In this paper we compute the stable rank and connected stable rank of some $C^*$-algebras regarded as noncommutative manifolds. Mainly, we consider the noncommutative spaces studied by Matsumoto and Tomiyama, their higher dimensional versions and the $C^*$-algebras of continuous fields of these noncommutative manifolds.

We shall give a representation formula of super-minimal J-holomorphic curves of a nearly Kähler 6-dimensional sphere and construct a deformation of such J-holomorphic curves.

In this paper we prove that the Hodge conjecture is true for any self-product of the jacobian variety $J(C_{p^\mu,q^\nu})$ of the curve $C_{p^\mu,q^\nu} : y^{q^\nu}=x^{p^\mu}-1$, where $p^\mu$ and $q^\nu$ are powers of distinct prime numbers $p$ and $q$. We also prove that the Hodge ring of $J(C_{p^\mu,q^\nu})$ is {\it not} generated by the divisor classes whenever $p^\mu q^\nu\neq 12$ and $(\mu,\nu)\neq(1,1)$.

The far point set of a self-map of a closed Riemannian manifold $M$ is defined to be the set of points mapped into their cut locus. We prove that the far point set of a map $f$ with Lefschetz number $L(f) \neq \chi(M)$ is infinite unless $M$ is a sphere. There are homology classes supported near $\text{Far}(f)$ which determine $L(f)-\chi(M).$ Using geometric representatives of Thom classes, we obtain a geometric integral formula for the the Lefschetz number, which specializes to the Chern-Gauss-Bonnet formula when $f=\text{Id}.$ We compute this formula explicitly for constant curvature metrics. Finally, we give upper and lower bounds for $L(f)$ in terms of the geometry and topology of $M$ and the differential of $f$.

In this work we use a commutative generalization of complex numbers, called bicomplex numbers, to introduce a holomorphic Riemann zeta function of two complex variables satisfying the complexified Cauchy-Riemann equations. Furthermore, we establish a bicomplex Riemann hypothesis equivalent to the complex Riemann hypothesis of one variable and we obtain a bicomplex Euler Product.

We prove that the anti-pluricanonical map $\Phi_{|-mK_{X}|}$ is birational when $m\geq 16$ for $5$-fold $X$ whose anticanonical divisor is nef and big.

In this short note, we consider the conjecture that the log canonical divisor (resp.\ the anti-log canonical divisor) $K_X + \Delta$ (resp.\ $-(K_X + \Delta)$) on a pair $(X, \Delta)$ consisting of a complex projective manifold $X$ and a reduced simply normal crossing divisor $\Delta$ on $X$ is ample if it is numerically positive. More precisely, we prove the conjecture for $K_X + \Delta$ with $\Delta \neq 0$ in dimension $4$ and for $-(K_X + \Delta)$ with $\Delta \neq 0$ in dimension $3$ or $4$.

Asymptotics of ODE appearing in the turning point problems can be characterized literally by its characteristic polygon. The Airy equation has a one-segment characteristic polygon. Fedoryuk ([4]), Nakano ([8], [12]), Nakano et al. ([13]), and Roos ([17], [18]) studied ODE's with a several-segement one. The more segments, the more complicated asymptotics. Here, we study an ODE with a many-segment one. Firstly, the ODE is reduced to the simpler ODE's in some subdomains, and then the reduced ODE's have the WKB solutions as their asymptotic solutions. Secondly, two sets of the WKB solutions in the neighboring subdomains are related by a matching matrix. In our analysis the stretching-matching method is applied and the Stokes curves play an important role. How to get the Stokes curve configuration for the reduced ODE's is analyzed precisely.

Let $f$ be a primitive form whose weight is greater than $2$. Weston [23, Theorem 1] showed that the mod $p$ representation $\bar{\rho}$ associated to $f$ is irreducible and the deformation problem for $\bar{\rho}$ is unobstructed for almost all $p$. The aim of this article is to give a simpler proof of his result in some cases.

For sphere bundles of the induced bundles of isometric immersions, harmonic sections which are normal to submanifolds and their index are studied. The lower bounds of the index of these sections are given in terms of intrinsic quantities of submanifolds.

We introduce a multiple analogue of the gamma function which differs from the one defined by Barnes [B]. Using this function, we give expressions of the multiple sine and cosine functions in terms of zeta regularized products. The expression of the multiple sine function can be interpreted as a reflection formula of this new multiple analogue of the gamma function.

We give a simple criterion for when an isometric immersion from the real two-plane into the two-dimensional complex hyperbolic space with non-zero parallel mean curvature vector is invariant under the action of some lattice of $\mathbf{R}^2$.