RESEARCH GROUP

MATHEMATICAL STRUCTURES OF THE UNIVERSE

#### Books, review articles, conference announcements

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 1. Micha³ Eckstein, Nicolas FrancoCausal structure for noncommutative geometryFrontiers of Fundamental Physics, vol. 14, pp. 138-xxx (2015). [abstract] [journal] Abstract:Noncommutative Geometry à la Connes offers a promising framework for models of fundamental interactions. To guarantee the correct signature, the theory of Lorentzian spectral triples has been developed. We will briefly summarise its main elements and show that it can accommodate a sensible notion of causality understood as a partial order relation on the space of states on an algebra. For almost-commutative algebras of the form $C^\infty \otimes \A_F$, with $\A_F$ being finite-dimensional, the space of (pure) states is a simple product of space-time $\M$ and an internal space. The exploration of causal structures in this context leads to a surprising conclusion: The motion in both space-time and internal space is restricted by a "finite speed of light" constraint. We will present this phenomena on 2 simple toy-models. 2. Nicolas Franco, Micha³ EcksteinNoncommutative geometry, Lorentzian structures and causality in “Mathematical Structures of the Universe”, eds. M. Eckstein, M. Heller, S.J. Szybka, Copernicus Center Press (2014), pp. 315-340 [abstract] [preprint] [journal] Abstract:The theory of noncommutative geometry provides an interesting mathematical background for developing new physical models. In particular, it allows one to describe the classical Standard Model coupled to Euclidean gravity. However, noncommutative geometry has mainly been developed using the Euclidean signature, and the typical Lorentzian aspects of space-time, the causal structure in particular, are not taken into account. We present an extension of noncommutative geometry \a la Connes suitable the for accommodation of Lorentzian structures. In this context, we show that it is possible to recover the notion of causality from purely algebraic data. We explore the causal structure of a simple toy model based on an almost commutative geometry and we show that the coupling between the space-time and an internal noncommutative space establishes a new speed of light constraint'. 3. Nicolas FrancoTowards a noncommutative version of GravitationAIP Conference Proceedings, vol. 1241, pp. 588-594 (2010). [abstract] [preprint] [journal] Abstract:Alain Connes' noncommutative theory led to an interesting model including both Standard Model of particle physics and Euclidean Gravity. Nevertheless, an hyperbolic version of the gravitational part would be necessary to make physical predictions, but it is still under research. We shall present the difficulties to generalize the model from Riemannian to Lorentzian Geometry and discuss key ideas and current attempts. 4. Nicolas FrancoSurvey of Gravity in Non-Commutative GeometryTopology, Quantum fields theory & Cosmology, Hermann, pp. 313-329 (2009). [abstract] [preprint] Abstract:We present a survey of the application of Cones' Non-Commutative Geometry to gravitation. Bases of the theory and Euclidian gravity models are reviewed. Then we discuss the problem of a Lorentzian generalization of the theory and review existing attempts of solution.
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