The original motivation for solving of the Einstein equation is to understand space-time in the absence of matter; see the essay by Tod in this volume for an overview of the mathematics of general relativity. The equation governs the manner in which space-time is inuenced by the sole force of gravity. As is well known, singularities such as black holes can occur. The study of the vacuum Einstein equations is a difficult problem in nonlinear hyperbolic systems; see the essay by Christodoulou in this volume. Complete space-times with nontrivial gravitational radiation are not well understood.

Weyl points are degenerate points on the spectral bands at which energy bands intersect conically. They are the origins of many novel physical phenomena and have attracted much attention recently. In this paper, we investigate the existence of such points in the spectrum of the 3-dimensional Schrödinger operator H=−Δ+V(x) with V(x) being in a large class of periodic potentials. Specifically, we give very general conditions on the potentials which ensure the existence of 3-fold Weyl points on the associated energy bands. Different from 2-dimensional honeycomb structures which possess Dirac points where two adjacent band surfaces touch each other conically, the 3-fold Weyl points are conically intersection points of two energy bands with an extra band sandwiched in between. To ensure the 3-fold and 3-dimensional conical structures, more delicate, new symmetries are required. As a consequence, new techniques combining more symmetries are used to justify the existence of such conical points under the conditions proposed. This paper provides comprehensive proof of such 3-fold Weyl points. In particular, the role of each symmetry endowed to the potential is carefully analyzed. Our proof extends the analysis on the conical spectral points to a higher dimension and higher multiplicities. We also provide some numerical simulations on typical potentials to demonstrate our analysis.

The aim of the paper is to derive spectral estimates into several classes of magnetic systems. They include three-dimensional regions with Dirichlet boundary as well as a particle in 3 confined by a local change of the magnetic field. We establish two-dimensional BerezinLiYau and LiebThirring-type bounds in the presence of magnetic fields and, using them, get three-dimensional estimates for the eigenvalue moments of the corresponding magnetic Laplacians.

We compute the particle spectrum and some of the Yukawa couplings for a family of heterotic compactifications on quintic threefolds X involving bundles that are deformations of TX+ O_X. These are then related to the compactifications with torsion found recently by Li and Yau. We compute the spectrum and the Yukawa couplings for generic bundles on generic quintics, as well as for certain stable non-generic bundles on the special Dwork quintics. In all our computations we keep the dependence on the vector bundle moduli explicit. We also show that on any smooth quintic there exists a deformation of the bundle TX+ O_X whose Kodaira-Spencer class obeys the Li-Yau non-degeneracy conditions and admits a non-vanishing triple pairing.

We study motion of a charged particle confined to a Dirichlet layer of a fixed width placed into a homogeneous magnetic field. If the layer is planar and the field is perpendicular to it, the spectrum consists of infinitely degenerate eigenvalues. We consider translationally invariant geometric perturbations and derive several sufficient conditions under which a magnetic transport is possible, that is, the spectrum, in its entirety or a part of it, becomes absolutely continuous.