One of the main challenges in theoretical physics over the last five decades has been to reconcile quantum mechanics with general relativity into a theory of quantum gravity. However, such a theory has been proved to be hard to attain due to
i) conceptual difficulties present in both the component theories (General Relativity (GR) and Quantum Theory);
ii) lack of experimental evidence, since the regimes at which quantum gravity is expected to be applicable are far beyond the range of conceivable experiments. Despite these difficulties, various approaches for a theory of Quantum Gravity have been developed.
In my research I focus on two such approaches: Loop Quantum Gravity and the Topos theoretic approach.
The choice fell on these approaches because, although they both reject the Copenhagen interpretation of quantum theory, their underpinning philosophical approach to formulating a quantum theory of gravity are radically different.
In particular LQG is a rather conservative scheme, inheriting all the formalism of both GR and Quantum Theory, as it tries to bring to its logical extreme consequences the possibility of combining the two. On the other hand, the Topos approach involves the idea that a radical change of perspective is needed in order to solve the problem of quantum gravity, especially in regard to the fundamental concepts of `space' and `time'. Given the partial successes of both approaches, the hope is that it might be possible to find a common ground in which each approach can enrich the other.
Unfortunately, to date, there is no topos (roughly speaking a topos is a category which is similar to Sets: fundamental mathematical properties (disjoint union, Cartesian product, etc) have a topos analogue) formulation of quantum gravity as such. However, there is well-developed idea on how topos theory can be used in general to describe theories of physics including, potentially, a theory of quantum gravity.
The key idea in the topos approach is that, constructing a theory of physics involves finding a representation, in a topos, of a certain formal language (a formal language is a deductive system of reasoning made of atomic variables, relations between such variables, and rules of inference. In this context it is assumed that each system has a formal language attached to it and which provides a deductive system based on intuitionistic logic), that is attached to the system under investigation. Thus the topos approach consists in first understanding at a fundamental level what a theory of physics and associated conceptual framework should look like and, then, applying these insights to quantum gravity. In this context, a radically new way of thinking about space, time is suggested: for example, the possibility that both GR and quantum theory are `emergent' theories.