Let $\frak g$ be a symmetrizable generalized Kac-Moody algebra and $\frak n_-$ the sum of all its negative root spaces. We obtain a formula for the twining characters of the Lie algebra homology modules of $\frak n_-$ with coefficients in the irreducible highest weight $\frak g$-module $L(\Lambda)$ of symmetric, dominant integral highest weight $\Lambda$. This formula gives a new (and convincing) proof of the formula for the twining character of $L(\Lambda)$ above.

We find new examples of harmaonic maps between compact Riemannian manifolds. A section of a Riemannian fibration is called harmonic if it is harmonic as a map from the base manifold into the total space. When the fibres are totally geodesic, the Euler-Lagrange equation for such sections is formulated. In the case of distributions, which are sections of a Grassmannian bundle, this formula is described in terms of the geometry of base manifolds. Examples of harmonic distributions are constructed when the base manifolds are homogeneous spaces and the integral submanifolds are totally geodesic. In particular, we show all the generalized Hopf-fibrations define harmonic maps into the Grassmannian bundles with the standard metric.

The solutions of the algebraic equation $y^{mn}+x y^{mp}-1=0$ with $n>p$ and $m\geq 2$ satisfy a generalized hypergeometric differential equation with imprimitive finite irreducible monodromy group. Thanks to this fact, we can determine the monodromy group and the Schwarz map of the differential equation.

We consider the integral of (the square of) the length of the normal curvature tensor for immersions of manifolds into real space forms, especially into spheres. The first variation formula is given and the Euler-Lagrange equation is expressed in terms of the isothermal coordinates when the submanifold is two-dimensional. The relations between the critical surfaces and Willmore surfaces are discussed. We also give formulas concerning the residue of logarithmic singularities of $S$-Willmore points or estimate it by a conformal invariant.

We show that if a compact critical surface satisfies certain conditions and the immersion is minimal, then the Gauss curvature is a non-negative constant and the immersion is a standard minimal immersion of a sphere or a constant isotropic minimal immersion of a flat torus. To prove this result, we study two-dimensional Riemannian manifolds admitting concircular scalar fields whose characteristic functions are polynomials of degree $2$. Moreover, the case that the characteristic functions are polynomials of degree $3$ is studied.

For a certain class of reversible random walks possibly with drift on an abelian covering graph of a finite graph, using the technique of twisted transition operator, we obtain the asymptotic behavior of the $n$-step transition probability $p_n(x,y)$ as $n \to \infty$ and give an expression of the constant which appears in the asymptotics.

We apply the braid orbit theorem to projective semilinear groups over the finite fields with $p^2$ elements and some almost simple groups of Lie type. The projective special linear groups $\psl_2(p^2)$ with $p\equiv\pm 3\; (\bmod\; 8)$, the Tits simple group, and some small simple groups occur regularly as Galois groups over the rationals.

We give an affirmative answer to the following conjecture of Ma, Kato: Let $S$ be a compact complex surface in Kodaira's class $\rm {VII_{0}}$ which contains a strictly positive number of rational curves being exactly equal to the second Betti number of $S$. Then $S$ admits a global spherical shell.

We define a riemannian version of the vortex filament equation. Using perturbation to a parabolic equation, we prove the short time unique existence of a solution for any initial closed curve.