For a polarized K3 surface $(X,H)$ of Picard number 1, we shall study the birational equivalence classes of moduli of stable sheaves in the case when $\left(H^2\right)$ and the dimension are small.

In this paper, we construct a right-angled $5$-polytope $\mathcal{P}$ of finite volume such that all the right-angled Coxeter groups with Fuchsian ends obtained from $\mathcal{P}$ are locally rigid. This is the first explicit example of locally rigid discrete subgroups of $\text{Isom}\left({\mathrm{ℍ}}^5\right)$ with Fuchsian ends.

In a proper hyperbolic geodesic space, it is well known that the sequential boundary can be identified as topological spaces with the geodesic boundary. We show that in a (not necessarily proper) hyperbolic geodesic space, the sequential boundary can be identified as topological spaces with the quasi-geodesic boundary.

In this paper, we consider a class of singular nonlinear first order partial differential equations $t(\partial u/\partial t)=F(t,x,u,\partial u/\partial x)$ with $(t,x)\in \mathrm{ℝ}\times \mathrm{ℂ}$ under the assumption that $F(t,x,z_1,z_2)$ is a function which is continuous in $t$ and holomorphic in the other variables. Under suitable conditions, we determine all the solutions of this equation in a neighborhood of the origin.

For many fibred systems the existence of an invariant measure can be proved but considerably less is known about the shape of the density. In this note various examples of invariant densities are discussed: Piecewise fractional linear maps with four branches and maps which are associated to continued fractions with increasing digits. There are ergodic maps with a non-integrable density which do not have an indifferent fixed point and maps such that the set of points which miss the digit $k=1$ has positive Lebesgue measure.

Let ${\Sigma }_{g,1}^1$ be a genus $g$ compact oriented surface with one boundary component and one marked point $x$. Let $G$ be the identity component of the group of symplectomorphisms that preserve the marked point $x$. By using the flux homomorphism and the short exact sequence $1\to G_{\text{rel}}\to G\to {\text{Diff}}_{+}\left(S^1\right)\to 1$, we construct a central $\mathrm{ℝ}$-extension of ${\text{Diff}}_{+}\left(S^1\right)$. We also show that the second cohomology class corresponding to the central $\mathrm{ℝ}$-extension is equal to the Euler class of ${\text{Diff}}_{+}\left(S^1\right)$ up to non-zero constant multiple.

Using theories of principal fibre bundles and connections in [11], we study curvature invariants as Riemannian submanifolds for equivariant isometric minimal immersions from Riemannian or Hermitian symmetric spaces of compact type into Grassmannian manifolds. First we calculate principal curvatures of all equivariant full isometric minimal immersions from the complex projective line to complex quadrics. It is shown that they do not depend on the parameter of the moduli space. According to the preceding research, it is important for the study of curvature invariants to examine the behavior of higher order covariant derivative of the second fundamental form. It also enables us to distinguish those maps by curvature invariants in the moduli space. In addition, we calculate principal curvatures of all equivariant full holomorphic isometric embeddings between the complex projective spaces.

In [5] we gave examples of triples $(g,h_1,h_2)$ of smooth functions to construct smooth $SU(p,q)$-actions on the projective space ${\mathbf{P}}_{\mathrm{ℂ}}^{p+q-1}$ $(p,q\ge 3)$ whose restricted $S\left(U\right(p)\times U(q\left)\right)$-action is the standard action. By using these examples of triples $(g,h_1,h_2)$, we shall construct smooth (resp. analytic) $S{O}_0(p,q)$-actions on $S^{p+q-1}$ $(p,q\ge 3)$ whose restricted $SO\left(p\right)\times SO\left(q\right)$-action is the standard action. Consequently, we shall show that for each positive integer $m$ there exist uncountably infinite smooth conjugacy classes of smooth $S{O}_0(p,q)$-actions on $S^{p+q-1}$ $(p,q\ge 3)$ with $m$ closed and $m+1$ open orbits whose restricted $SO\left(p\right)\times SO\left(q\right)$-action is the standard action. On the other hand, we shall show that there exist exactly countably many analytic conjugacy classes of analytic $S{O}_0(p,q)$-actions on $S^{p+q-1}$ $(p,q\ge 3)$ with one closed and two open orbits whose restricted $SO\left(p\right)\times SO\left(q\right)$-action is the standard action.

Brown has defined generalised Thompson’s groups $F_n$, $T_n$, $V_n$ where $n$ is an integer at least $2$ and Thompson’s groups $F=F_2$, $T=T_2$, $V=V_2$ in the 80’s. Burillo, Cleary and Stein have found that there is a quasi-isometric embedding from $F_n$ to $F_m$ where $n$ and $m$ are positive integers at least 2. We show that there is a quasi-isometric embedding from $T_n$ to $T_2$ for any $n\ge 2$ and no embeddings from $T_2$ to $T_n$ for $n\ge 3$.

To investigate the interrelationship between the subgroup posets of groups, we introduce poset-blowdowns between subgroup posets, motivated by blowdowns in algebraic geometry. Poset-blowdowns are divided into three types (tame, wild, hybrid) depending on their combinatorial properties. We provide various criteria for determining types. Moreover we completely classify tame types for finite groups: they are associated with either generalized quaternion groups or cyclic groups of prime power orders.

For an elliptic curve $E$ over $\mathrm{\mathbb{Q}}$, putting $K=\mathrm{\mathbb{Q}}\left(E\right[p\left]\right)$ which is the $p$-th division field of $E$ for an odd prime $p$, we study the ideal class group ${\text{Cl}}_K$ of $K$ as a $\text{Gal}(K/\mathrm{\mathbb{Q}})$-module. More precisely, for any $j$ with $1⩽j⩽p-2$, we give a condition that ${\text{Cl}}_K\otimes {\mathbb{F}}_p$ has the symmetric power ${\text{Sym}}^{j}E\left[p\right]$ of $E\left[p\right]$ as its quotient $\text{Gal}(K/\mathrm{\mathbb{Q}})$-module, in terms of Bloch-Kato’s Tate-Shafarevich group of ${\text{Sym}}^j{V}_{p}E$. Here $V_{p}E$ denotes the rational $p$-adic Tate module of $E$. This is a partial generalization of a result of Prasad and Shekhar for the case $j=1$.

Using a descent method, we construct certain power series generated by linear recurrences, each of which possesses the following property: The infinite set consisting of all its values and all the values of its derivatives of any order, at any nonzero algebraic numbers within its domain of existence, is algebraically independent. The main theorems of this paper assert that the power series of the form $\sum _{k=0}^{\infty }{z}^{e_k}$, where $\{e_k{\}}_{k\ge 0}$ is a linear recurrence with certain admissible properties, have this property. In particular, Main Theorem 1.16 provides a class of $\{e_k{\}}_{k\ge 0}$ which is simpler than ever before.

In this paper we show a reconstruction formula of the convex hull of the defect $D$ from the Dirichlet to Neumann map associated with the magnetic Schrödinger operator by using the enclosure method proposed by Ikehata [6], assuming certain higher regularity for the potentials of the magnetic Schrödinger operator, under the Dirichlet condition or the Robin condition on the boundary $\partial D$ in the two and three dimensional case.