We give the functorial moduli construction of pure parabolic sheaves in the sense of Álvarez-Cónsul and A. King by using the moduli of filtered Kronecker modules which we introduced in our earlier work. We also use a version of S. G. Langton’s result to deduce the projectivity of the moduli of pure parabolic sheaves of maximal dimension. As an application of functorial moduli construction, we can get the morphisms at the level of moduli stacks.

We consider the global Cauchy problem for a two-component system of cubic semilinear wave equations in two space dimensions. We give a criterion for large-time nondecay of the energy for small amplitude solutions in terms of the radiation fields associated with the initial data.

We establish the existence of congruences between a fixed endoscopic, globally generic, cuspidal automorphic representation Π of $\mathrm{GSp}(4)$ of square-free conductor and stable cuspidal automorphic representations of the same weight modulo certain prime factors of the value at 1 of the adjoint L-function of Π, normalized by a suitable period.

We obtain the boundedness of Calderón–Zygmund operators on the entire Hardy–Morrey spaces with variable exponents. We obtain this result by refining the extrapolation theory. This refined extrapolation theory also gives the boundedness of Fourier multipliers, Bochner–Riesz means, and parametric Marcinkiewicz integrals on Hardy–Morrey spaces with variable exponents.

We analyze the linearized operator for nonradial oscillations of spherically symmetric self-gravitating gaseous stars in view of the functional analysis. The evolution of the star is supposed to be governed by the Euler–Poisson equations under the equation of state of the ideal gas, and the motion is supposed to be adiabatic. We consider the case of not necessarily isentropic, that is, not barotropic motions. Basic theory of self-adjoint realization of the linearized operator is established. Some problems in the investigation of the concrete properties of the spectrum of the linearized operator are proposed. The existence of eigenvalues which accumulate to zero is proved in a mathematically rigorous fashion. The absence of continuous spectra and the completeness of eigenfunctions for the operators reduced by spherical harmonics is discussed.

We construct transcendental automorphims of ${\mathbb{C}}^2$ that have an unbounded and regular Siegel domain.

We prove that the braided Thompson group $BV$ has a linear divergence function. By the work of Druţu, Mozes, and Sapir [12], this implies none of the asymptotic cones of $BV$ has a cut point.