We reconstruct Brumer's family with 3-parameters of curves of genus two whose jacobian varieties admit a real multiplication of discriminant 5. Our method is based on the descent theory in geometric Galois theory which can be compared with a classical problem of Noether. Namely, we first construct a 3-parameter family of polynomials $f(X)$ of degree 6 whose Galois group is isomorphic to the alternating group $A_5$. Then we study the family of curves defined by $Y^2=f(X)$, showing that they are equivalent to Brumer's family. The real multiplication will be described in three distinct ways, i.e., by Humbert's modular equation, by Poncelet's pentagon, and by algebraic correspondences.

We present a theory of intersection on the complex projective line for homology and cohomology groups defined by connections which are regular or not. We apply this theory to confluent hypergeometric functions, and obtain, as an analogue of period relations, quadratic relations satisfied by confluent hypergeometric functions.

A cyclic surface in the Lorentz-Minkowski three-space is one that is foliated by circles. We classify all maximal cyclic timelike surfaces in this space, obtaining different families of non-rotational maximal surfaces. When the mean curvature is a non-zero constant, we prove that if the surface is foliated by circles in parallel planes, then it must be rotational. In particular, we obtain all timelike surfaces of revolution with constant mean curvature.

Normal coordinate systems for pseudo-Riemannian metrics are investigated from a viewpoint of the theory of partial differential equations. Given a cartesian coordinate system $x$, a local metric for which $x$ is a normal coordinate system is determined by a metric tensor at the origin and any one of certain three matrix functions. These are related one another by three partial differential equations. Solvability of these equations in $C^{\infty}$ framework and power series expansion of solutions are discussed.

We investigate configurations of conics in the projective plane which have the property that the number of tacnodes is equal or close to the upper bound obtained from the Miyaoka-Yau inequality. We show that for 5 conics there are exactly 3 configurations, including 2 new ones, achieving the maximum 17 tacnodes, and for 6 conics the maximum number of tacnodes is 22, which together with previous results implies that the Miyaoka-Yau bound can never be achieved.

We prove a vanishing theorem for the Hodge number $h^{2,1}$ of projective toric varieties provided by a certain class of polytopes. We explain how this Hodge number also gives information about the deformation theory of the toric Gorenstein singularity derived from the same polytope. In particular, the vanishing theorem for $h^{2,1}$ implies that these deformations are unobstructed.

Recently, McIntosh develops a method of constructing all non-isotropic harmonic tori in a complex projective space in terms of their spectral data. In this paper, we classify all spectral data whose spectral curves are smooth rational or elliptic curves. We also construct explicitly corresponding harmonic maps.