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We review our recent relativistic generalization of the Gutzwiller–Duistermaat–Guillemin trace formula and Weyl law on globally hyperbolic stationary space-times with compact Cauchy hypersurfaces. We also discuss anticipated generalizations to non-compact Cauchy hypersurface cases.

Paleo-Sciences including palaeoclimatology and palaeoecology have accumulated numerous records related to climatic changes. The researchers have usually tried to identify periodic and quasi-periodic processes in these paleoscientific records. In this paper, we show that this analysis is incomplete. As follows from our results, random processes, namely processes with a single-time-constant${\tau }_0$ (noise with a Lorentzian noise spectrum), play a very important and, perhaps, a decisive role in numerous natural phenomena. For several of very important natural phenomena the characteristic time constants ${\tau }_0$ are very similar and equal to $(5-8)\times 10^3$ years. However, this value of ${\tau }_0$ is not universal. For example, the spectral density fluctuations of the atmospheric radiocarbon ${\delta }^{14}$C are characterized by a Lorentzian with ${\tau }_0\approx 300$ years. The frequency dependence of spectral density fluctuations for benthic ${\delta }^{18}$O records contains two Lorentzians with ${\tau }_0\approx 8000$ years and ${\tau }_0>10^5$ years.

Flat Lorentz (3, 1) space is the natural home for Einstein's Special Theory of Relativity, with three "space" dimensions and one "time" dimension. The geometry naturally encodes the ideas of inertial frames, time and space dilation. Much of the mathematical terminology and research interests are infused with physical considerations, but purely mathematical questions are at the core of this treatment.

This is an introduction to Lorentzian (n, 1) geometry for all n ≥ 1, examining the constant curvature spaces: flat, de Sitter (positive curvature) and anti-de Sittter spaces. The conformal boundaries of these Lorentz spacetimes will also be constructed. The connection between Lorentz (2, 1) geometry and geometry of the hyperbolic plane will also be investigated.