The standard "big-bang theory" sets the origin of the Universe at an infinitesimal limit called the Planck scale (10 power - 33 cm). However, even if it is "very small", this limit is not zero. Therefore, the question is the following : does it exist "something" under the Planck scale? In other words, does it exist a "zero instant" in the remote history of the Universe?


 The objective of the Mathematical Center of Riemannian Cosmosology (MCRC) is to demonstrate that there exists a "zero point" corresponding to the origin. Such a zero point is not a physical object but has a pure mathematical content and meaning. From this view, one can say that this singular zero point, could be heuristically compared to a sort of "cosmological code" of the Universe.  MCRC currently works on the following domains :

















One of the main objectives of our work was to establish, in terms of quantum groups, the existence of a natural link between "q_deformation", quantisation of spacetime and "deformation" of the signature of the metric. We showed that in dimension D = 4, the Lorentzian and the Euclidean structures are related by twisting. The only natural signatures at the Planck scale are then the deformations of the Lorentzian (+ + + - ) and Euclidean (+ + + +) signatures. Our results then suggest that, in order to be compatible with noncommutative geometry, only the case of superposition of signatures (+ + + ±) should be considered at the scale of quantum gravity, whereas the ultra-hyperbolic case (+ + - - ) should be excluded. We show this from the viewpoints both of q-Lorentzian symmetry and q-spacetime. Our main mathematical result is then the construction of the new cocycle bicrossproduct   where H is a quantum group and c a 2-cocycle of "twisting" type. Such a construction has allowed us to construct the unification of the Lorentzian and Euclidean signatures within a unique structure of quantum group, of the form Uq(so(4)) //Uq(so(3, 1)). We also suggest that the Majid "semidualisation" describes the q-Euclidean to q-Lorentzian transition.   In the same way, we study the structures of the q-Poincaré group and relate it to the k -Poincaré group P k as well as to their different deformations by twisting. We discuss then the conjecture according to which it might exist a general link between deformation cocycle c , curvature (generally of the space of the phases of the system but here curvature of pre-spacetime) and anomalies of the theory. All this work is developed in the paper Construction of Cocycle Bicrossproducts by Twisting;






After having established the superposition of qSO(3,1) and qSO(4) within a unique quantum group structure, we have applied this situation within a physical context. The recent literature contains many descriptions regarding the physical state of the universe at the vicinity of the Planck scale. Non commutative geometry, string theory,   supergravity or quantum gravity have contributed, independently of each others, to establish on solid basis the data of a "transition phase" in the physical content and the geometric structures of the (pre)spacetime at such a scale. But what is the nature of this dramatic change?   We have proposed a novel approach, based on one of the most natural and realistic physical condition predicted by the Standard Model for the (pre)universe. In factn we have shown for the first time that   the spacetime -consideredc as a global system- should be expected   to reach a thermal equilibrium   at the Planck scale. Then as a direct consequence, the spacetime system   must be considered   as subject to the Kubo-Martin-Schwinger (KMS) condition. Therefore, in the interior of the KMS strip, i.e. from the scale ß = 0 to the scale ß = Planck , the fourth coordinate g 44   must be considered as complex, the two real poles being   ß = 0 and ß = Planck . As a result,   within the limits of the KMS strip, the Lorentzian and the Euclidean metric are in a "quantum superposition state" (or coupled), this entailing a "unification" (or coupling) between the topological (Euclidean) and the physical (Lorentzian) states of space-time. This KMS state of spacetime at the Planck scale has been developed in several of our papers, amongst which Thermal Equilibrium and KMS Condition at the Planck Scale





As we saw in the Quantum group part of our work, the bicrossproduct that we have set up suggests an unexpected kind of "unification" between the Lorentzian and the Euclidean Hopf algebras at the Planck scale and yields the possibility of a "q-deformation" of the signature from the Lorentzian (physical) mode   to the Euclidean (topological) mode. We also showed that this situation defines implicitly a (semi)duality transformation between Lorentzian and Euclidean quantum groups. Such a dualisation corresponds (as we saw page 3) to the KMS state of spacetime at the Planck scale. But let's considere now the 0 scale. What theory should then prevail at this scale? In our approach, we consider that the Euclidean situation corresponds to the simplest topological field theory. The topological limit of quantum field theory, described in particular by the Witten invariant   Z = Tr (-1) n is then given by the usual quantum statistical partition function taken over the (3+1) Minkowskian space-time   Z = Tr (-1) n   e - ßH, with ß = 1/kt   and n being the zero energy states number of the theory, for example the fermion number in supersymmetric theories. Then Z describes all zero energy states for null values of the Hamiltonian H.


Now, we propose in our approach a new topological limit   of quantum field theory, non-trivial (i.e. corresponding to the non-trivial minimum of the action). Built from scale ß = 0 and independent   of H, this unexpected topological limit (in 4D dimensions) is then given by the temperature limit (Hagedorn temperature) of the physical system (3+1)D.Indeed, when applied to quantum space-time, the KMS properties are such that the time-like direction of the system, within the limits of the "KMS strip" (i.e. between the zero scale and the Planck scale) should be considered as complex : +++±. In this case, on the ß = 0   limit, the theory is projected onto the pure imaginary boundary   of the KMS strip. Then the partition function gives the pure topological state connected with the zero mode of the scale : Z (ß = 0)   = Tr (-1) s, where s represents the instantonic number. This new "singularity invariant", isomorphic to the Witten index Z = Tr (-1) F , can be connected with the initial singularity of space-time, reached for ß = 0 in the partition function Z = Tr (-1) s e - ßH .Then the possible resolution of the initial singularity in the framework of topological theory allows us to envisage the existence, before the Planck scale, of a purely topological first phase of expansion of space-time, parameterized by the growth of the dimension of moduli space dim M   and described by the Euclidean "pseudo-dynamic" which corresponds to an evolution in imaginary time :  s ß (M0,1/Top) =   e - ß H   ( M0,1/Top)   e ß H   


This approach is developped in our paper: Toplogical Field Theory of the Initial Singularity of Spacetime



< Here are some of our preprints :


CERN Server :


Construction of Cocycle Bicrossproducts by Twisting ( CERN)


The KMS State of Space-Time at the Planck Scale ( CERN)


Topological Origin of Inertia ( CERN)


Topological Field Theory of the Initial Singularity of Space-Time ( CERN)


ArXiv :


Construction of Cocycle Bicrossproducts by Twisting (Arxiv)








Here are some links to our published papers and works in the field of theoretical physics and mathematics :


Topological Field Theory of the Initial Singularity of Spacetime (Classical&Quantum Gravity)


Kms State of Spacetime at the Planck Scale (Chinese Journal of Physics)


Topological Origin of Inertia (Czechoslovak Journal of Physics)


Spacetime Metric and the KMS Condition at the Planck Scale (Annals of Physics)


KMS Spacetime at the Planck Scale (Nuovo Cimento B)


Thermal Equilibrium and KMS Condition at the Planck Scale (Chinese Annals of Mathematics)




Our doctorate thesis can be found on the CNRS / CCSD server :


Fluctuations Quantiques de la MŽtrique ˆ l'Echelle de Planck (CNRS Server/ Thses de MathŽmatique)


Etat Topologique de l'Espace-Temps ˆ l'Echelle ZŽro (CNRS Server/ Thses de Physique ThŽorique et MathŽmatique)




Here is the link to HEP names with other links to papers and citations :




Here is our response to FabienBesnard :



Here are the Thesis Reports in PDF Reports-I.G.Bogdanoff.pdf 



MISCELANOUS DOCUMENTS'Affaire-B.htm Reports-Bogdanoff.htméponse à Fabien Besnard.pdf