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It is a long-standing problem in the foundations of physics to find a theory of quantum gravity that would unify or reconcile quantum theory with general relativity. Despite huge efforts, today no predictive theory of quantum gravity exists.At first sight, one might expect that both quantum theory and general relativity must be revised before a successful unification can be achieved. In practice, however, most research programmes use more-or-less standard quantum theory, including the major ones (string theory and loop quantum gravity). This may very well be conceptually wrong.It is commonly expected that at very small distances (which are relevant for quantum gravity) the continuum picture of space-time breaks down. Thus, for quantum gravity it is desirable to have a mathematical formalism that does not fundamentally depend on the use of the continuum in the form of the real or complex numbers. Secondly, the usual interpretation of quantum theory is instrumentalist, i.e., it depends on measurements and observers external to the quantum system itself. In a future theory of quantum gravity and cosmology, the whole universe will be treated as a quantum system. Clearly, there is no external observer, and a more realist description of quantum systems is necessary.In the last few years, we have made major progress in developing a genuinely new mathematical framework for physical theories which (a) is realist in the sense that measurements play no special rle and (b) does not fundamentally depend on the use of the continuum, i.e., the real or complex numbers. The main mathematical tool is topos theory, a highly developed branch of category theory.Usually, our mathematical structures are built from sets and functions between them. One says that sets and functions form a category Set. For a given point x and a subset S, the proposition x lie in S is either true or false. A topos provides all the necessary structures that allow to do mathematics : one can define mathematical objects in a topos just as in Set (which is a topos itself, of course), and each topos has an internal logic. This logic is intuitionistic, which means that the law of excluded middle need not hold. One can do proofs in a topos using the internal logic. Thus each topos is a mathematical universe of its own. It is intriguing that this structure from the foundations of mathematics plays a rle in the foundations of physics.Today, we can describe the states of a physical system and the physical quantities topos-internally. The main open problem is the description of physical space and space-time. Clearly, it is highly relevant for any approach to quantum gravity to describe in detail how space and space-time are encoded mathematically, or how they possibly arise from more fundamental structures. The proposed research will consider several approaches to these questions. There will be major extensions to our current framework.In the first, longer part of the research (A. Space-like structures in a topos of presheaves, months 1-18), we will use several mathematical objects that have proven useful in the description of topos quantum theory and will consider space-like structures defined from them. Other results of category theory will be important in this research.In the second part (B. Other relevant theories, months 19-30), we will consider related mathematical theories and develop and clarify their physical content. In particular, we want to consider how the emergence of space and time from more basic, categorical structures can be described.In parallel to A. and B., we will consider physical models (C. Models), in order to concretise and test our ideas. The topos methods allow the development of genuinely new physical models. One aim is a simple model of quantum cosmology.
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