Uses of non-commutative geometry in quantum gravity

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The theory of general relativity encodes our understanding of the classical nature of space-time. This includes gravity, but it does not describe the fundamental quantum nature of space-time. For example, close to the Big Bang, and close to the singularity in a black hole, the theory ceases to be valid because quantum effects set in. The development of a quantum gravity theory is one of the most challenging problems faced by contemporary theoretical physics. Non-commutative geometry and loop quantum gravity are two approaches to tackle this issue. They propose promising candidates for the notion of quantum spacetime using a variety of mathematical techniques. The quantum gravity challenge involves not only theory, but also now experiments. Due to the extreme nature of the physical phenomena that are central to quantum gravity, it has been thought for many years that it would impossible to measure quantum gravitational effects. The field has reached an exciting point, however, with the opening of a new experimental window: the launching in early 2009 of the FERMI satellite. It is currently observing very-high-energy gamma-ray bursts which are able to carry some signatures of the quantum gravity regime. These signatures could give us the ability to indirectly observe for the first time quantum gravitational effects, with the prospect of allowing us to test and push forward different quantum gravity models. My research focuses firstly on understanding the physics behind the non-commutative geometry and loop quantum gravity approaches. To do so, I am proposing to explore new mathematical bridges between these two approaches that I have identified in my recent work. It aims secondly at identifying and predicting some concrete signatures of quantum gravitational phenomena that could be observed by some upcoming experiments such as FERMI.