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We review recent developments in structural stability as applied to key topics in general relativity. For a nonlinear dynamical system arising from the Einstein equations by a symmetry reduction, bifurcation theory fully characterizes the…
A class of second-order differential equations commonly arising in physics applications are considered, and their explicit hypergeometric solutions are provided. Further, the relationship with the Generalized and Universal Associated…
In this paper, the key ideas of characterizing universality classes of dissipation-free (incompressible) quantum Hall fluids by mathematical objects called quantum Hall lattices are reviewed. Many general theorems about the classification…
Recent developments point to a breakdown in the generalized second law of thermodynamics for theories with Lorentz symmetry violation. It appears possible to construct a perpetual motion machine of the second kind in such theories, using a…
In this paper the factorization method is used in order to obtain the eigenvalues and eigenfunctions of a quantum particle confined in a one-dimensional infinite well. The output results from the mentioned approach allows us to explore an…
Six major frameworks have emerged attempting to describe particle physics beyond the Standard Model. Despite their different theoretical genera, these frameworks have a number of common phenomenological features and problems. While it will…
We discuss universal properties of axisymmetric and stationary configurations consisting of a central black hole and surrounding matter in Einstein-Maxwell theory. In particular, we find that certain physical equations and inequalities…
A classification of integrable two-component systems of non-evolutionary partial differential equations that are analogous to the Camassa-Holm equation is carried out via the perturbative symmetry approach. Independently, a classification…
A complete classification of 2D superintegrable systems on two-dimensional conformally flat spaces has been performed over the years and 58 models, divided into 12 equivalence classes, have been obtained. We will re-examine two…
The fate of matter forming a black hole is still an open problem, although models of quantum gravity corrected black holes are available. In loop quantum gravity (LQG) models were presented, which resolve the classical singularity in the…
In the harmonic description of general relativity, the principle part of Einstein's equations reduces to 10 curved space wave equations for the componenets of the space-time metric. We present theorems regarding the stability of several…
Extended gravitational models have gained large attention in the last couple of decades. In this work, we examine the solution space of vacuum, static, and spherically symmetric spacetimes within $F(R)$ theories, introducing novel methods…
The linearizability of differential equations was first considered by Lie for scalar second order semi-linear ordinary differential equations. Since then there has been considerable work done on the algebraic classification of linearizable…
In the framework of deterministic finslerian models, a mechanism producing dissipative dynamics at the Planck scale is discussed. It is based on a geometric evolution from Finsler to Riemann structures defined on the fiber bundle ${ TM}\to…
Cosmology is living through fascinating times, where new observations from ground and space telescopes are questioning the established paradigm, the so-called Lambda Cold Dark Matter model. The particle nature of Dark Matter is severely…
The gravity-scalar field system in spherical symmetry provides a natural setting for exploring gravitational collapse and its aftermath in quantum gravity. In a canonical approach, we give constructions of the constraint and Hamiltonian…
We study mirror symmetry (A-side vs B-side) in the framework of quantum differential systems. We focuse on the logarithmic and non-resonant case, which describes the geometric situation. We show that quantum differential systems provide a…
Motivated by generalized uncertainty principle, we derive a discrete picture of the space that respects Lorentz symmetry as well as gauge symmetry through setting an equivalency between linear GUP correction term and electromagnetic…
Symmetry plays a central role in quantum field theory. Recent developments include symmetries that act on defects and other subsystems, and symmetries that are categorical rather than group-like. These generalized notions of symmetry allow…
Following a general introduction to open questions beyond the Standard Model, the prospects for addressing them in the new era opened up by the LHC are reviewed. Sample highlights are given of ways in which the LHC is already probing beyond…