We prove lower bound estimate for the first nonzero eigenvalue of the $V$-Laplacian on a compact Riemannian manifold without boundary or with a smooth convex boundary and Neumann condition.

Nontrivial examples of Teichmüller curves have been studied systematically with notions of combinatorics invariant under affine homeomorphisms. An origami (square-tiled surface) induces a Teichmüller curve for which the absolute Galois group acts on the embedded curve in the moduli space. In this paper, we study general origamis not admitting pure half-translation structure. Such a flat surface is given by a cut-and-paste construction from origami that is a translation surface. We present an algorithm for the simultaneous calculation of the Veech groups of origamis of given degree. We have calculated the equivalence classes, the $PSL(2,\mathbf{Z})$-orbits, and some Galois invariants for all the patterns of origamis of degree $d \leq 7$.

In this paper, we derive the generalized hypergeometric functions used in mirror computation of degree $k$ hypersurface in $CP^{N-1}$ as generating functions of intersection numbers of the moduli space of quasimaps from $CP^1$ with two marked points to $CP^{N-1}$.

In this paper, we study étale cohomologies of quadrics over $\mathbf{R}$. An element in the étale cohomology is called algebraic, if it is in the image of the cycle map from the Chow ring. In this paper, we compute the étale cohomology of norm quadrics, and give examples which have many non-algebraic elements.

We give a list of hyperbolic two-bridge links which includes all such links with complete exceptional surgeries, i.e., Dehn surgeries on both components which yield non-hyperbolic manifolds but whose all the proper sub-fillings give hyperbolic manifolds. Also all the candidate slopes of complete exceptional surgeries for them are enumerated in our lists.

We study the relation between the vanishing of André-Quillen homology and complete intersection dimensions and we extend some of the existing results in the literature. As an application of one of our results, we give a characterization of algebra retracts of finite André-Quillen dimension with respect to complete intersection dimensions and complexity. We also investigate the relation between André-Quillen homology and an unpublished question of Foxby about complete intersection dimensions.

Let $R \ltimes M$ be a trivial extension of a commutative ring $R$ by an $R$-$R$-bimodule $M$. We first investigate some homological properties over $R \ltimes M$. Then we provide a way to construct new semidualizing $R \ltimes M$-modules from given semidualizing $R$-modules. It is proven that $C$ is a semidualizing $R$-module and $M$ belongs to the Auslander class $\mathcal{A}_{C}(R)$ if and only if $\mathbf{T}(C)$ is a semidualizing $R \ltimes M$-module and $\textrm{Tor}_{i}^{R}(C,M) = 0$ for any $i \geq 1$. Finally, we describe when $\mathbf{T}(C)$ is a dualizing $R \ltimes M$-module.

Yoshihara's definition of Galois points for irreducible plane curves is extended to reducible plane curves. We also define simultaneous Galois points, weakening the conditions of the definition. We study the number of simultaneous Galois points for a reduced plane curve with nonsingular components.