5th workshop on the analysis of PDEs, with a strong emphasis on the study of inverse problems and control theory, to be held at the Fédération des Unités de Mathématiques de l’Agglomération Marseillaise (Frumam) in Marseilles, France, on November 23-24, 2017.

co-organized by E. Soccorsi (CPT)

The so-called Ponzano-Regge model is an attempt to define a quantum theory of gravity. Using the formalism of tetrades, the partition function of such a theory can be written in terms of a SU(2) BF action then can be reduced to a sum over the representations of SU(2) which is thus ill-defined. This is regularized replacing the representations of SU(2) by a period of periodic representations of Uq(SU(2)). This regularization is ad hoc since until then no one has been able to deal properly with path integral. The result is then written under the form of Turaev-Viro topological invariant.

We will see that, in the case of abelian BF theory, however, thanks to Deligne-Beilinson cohomology, we can clearly identify the degrees of freedom of the theory and extract the relevant toplogical information. The regularization appearing is then natural and is related to an abelian version of Turaev-Viro invariant in a way analogous to the so-called Deloup-Turaev reciprocity formula that connects abelian Chern-Simons theory on the one hand and an abelian version of Reshetikhin-Turaev invariant on the other hand.