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The objective of the current paper is essentially twofold. Firstly, to make clear the difference between two notions of rolling a Riemannian manifold over another, using a language accessible to a wider audience, in particular to readers…

We discuss a variety of applications of the relative trace formula of Kuznetsov type in analytic number theory and the theory of automorphic forms.

We study finitely cyclic self-adjoint operators in a Hilbert space, i.e. self-adjoint operators that posses such a finite subset in the domain that the orbits of all its elements with respect to the operator are linearly dense in the space.…

A little-known and highly economical characterization of the real interval [0, 1], essentially due to Freyd, states that the interval is homeomorphic to two copies of itself glued end to end, and, in a precise sense, is universal as such.…

A simple visual representation of Minkowski spacetime appropriate for a student with a background in geometry and algebra is presented. Minkowski spacetime can be modeled with a Euclidean 4-space to yield accurate visualizations as…

We elaborate an explicit version of the relative trace formula on $\PGL(2)$ over a totally real number field for the toral periods of Hilbert cusp forms along the diagonal split torus. As an application, we prove (i) a spectral…

A number of recent papers treated the representation theory of partially ordered sets in unitary spaces with the so called orthoscalar relation. Such theory generalizes the classical theory which studies the representations of partially…

We review the developments of a recently proposed approach to study integrable theories in any dimension. The basic idea consists in generalizing the zero curvature representation for two-dimensional integrable models to space-times of…

It is argued that the standard quantum mechanical description of the Bell correlations between entangled subsystems is in conflict with relativistic space-time symmetry. Proposals to abandon relativistic symmetry, in the sense of explicitly…

The different kinds of self-similarity in general relativity are discussed, with special emphasis on similarity of the ``first'' kind, corresponding to spacetimes admitting a homothetic vector. We then survey the various classes of…

Embedding diagrams prove to be quite useful when learning general relativity as they offer a way of visualizing spacetime curvature through warped two dimensional (2D) surfaces. In this manuscript we present a different 2D construct that…

Using the complete orthonormal sets of radial parts of nonrelativitistic exponential type orbitals (2,1, 0, 1, 2, ...) and spinor type tensor spherical harmonics of rank s the new formulae for the 2(2s+1)-component relativistic spinors…

Having developed a description of indefinite extrinsic symmetric spaces by corresponding infinitesimal objects in the preceding paper we now study the classification problem for these algebraic objects. In most cases the transvection group…

We study families of submanifolds in symmetric spaces of compact type arising as exponential images of s-orbits of variable radii. Special attention is given to the cases where the s-orbits are symmetric.

We investigate the conformal geometry of spherically symmetric spacetimes in general without specifying the form of the matter distribution. The general conformal Killing symmetry is obtained subject to a number of integrability conditions.…

We establish a transfer of unitarity for a Bernstein component of the category of smooth representations of a reductive p-adic group to the associated Hecke algebra, in the framework of the theory of types, whenever the Hecke algebra is an…

At present, there is practically no doubt that general relativity is closely related to gravity. Moreover, after the work of Jacobson, Padmanabhan and others, it became clear that a thermodynamic interpretation of Einstein's relativistic…

The arithmetic fundamental lemma conjecture of the third author connects the derivative of an orbital integral on a symmetric space with an intersection number on a formal moduli space of $p$-divisible groups of Picard type. It arises in…

We initiate the study of analogues of symmetric spaces for the family of finite dihedral groups. In particular, we investigate the structure of the automorphism group, characterize the involutions of the automorphism group, and determine…

We study properties of a map from a certain unitary group in $n$ variables to a related unitary group in $\binom{n}{k}$ variables. We explain how it gives rise to a map between canonical models of Shimura varieties and we prove that it…