Final Activity Report Summary - ADG IN QG (Applications of Abstract Differential Geometry to Quantum Gravity: a Sheaf and Category-Theoretic Approach)
An entirely algebraic, i.e. sheaf-theoretic, approach to gravity, both classical and quantum, was developed.
With respect to classical gravity, we showed how the abstract differential geometry (ADG) technology could be applied, in a finitistic-algebraic setting, so as to completely avoid the singularities and their associated unphysical infinities assailing gravity (GR) due to the latters assumption of a smooth base space-time manifold, which was not present at all in ADG. With respect to quantum gravity (QG) we showed how ADG could be applied to formulate non-perturbative QG as a pure quantum gauge theory in a manifestly background independent way.
In addition, during the course of development of our work, homological algebra, i.e. sheaf, category and topos-theoretic ideas were infused into more mainstream QG research channels, while we also made contact with other contemporary QG research trends, such as the application of ideas from the consistent histories approach to quantum mechanics, to quantum space and time structure, such as quantum space-time topology, and to gravity.
With respect to classical gravity, we showed how the abstract differential geometry (ADG) technology could be applied, in a finitistic-algebraic setting, so as to completely avoid the singularities and their associated unphysical infinities assailing gravity (GR) due to the latters assumption of a smooth base space-time manifold, which was not present at all in ADG. With respect to quantum gravity (QG) we showed how ADG could be applied to formulate non-perturbative QG as a pure quantum gauge theory in a manifestly background independent way.
In addition, during the course of development of our work, homological algebra, i.e. sheaf, category and topos-theoretic ideas were infused into more mainstream QG research channels, while we also made contact with other contemporary QG research trends, such as the application of ideas from the consistent histories approach to quantum mechanics, to quantum space and time structure, such as quantum space-time topology, and to gravity.