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The two surprising features of gravity are (a) the principle of equivalence and (b) the connection between gravity and thermodynamics. Using principle of equivalence and special relativity in the {\it local inertial frame}, one could obtain…

The importance of quantum effects for exotic nuclear shapes is demonstrated. Based on the example of a sheet of nuclear matter of infinite lateral dimensions but finite thickness, it is shown that the quantization of states in momentum…

A model for quantized gravity coupled to matter in the form of a single scalar field is investigated in four dimensions. For the metric degrees of freedom we employ Regge's simplicial discretization, with the scalar fields defined at the…

We study a natural model of random 2-dimensional cubical complex which is a subcomplex of an n-dimensional cube, and where every possible square $2$-face is included independently with probability p. Our main result is to exhibit a sharp…

In this thesis we investigate the importance of causality in non-perturbative approaches to quantum gravity. Firstly, causal sets are introduced as a simple kinematical model for causal geometry. It is shown how causal sets could account…

We analyze the nonlinear optics of quasi one-dimensional quantum graphs and manipulate their topology and geometry to generate for the first time nonlinearities in a simple system approaching the fundamental limits of the first and second…

The recently proposed gravitational entropy generalize the usual black hole entropy to Euclidean solutions without U(1) symmetry in the framework of Einstein gravity. The entropy of such smooth configuration is given by the area of minimal…

Here are described the geometric structures of the lines of principal curvature and the partially umbilic singularities of the tridimensional non compact generic quadric hypersurfaces of ${\mathbb R}^4$. This includes the ellipsoidal…

Non-perturbative theories of quantum gravity inevitably include configurations that fail to resemble physically reasonable spacetimes at large scales. Often, these configurations are entropically dominant and pose an obstacle to obtaining…

I will briefly discuss three cosmological models built upon three distinct quantum gravity proposals. I will first highlight the cosmological role of a vector field in the framework of a string/brane cosmological model. I will then present…

We discuss scaling relations in four dimensional simplicial quantum gravity. Using numerical results obtained with a new algorithm called ``baby universe surgery'' we study the critical region of the theory. The position of the phase…

In the event symmetric approach to quantum gravity it is assumed that the fundamental laws of physics must be invariant under exchange of any two space-time events. The fact that this symmetry if obviously not observed is attributed to the…

In a quantum gravity theory, spacetime at mesoscopic scales can acquire a novel structure very different from the classical concept of general relativity. A way to effectively characterize the quantum nature of spacetime is through a…

We show that, in four-dimensional spacetimes with an arbitrary Einstein metric, with and without a cosmological constant, perturbative dynamical degrees of freedom in generic quadratic-curvature gravity can be decoupled into massless and…

We consider surfaces in Euclidean space parametrized on an annular domain such that the first fundamental form and the principal curvatures are rotationally invariant, and the principal curvature directions only depend on the angle of…

We give the classification of constant mean curvature rotational surfaces of elliptic, hyperbolic, and parabolic type in the four-dimensional pseudo-Euclidean space with neutral metric.

We develop a perturbation theory of four-dimensional topological 2-form gravity without cosmological constant. A 2-form and an $SU(2)$ connection 1-form are used as fundamental variables instead of metric. There is no quantum correction…

A class of 3d $\mathcal{N}=2$ supersymmetric gauge theories are constructed and shown to encode the simplicial geometries in 4-dimensions. The gauge theories are defined by applying the Dimofte-Gaiotto-Gukov construction in 3d/3d…

We derive an effective Hamiltonian for a quantum system constrained to a submanifold (the constraint manifold) of configuration space (the ambient space) by an infinite restoring force. We pay special attention to how this Hamiltonian…

We investigate the geometry of a quantum universe with the topology of the four-torus. The study of non-contractible geodesic loops reveals that a typical quantum geometry consists of a small semi-classical toroidal bulk part, dressed with…