In this paper we construct a new time-periodic solution of the vacuum Einsteins field equations, this solution possesses physical singularities, i.e., the norm of the solutions Riemann curvature tensor takes the infinity at some points. We show that this solution is intrinsically time-periodic and describes a time-periodic universe with the time-periodic physical singularity. By calculating the Weyl scalars of this solution, we investigate new physical phenomena and analyze new singularities for this universal model.

We introduce a new model for elliptic fibrations endowed with a Mordell-Weil group of rank one. We call it a Q _7 (\mathscr {L},\mathscr {S}) model.

We study the quantum sheaf cohomology of flag manifolds with deformations of the tangent bundle and use the ring structure to derive how the deformation transforms under the biholomorphic duality of flag manifolds. Realized as the OPE ring of A/2-twisted twodimensional theories with (0,2) supersymmetry, quantum sheaf cohomology generalizes the notion of quantum cohomology. Complete descriptions of quantum sheaf cohomology have been obtained for abelian gauged linear sigma models (GLSMs) and for nonabelian GLSMs describing Grassmannians. In this paper we continue to explore the quantum sheaf cohomology of nonabelian theories. We first propose a method to compute the generating relations for (0,2) GLSMs with (2,2) locus. We apply this method to derive the quantum sheaf cohomology of products of Grassmannians and flag manifolds. The dual deformation associated with the biholomorphic duality gives rise to an explicit IR duality of two A/2-twisted (0,2) gauge theories.

We consider a kinetic model of self-propelled particles with alignment interaction and with precession about the alignment direction. We derive a hydrodynamic system for the local density and velocity orientation of the particles. The system consists of the conservative equation for the local density and a non-conservative equation for the orientation. First, we assume that the alignment interaction is purely local and derive a first order system. However, we show that this system may lose its hyperbolicity. Under the assumption of weakly non-local interaction, we derive diffusive corrections to the first order system which lead to the combination of a heat flow of the harmonic map and Landau-Lifschitz-Gilbert dynamics. In the particular case of zero self-propelling speed, the resulting model reduces to the phenomenological Landau-Lifschitz-Gilbert equations. Therefore the present theory provides a kinetic formulation of classical micromagnetization models and spin dynamics.

We consider the scalar wave equation in the Kerr geometry for Cauchy data which is smooth and compactly supported outside the event horizon. We derive an integral representation which expresses the solution as a superposition of solutions of the radial and angular ODEs which arise in the separation of variables. In particular, we prove completeness of the solutions of the separated ODEs.