct the period mapping for a family $\mathcal{F}=\left\{S\right(X,Y\left)\right\}$ of $K3$ surfaces with $2$ complex parameters $X$ and $Y$. The inverse correspondence of the period mapping gives a system of generators of Hilbert modular functions for $\mathbb{Q}\left(\sqrt{5}\right)$. Moreover, we show an explicit expression of this inverse correspondence by theta constants.

Kurihara described all higher Fitting ideals of the minus part of Iwasawa modules of ideal class groups over totally real fields by using Stickelberger elements and Euler systems of “Gauss sums.” In this paper, we obtain some partial results for elliptic units which are analogues of his result. By using Kolyvagin derivative classes of Euler systems of elliptic units, we construct some ideals ${\mathfrak{C}}_{i,\chi }^{\mathrm{ell}}$ of Iwasawa algebras and prove that they give “upper bounds” of higher Fitting ideals of one- and two-variable Iwasawa modules of ideal class groups over imaginary quadratic fields.