RESEARCH GROUP

MATHEMATICAL STRUCTURES OF THE UNIVERSE

#### Selected publications

 Author:Title:Abstract:Journal:Full Record: ANDORAND NOT Author:Title:Abstract:Journal:Full Record: ANDORAND NOT Author:Title:Abstract:Journal:Full Record: from to   type ALL researchconferencereviewsbookspopular res per page (134 res)
 7. Sebastian J. SzybkaRównania Einsteina i efekt niejednorodno¶ci w kosmologiiW ,,Ogólna teoria wzglêdno¶ci a filozofia - sto lat interakcji'', red. P. Polak, J. M±czka, CCPress, pp. 127-142 [journal] Abstract: 8. Nicolas Franco, Micha³ EcksteinCausality in noncommutative two-sheeted space-times, vol. xxx, pp. xxx-xxx (2015). [abstract] [preprint] Abstract:We investigate the causal structure of two-sheeted space-times using the tools of Lorentzian spectral triples. We show that the noncommutative geometry of these spaces allows for causal relations between the two sheets. The computation is given in details when the sheet is a 2- or 4-dimensional globally hyperbolic spin manifold. The conclusions are then generalised to a point-dependent distance between the two sheets resulting from the fluctuations of the Dirac operator. 9. 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. 10. Edited by James Ladyman, Stuart Presnell, Gordon McCabe, Micha³ Eckstein, Sebastian J. SzybkaRoad to Reality with Roger PenroseCCPress [abstract] [preprint] [journal] Abstract:Where does the road to reality lie? This fundamental question is addressed in this collection of essays by physicists and philosophers, inspired by the original ideas of Sir Roger Penrose. The topics range from black holes and quantum information to the very nature of mathematical cognition itself. 11. A. Woszczyna, W. Czaja, K. G³ód, Z. A. Golda, R. A. Kycia, A. Odrzywo³ek, P. Plaszczyk, L. M. Soko³owski, S. J. Szybkaccgrg: The symbolic tensor analysis package with tools for general relativityWolfram Library Archive, vol. 8848 (2014). [abstract] [journal] Abstract:Riemann and Weyl curvature, covariant derivative, Lie derivative, the first and the second fundamental form on hyper-surfaces, as well as basic notions of relativistic hydrodynamics (expansion, vorticity, shear) are predefined functions of the package. New tensors are easy to define. Instructions, basic examples, and some more advanced examples are attached to the package. Characteristic feature of the ccgrg package is the specific coupling between the functional programming and the Parker-Christensen index convention. This causes that no particular tools to rising/lowering tensor indices neither to the tensor contractions are needed. Tensor formulas are written in the form close to that of classical textbooks in GRG, with the only difference that the summation symbol appears explicitly. Tensors are functions, not matrixes, and their components are evaluated lazily. This means that only these components which are indispensable to realize the final task are computed. The memoization technique prevents repetitive evaluation of the same quantities. This saves both, time and memory. 12. Nicolas FrancoTemporal Lorentzian Spectral TriplesRev. Math. Phys., vol. 26, 8, p. 14300076 (2014). [abstract] [preprint] [journal] Abstract:We present the notion of temporal Lorentzian spectral triple which is an extension of the notion of pseudo-Riemannian spectral triple with a way to ensure that the signature of the metric is Lorentzian. A temporal Lorentzian spectral triple corresponds to a specific 3+1 decomposition of a possibly noncommutative Lorentzian space. This structure introduces a notion of global time in noncommutative geometry. As an example, we construct a temporal Lorentzian spectral triple over a Moyal--Minkowski spacetime. We show that, when time is commutative, the algebra can be extended to unbounded elements. Using such an extension, it is possible to define a Lorentzian distance formula between pure states with a well-defined noncommutative formulation.
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