Abstract:
Atmospheric neutrino and solar neutrino data from the first phase of Super-Kamiokande (SK-I) are presented. The observed data are used to study atmospheric and solar neutrino oscillations. Zenith angle distributions from various atmospheric neutrino data samples are used to estimate the neutrino oscillation parameter region. In addition, a new result of the $L/E$ measurement is presented. A dip in the $L/E$ distribution was observed in the data, as predicted from the sinusoidal flavor transition probability of neutrino oscillation. The energy spectrum and the time variation such as day/night and seasonal differences of solar neutrino flux are measured in Super-Kamiokande. The neutrino oscillation parameters are strongly constrained from those measurements.

Abstract:
We developed a cryogenic system on a rotating table that achieves sub-Kelvin conditions. The cryogenic system consists of a helium sorption cooler and a pulse tube cooler in a cryostat mounted on a rotating table. Two rotary-joint connectors for electricity and helium gas circulation enable the coolers to be operated and maintained with ease. We performed cool-down tests under a condition of continuous rotation at 20 rpm. We obtained a temperature of 0.23 K with a holding time of more than 24 hours, thus complying with catalog specifications. We monitored the system's performance for four weeks; two weeks with and without rotation. A few-percent difference in conditions was observed between these two states. Most applications can tolerate such a slight difference. The technology developed is useful for various scientific applications requiring sub-Kelvin conditions on rotating platforms.

Abstract:
For each nonsingular hyperelliptic curve of arbitrary genus, we construct a natural injection from the Galois cohomology of 2-torsion subgroups of Jacobian varieties of the curve to the set of isomorphism classes of nonsingular complete intersections of two quadrics. This gives a generalization of the result of Flynn and Skorobogatov.

Abstract:
Can a smooth plane cubic be defined by the determinant of a square matrix with entries in linear forms in three variables? If we can, we say that it admits a linear determinantal representation. In this paper, we investigate linear determinantal representations of smooth plane cubics over various fields, and prove that any smooth plane cubic over a large field (or an ample field) admits a linear determinantal representation. Since local fields are large, any smooth plane cubic over a local field always admits a linear determinantal representation. As an application, we prove that a positive proportion of smooth plane cubics over Q, ordered by height, admit linear determinantal representations. We also prove that, if the conjecture of Bhargava-Kane-Lenstra-Poonen-Rains on the distribution of Selmer groups is true, a positive proportion of smooth plane cubics over Q fail the local-global principle for the existence of linear determinantal representations.

Abstract:
It is well-known that theta characteristics on smooth plane curves over a field of characteristic different from two are in bijection with certain smooth complete intersections of three quadrics. We generalize this bijection to possibly singular hypersurfaces of any dimension over arbitrary fields including those of characteristic two. It is accomplished in terms of linear orbits of tuples of symmetric matrices instead of smooth complete intersections of quadrics. As an application of our methods, we give a description of the projective automorphism groups of complete intersections of quadrics generalizing Beauville's results.

Abstract:
We discuss the ascending chain condition for lengths of extremal rays. We prove that the lengths of extremal rays of $n$-dimensional $\mathbb Q$-factorial toric Fano varieties with Picard number one satisfy the ascending chain condition.

Abstract:
We prove that a smooth plane curve over a global field of characteristic two is defined by the determinant of a symmetric matrix with entries in linear forms in three variables if and only if such a symmetric determinantal representation exists over the completion at each place of the base field. We also prove a stronger assertion when the degree of the plane curve is odd or the plane curve has a rational point. The canonical theta characteristics on smooth algebraic curves, which exist only in characteristic two, play an important role in the proof. It seems a rather curious phenomenon in characteristic two because we have counterexamples in other characteristics..

Abstract:
A smooth plane curve is said to admit a symmetric determinantal representation if it can be defined by the determinant of a symmetric matrix with entries in linear forms in three variables. We study the local-global principle for the existence of symmetric determinantal representations of smooth plane curves over a global field of characteristic different from two. When the degree of the plane curve is less than or equal to three, we relate the problem of finding symmetric determinantal representations to more familiar Diophantine problems on the Severi-Brauer varieties and mod 2 Galois representations, and prove that the local-global principle holds for smooth plane conics and smooth plane cubics. We also construct counterexamples to the local-global principle for smooth plane quartics.

Abstract:
We prove that the defining equations of the Fermat curves of prime degree cannot be written as the determinant of symmetric matrices with entries in linear forms in three variables with rational coefficients. In the proof, we use a relation between symmetric matrices with entries in linear forms and non-effective theta characteristics on smooth plane curves. We also use some results of Gross-Rohrlich on the rational torsion points on the Jacobian varieties of the Fermat curves of prime degree.

Abstract:
We give classifications of linear orbits of pairs of square matrices with non-vanishing discriminant polynomials over a field in terms of certain coherent sheaves with additional data on closed subschemes of the projective line. Our results are valid in a uniform manner over arbitrary fields including those of characteristic two. This work is based on the previous work of the first author on theta characteristics on hypersurfaces. As an application, we give parametrizations of orbits of pairs of symmetric matrices under special linear groups with fixed discriminant polynomials generalizing some results of Wood and Bhargava-Gross-Wang.