Let <i>H</i> denote a spherical subgroup within a semisimple algebraic group <i>G</i>. In this paper we study the closures of the finitely many <i>H</i>-orbits in the flag variety of <i>G</i>. Using the language of Frobenius splitting we provide a criterion for these closures to have nice geometric and cohomological properties. We then show how the criterion applies to the spherical subgroups of minimal rank studied by N. Ressayre. Finally, we also provide applications of the criterion to orbit closures which are not multiplicity-free in the sense defined by M. Brion.

We propose an approach to find constant curvature metrics on triangulated closed 3-manifolds using a finite dimensional variational method whose energy function is the volume. The concept of an angle structure on a tetrahedron and on a triangulated closed 3-manifold is introduced following the work of Casson, Murakami and Rivin. It is proved by A. Kitaev and the author that any closed 3-manifold has a triangulation supporting an angle structure. The moduli space of all angle structures on a triangulated 3-manifold is a bounded open convex polytope in a Euclidean space. The volume of an angle structure is defined. Both the angle structure and the volume are natural generalizations of tetrahedra in the constant sectional curvature spaces and their volume. It is shown that the volume functional can be extended continuously to the compact closure of the moduli space. In particular, the maximum point of the

The existence of a discrete travelling wave is proved for the relaxing scheme. The main idea is to change the original scheme such that the resulting scheme is monotonic to which Jennings' result can be applied. The equivalence of the resulting scheme and the original one is shown when = 1 q. The u-component of the discrete travelling wave thus obtained is a discrete shock for a monotone conservative difference scheme, which approximates the corresponding conservation law.

We prove the nonlinear asymptotic stability of the gravitational hydrostatic equilibrium for the general equation of state of pressure-density relation in the framework of vacuum free boundary problem of spherically symmetric compressible Navier-Stokes-Poisson equations in three dimensions.The results apply to white dwarfs and polytropes with γ≥2, for which even the linearized asymptotic stability results are not available in literature, to the best of the authors' knowledge. Detailed decay rates of perturbations are given.

Fractional statistics is one of the most intriguing features of topological phases in 2D. In particular, the so-called non-Abelian statistics plays a crucial role towards realizing universal topological quantum computation. Recently, the study of topological phases has been extended to 3D and it has been proposed that loop-like extensive objects can also carry fractional statistics. In this work, we systematically study the so-called three-loop braiding statistics for loop-like excitations for 3D fermionic topological phases. Most surprisingly, we discovered new types of non-Abelian three-loop braiding statistics that can only be realized in fermionic systems (or equivalently bosonic systems with fermionic particles). The simplest example of such non-Abelian braiding statistics can be realized in interacting fermionic systems with a gauge group Z2×Z8 or Z4×Z4, and the physical origin of non-Abelian statistics can be viewed as attaching an open Majorana chain onto a pair of linked loops, which will naturally reduce to the well known Ising non-Abelian statistics via the standard dimension reduction scheme. Moreover, due to the correspondence between gauge theories with fermionic particles and classifying fermionic symmetry-protected topological (FSPT) phases with unitary symmetries, our study also give rise to an alternative way to classify FSPT phases with unitary symmetries. We further compare the classification results for FSPT phases with arbitrary Abelian total symmetry G^f and find systematical agreement with previous studies using other methods. We believe that the proposed framework of understanding three-loop braiding statistics (including both Abelian and non-Abelian cases) in interacting fermion systems applies for generic fermonic topological phases in 3D.