We consider a claim mentioned in [33] p.187 about the relation between a symplectic chain complex with $\omega$-compatible bases and Reidemeister Torsion of it. This is an explanation of it.

In this article, we study the group structure of the Galois group of the maximal pro-$p$ extension unramified outside $p$ of imaginary quadratic fields by using Iwasawa theory.

C.A. Giller defined a crossing change for surfaces in $4$-space, and proved an unknotting theorem. In this paper, we present such an unknotting theorem for singular surface braids, extending S. Kamada's result for those without branch points. As a consequence, we recover Giller's unknotting theorem. We also study finite type invariants for singular surface braids associated with the crossing changes.

We study the cohomological properties of reflexive rank 2 sheaves on smooth projective threefolds. Applications are given to the relationship between moduli of reflexive sheaves and Hilbert schemes of associated elliptic curves on Fano threefolds.

We observe Naruki's cross ratio variety from the viewpoint of the root system of type $3A_{2}$. We construct some kind of models as moduli space of marked cubic surfaces on which the action of a Weyl subgroup of type $3A_{2}$ and its normalizer can be easily observed. We describe the structure of our model and its relationship to the cross ratio variety.

We answer the question ``Does the Y-triangle move preserve intrinsic knottedness?'' in the negative by giving an example of a graph that is obtained from the intrinsically knotted graph $K_{7}$ by triangle-Y and Y-triangle moves but is not intrinsically knotted.

The gonality is one of important invariants in the study of linear systems on curves. The gonality conjecture which was posed by Green and Lazarsfeld predicts that we can read off the gonality of a curve from any one line bundle of sufficiently large degree on the curve. This conjecture had been proved for curves on Hirzebruch surfaces by Aprodu. In this artlcle, we will extend this result for curves on certain toric surfaces.

We consider a generalization of Calabi-Yau structures in the context of Sasakian manifolds. We study deformations of a special class of Legendrian submanifolds and classify invariant contact Calabi-Yau structures on 5-dimensional nilmanifolds. Finally we generalize to codimension $r$.

We express the Casson-Walker invariants for the cyclic covering spaces of the three-dimensional sphere branched along satellite knots in terms of companions, patterns, and winding numbers.

We prove that the Cartan matrices of the symmetric artin algebras whose Auslander-Reiten quivers admit a generalized standard stable tube are singular and derive some consequences.

In this paper we study geometric properties of the slowness surface of the system of crystal acoustics for cubic crystals. In particular we shall study curvature properties of the surface and the behaviour of the surface near singular points. The main result is that in the generic nearly isotropic case there are no planes which are tangent to the surface along entire curves. This is in contrast with what happens for the slowness surface of the system of crystal optics for bi-axial crystals. Geometric information of the type we shall obtain is needed to understand the long-time behaviour of global solutions of the system of crystal acoustics.

We present a generalization of the div-curl lemma to a Banach space framework which is not included in the almost existing generalizations. An example is shown where this generalization is needed.

We develop the twistor theory of $G$-structures for which the (linear) Lie algebra of the structure group contains an involution, instead of a complex structure. The twistor space $Z$ of such a $G$-structure is endowed with a field of involutions $\mathcal{J}\in \Gamma (\End TZ)$ and a $\mathcal{J}$-invariant distribution $\mathcal{H}_{Z}$. We study the conditions for the integrability of $\mathcal{J}$ and for the (para-)holomorphicity of $\mathcal{H}_{Z}$. Then we apply this theory to para-quaternionic Kähler manifolds of non-zero scalar curvature, which admit two natural twistor spaces $(Z^{\epsilon},\mathcal{J},\mathcal{H}_{Z})$, $\epsilon=\pm 1$, such that $\mathcal{J}^{2}=\epsilon \Id$. We prove that in both cases $\mathcal{J}$ is integrable (recovering results of Blair, Davidov and Mu\u{s}karov) and that $\mathcal{H}_{Z}$ defines a holomorphic ($\epsilon=-1$) or para-holomorphic ($\epsilon=+1$) contact structure. Furthermore, we determine all the solutions of the Einstein equation for the canonical one-parameter family of pseudo-Riemannian metrics on $Z^{\epsilon}$. In particular, we find that there is a unique Kähler-Einstein ($\epsilon=-1$) or para-Kähler-Einstein ($\epsilon=+1$) metric. Finally, we prove that any Kähler or para-Kähler submanifold of a para-quaternionic Kähler manifold is minimal and describe all such submanifolds in terms of complex ($\epsilon=-1$), respectively, para-complex ($\epsilon=+1$) submanifolds of $Z^{\epsilon}$ tangent to the contact distribution.

Using Perelman's results on Kähler-Ricci flow, we prove that the $K$-energy is bounded from below if and only if the $F$-functional is bounded from below in the canonical Kähler class.