A general extension theorem for $L^2$ holomorphic bundle-valued top forms is formulated. Although its proof is based on a principle similar to Ohsawa-Takegoshi's extension theorem, it explains previous $L^2$ extendability results systematically and bridges extension theory and division theory.

We present a complete list of extremal elliptic $K3$ surfaces (Theorem 1.1). As an application, we give a sufficient condition for the topological fundamental group of complement to an $ADE$-configuration of smooth rational curves on a $K3$ surface to be trivial (Proposition 4.1 and Theorem 4.3).

We develop some symplectic techniques to control the behavior under symplectic transformation of analytic discs $A$ of $X = \mathbb{C}^{n}$ tangent to a real generic submanifold $R$ and contained in a wedge with edge $R$.

We show that if $A^{*}$ is a lift of $A$ to $T^{*}X$ and if $\chi$ is a symplectic transformation between neighborhoods of $p_{o}$ and $q_{o}$, then $A$ is orthogonal to $p_{o}$ if and only if $\widetilde{A} := \pi\chi A^{*}$ is orthogonal to $q_{o}$. Also we give the (real) canonical form of the couples of hypersurfaces of $\mathbb{R}^{2n} \simeq \mathbb{C}^{n}$ whose conormal bundles have clean intersection. This generalizes [10] to general dimension of intersection.

Combining this result with the quantized action on sheaves of the "tuboidal" symplectic transformation, we show the following: If $R$, $S$ are submanifolds of $X$ with $R \subset S$ and $p_{o} \in T^{*}_{S}X|_{R}$ but $ip_{o} \notin T^{*}_{R}X$, then the conditions $\text{cod}_{T^{\mathbb{C}}S}(T^{\mathbb{C}}R)=\text{cod}_{TS}(TR) (\text{resp. cod}_{T^{\mathbb{C}}S}(T^{\mathbb{C}}R) = 0)$ can be characterized as opposite inclusions for the couple of closed half-spaces with conormal bundles $\chi(T^{*}_{R}X)$ and $\chi(T^{*}_{S}X)$ at $\chi(p_{o})$.

In $\S$3 we give some partial applications of the above result to the analytic hypoellipticity of $CR$ hyperfunctions on higher codimensional manifolds by the aid of discs (cf. [2], [3] as for the case of hypersurfaces).

The $2^{g}$ theta constants of second kind of genus $g$ generate a graded ring of dimension $g(g+1)/2$. In the case $g \geq 3$ there must exist algebraic relations. In genus $g = 3$ it is known that there is one defining relation. In this paper we give a relation in the case $g = 4$. It is of degree $24$ and has the remarkable property that it is invariant under the full Siegel modular group and whose $\Phi$-image is not zero. Our relation is obtained as a linear combination of code polynomials of the $9$ self-dual doubly-even codes of length $24$.

On the half space ${\bf R}^n\times {\bf R}_+\,$, it has been known that harmonic Bergman space $b^p$ can contain a positive function only if $p>1+\frac 1n$. Thus, for $1\le p\le 1+\frac 1n$, Poisson integrals can be $b^p$-functions only by means of their boundary cancellation properties. In this paper, we describe what those cancellation properties explicitly are. Also, given such cancellation properties, we obtain weighted norm inequalities for Poisson integrals. As a consequence, under weighted integrability condition given by our weighted norm inequalities, we show that our cancellation properties are equivalent to the $b^p$-containment of Poisson integrals for $p$ under consideration. Our results are sharp in the sense that orders of our weights cannot be improved.

For a magnetic Schrödinger operator on a graph, which is a generalization of classical Harper operator, we study some spectral properties: the Bloch property and the behaviour of the bottom of the spectrum with respect to magnetic fields. We also show some examples which have interesting properties.

Some results on stabilization of (deterministic and stochastic) partial differential equations are established. In particular, some stability criteria from Chow [4] and Haussmann [6] are improved and subsequently applied to certain situations, on which the original criteria commonly do not work, to ensure almost sure exponential stability. This paper also extends to infinite dimension some results due to Mao [9] on stabilization of differential equations in finite dimension.

Let $\chi$ be a nontrivial Hecke character on a (strict) ray class group of a totally real number field ${\bf L}$ of discriminant $d_{\bf L}$. Then, $L(0,\chi )$ is an algebraic number of some cyclotomic number field. We develop an efficient technique for computing the exact values at $s=0$ of such abelian Hecke $L$-functions over totally real number fields ${\bf L}$. Let $f_\chi$ denote the norm of the finite part of the conductor of $\chi$. Then, roughly speaking, we can compute $L(0,\chi )$ in $O((d_{\bf L}f_\chi )^{0.5 +\epsilon})$ elementary operations. We then explain how the computation of relative class numbers of CM-fields boils down to the computation of exact values at $s=0$ of such abelian Hecke $L$-functions over totally real number fields ${\bf L}$. Finally, we give examples of relative class number computations for CM-fields of large degrees based on computations of $L(0,\chi )$ over totally real number fields of degree $2$ and $6$.

Introducing the idea of weighted sharing of values we prove some uniqueness theorems for meromorphic functions which improve some existing results.