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At present we have only the very successful but phenomenological Einstein geometrical modelling of the spacetime phenomenon. This geometrical model provides a `container' for other theories, in particular the quantum field theories. Here we…
Projective geometry provides the preferred framework for most implementations of Euclidean space in graphics applications. Translations and rotations are both linear transformations in projective geometry, which helps when it comes to…
We define symmetric spaces in arbitrary dimension and over arbitrary non-discrete topological fields $\K$, and we construct manifolds and symmetric spaces associated to topological continuous quasi-inverse Jordan pairs and -triple systems.…
We state the fundamental theorem of projective geometry for semimodules over semirings, which is facilitated by recent work in the study of bases in semimodules defined over semirings. In the process we explore in detail the linear algebra…
From descent theory to higher geometry, the idea of gluing has been embedded in many elegant and powerful techniques, proving instrumental for the solution of many problems. In this paper, we introduce a framework that allows to link…
We present a deductive theory of space-time which is realistic, objective, and relational. It is realistic because it assumes the existence of physical things endowed with concrete properties. It is objective because it can be formulated…
I investigate a class of dynamical systems in which finite pieces of spacetime contain finite amounts of information. Most of the guiding principles for designing these systems are drawn from general relativity: the systems are…
This thesis is devoted to studying various aspects of quantum mechanics on non-commutative space-time and to capture some of the surviving aspects of symmetries of quantum field theory on such space-time, illustrated through toy models in…
We remark the importance of adding suitable pre-geometric content to tensor models, obtaining what has recently been called tensorial group field theories, to have a formalism that could describe the structure and dynamics of quantum…
Background independence is often emphasized as an important property of a quantum theory of gravity that takes seriously the geometrical nature of general relativity. In a background-independent formulation, quantum gravity should determine…
Noncommutative geometry is a mathematical framework that expresses the structure of space-time in terms of operator algebras. By using the tools of quantum mechanics to describe the geometry, noncommutative space-times are expected to give…
At present, our notion of space is a classical concept. Taking the point of view that quantum theory is more fundamental than classical physics, and that space should be given a purely quantum definition, we revisit the notion of Euclidean…
Following recent assumptions to unify quantum mechanics and general relativity, the structure of spacetime is suppose to be a consequence of the relations among some fundamental objects, and its concept can be formulated without the…
The use of geometric invariants has recently played an important role in the solution of classification problems in non-commutative ring theory. We construct geometric invariants of non-commutative projectivizations, a significant class of…
In models of emergent gravity the metric arises as the expectation value of some collective field. Usually, many different collective fields with appropriate tensor properties are candidates for a metric. Which collective field describes…
We consider spacetime to be a 4-dimensional differentiable manifold that can be split locally into time and space. No metric, no linear connection are assumed. Matter is described by classical fields/fluids. We distinguish electrically…
The purpose of this paper is to present a model of a quantum space-time in which the global symmetries of space-time are unified in a coherent manner with the internal symmetries associated with the state space of quantum-mechanics. If we…
Projective modules play an important role in the study of the category of modules over rings and in the characterization of various classes of rings. Several characterizations of projective objects which are equivalent for modules over…
A general formal definition of a theory of space and time compatible with the inertia principle is given. The formal definition of reference frame and inertial equivalence between reference frames are used to construct the class of inertial…
These are expanded lecture notes of a mini-course whose objectives were to introduce the basic concepts, constructions and techniques of noncommutative geometry, as well as their uses as a framework for modelling quantum spacetime. Key…