Abstract:
The ``geometry'', in the sense of the classical differential geometry of smooth manifolds (CDG), is put under scrutiny from the point of view of Abstract Differential Geometry (ADG), along with resulting, thereby, potential physical consequences, in what, in particular, concerns physical ``gauge theories'', when the latter are viewed as being, anyway, of a ``geometrical character''. Yet, ``physical geometry'', in connection with physical laws and the associated with them, within the context of ADG, ``differential'' equations (whence, no background spacetime manifold is needed thereat), are also under discussion.

Abstract:
The paper concerns the fictitious entanglement of the so-called ``singularities'' in problems, pertaining to quantum gravity, due, in point of fact, to the way we try to employ, in that context, differential geometry, the latter being associated, in effect, by far, classically (:smooth manifolds), on the basis of an erroneous correspondence between what we may call/understand, as ``physical space'' and the ``cartesian-newtonian'' one.

Abstract:
Through the subsequent discussion we consider a certain particular sort of (topological) algebras, which may substitute the `` structure sheaf algebras'' in many--in point of fact, in all--the situations of a geometrical character that occur, thus far, in several mathematical disciplines, as for instance, differential and/or algebraic geometry, complex (geometric) analysis etc. It is proved that at the basis of this type of algebras lies the sheaf-theoretic notion of (functional) localization, which, in the particular case of a given topological algebra, refers to the respective ``Gel'fand transform algebra'' over the spectrum of the initial algebra. As a result, one further considers the so-called ``geometric topological algebras'', having special cohomological properties, in terms of their ``Gel'fand sheaves'', being also of a particular significance for (abstract) differential-geometric applications; yet, the same class of algebras is still ``closed'', with respect to appropriate inductive limits, a fact which thus considerably broadens the sort of the topological algebras involved, hence, as we shall see, their potential applications as well.

Abstract:
We present herewith certain thoughts on the important subject of nowadays physics, pertaining to the so-called ``singularities'', that emanated from looking at the theme in terms of ADG (: abstract differential geometry). Thus, according to the latter perspective, we can involve ``singularities'' in our arguments, while still employing fundamental differential-geometric notions such as connections, curvature, metric and the like, retaining also the form of standard important relations of the classical theory (e.g. Einstein and/or Yang-Mills equations, in vacuum), even within that generalized context of ADG. To wind up, we can extend (in point of fact, {calculate) over singularities classical differential-geometric relations/equations, without altering their forms and/or changing the standard arguments; the change concerns thus only the way, we employ the usual differential geometry of smooth manifolds, so that the base ``space'' acquires now quite a secondary role, not contributing at all (!) to the differential-geometric technique/mechanism that we apply. Thus, the latter by definition refers directly to the objects being involved--the objects that ``live on that space'', which by themselves are not, of course, ipso facto ``singular''!

Abstract:
Applying the classical Serre-Swan theorem, as this is extended to topological (non-normed) algebras, one attains a classification of elementary particles via their spin-structure. In this context, our argument is virtually based on a ``correspondence principle'' of S. A. Selesnick, formulated herewith in a sheaf-theoretic language, presisely speaking, in terms of vector sheaves. This then leads directly to second quantization, as well as, to other applications of geometric (pre)quantization theory.

Abstract:
The homological K？hler-de Rham differential mechanism models the dynamical behavior of physical fields by purely algebraic means and independently of any background manifold substratum. This is of particular importance for the formulation of dynamics in the quantum regime, where the adherence to such a fixed substratum is problematic. In this context, we show that the functorial formulation of the K？hler-de Rham differential mechanism in categories of sheaves of commutative algebras, instantiating generalized localization environments of physical observables, induces a consistent functorial framework of dynamics in the quantum regime. 1. Introduction The basic conceptual and technical issue pertaining to the current research attempts towards the construction of a viable quantum theory of the gravitational field, refers to the problem of independence of this theory from a fixed spacetime manifold substratum. In relation to this problem, we have argued about the existence and functionality of a homological schema of functorial general relativistic dynamics, constructed by means of connection inducing functors and their associated curvatures, which is, remarkably, independent of any background substratum [1]. More precisely, the homological dynamical mechanism is based on the modeling of the notion of physical fields in terms of connections, which effectuate the functorial algebraic process of infinitesimal scalars extensions, due to interactions caused by these fields. The appealing property of this schema lies on the fact that the induced field dynamics is not dependent on the codomain of representability of the observables and most importantly is only subordinate to the algebra-theoretic characterization of their structures. In this perspective, the absolute representability principle of classical general relativity, in terms of real numbers, may be relativized without affecting the functionality of the algebraic mechanism. Consequently, it is possible to describe the dynamics of gravitational interactions in generalized localization environments, instantiated by suitable categories of presheaves or sheaves. In particular, according to this strategy, the problem of quantization of gravity is equivalent to forcing the algebraic K？hler-de Rham general relativistic dynamical mechanism of the gravitational connection functorial morphism inside an appropriate sheaf-theoretic localization environment, which is capable of incorporating the localization properties of observables in the quantum regime. The only cost to be paid for this sheaf-theoretic

Abstract:
The mechanism of differential geometric calculus is based on the fundamental notion of a connection on a module over a commutative and unital algebra of scalars defined together with the associated de Rham complex. In this communication, we demonstrate that the dynamical mechanism of physical fields can be formulated by purely algebraic means, in terms of the homological K？hler-De Rham differential schema, constructed by connection inducing functors and their associated curvatures, independently of any background substratum. In this context, we show explicitly that the application of this mechanism in General Relativity, instantiating the case of gravitational dynamics, is related with the absolute representability of the theory in the field of real numbers, a byproduct of which is the fixed background manifold construct of this theory. Furthermore, the background independence of the homological differential mechanism is of particular importance for the formulation of dynamics in quantum theory, where the adherence to a fixed manifold substratum is problematic due to singularities or other topological defects. 1. Introduction From a category-theoretic viewpoint, the generative mechanism of differential geometric calculus is a consequence of the existence of a pair of adjoint functors, expressing the conceptually inverse algebraic processes of infinitesimally extending and restricting the scalars. Classically, the algebraic differential mechanism is based on the fundamental notion of a connection on a module over a commutative and unital algebra of scalars defined together with the associated de Rham complex [1–3]. A connection on a module induces a process of infinitesimal extension of the scalars of the underlying algebra, which is interpreted geometrically as a process of first-order parallel transport along infinitesimally variable paths in the geometric spectrum space of this algebra. The next stage of development of the differential mechanism involves the satisfaction of appropriate global requirements referring to the transition from the infinitesimal to the global level. These requirements are of a homological nature and characterize the integrability property of the variation process induced by a connection. Moreover, they are properly addressed by the construction of the De Rham complex associated to an integrable connection. The nonintegrability of a connection is characterized by the notion of curvature bearing the semantics of observable disturbances to the process of cohomologically unobstructed variation induced by the corresponding

Abstract:
The smooth gravitational singularities of the differential spacetime manifold based General Relativity (GR) are viewed from the perspective of the background manifold independent and, in extenso, Calculus-free Abstract Differential Geometry (ADG). In particular, the inner Schwarzschild singularity is being `resolved' ADG-theoretically in two different ways. A plethora of important mathematical, physical and philosophical issues in current classical and quantum gravity research are addressed and tackled.

Abstract:
In the current debate referring to the construction of a tenable background independent theory of Quantum Gravity we introduce the notion of topos-theoretic relativization of physical representability and demonstrate its relevance concerning the merging of General Relativity and Quantum Theory. For this purpose we show explicitly that the dynamical mechanism of physical fields can be constructed by purely algebraic means, in terms of connection inducing functors and their associated curvatures, independently of any background substratum. The application of this mechanism in General Relativity is constrained by the absolute representability of the theory in the field of real numbers. The relativization of physical representability inside operationally selected topoi of sheaves forces an appropriate interpretation of the mechanism of connection functors in terms of a generalized differential geometric dynamics of the corresponding fields in the regime of these topoi. In particular, the relativization inside the topos of sheaves over commutative algebraic contexts makes possible the formulation of quantum gravitational dynamics by suitably adapting the functorial mechanism of connections inside that topos.

Abstract:
We continue recent work and formulate the gravitational vacuum Einstein equations over a locally finite spacetime by using the basic axiomatics, techniques, ideas and working philosophy of Abstract Differential Geometry. The whole construction is `fully covariant', `inherently quantum' (both expressions are analytically explained in the paper) and genuinely smooth background spacetime independent.