Schedule

The goal of this talk is twofold:

— to give practical information about the organization of the school;

— to present a brief summary of what is Noncommutative Geometry, in order for the participants to get an overview of the subjects of the lectures that will be delivered.

In this talk, we exhibit the non-formal star-exponential of the Lie group $SL(2,\mathbb{R})$ realized geometrically on the curvature contraction of its one-sheeted hyperboloid orbits endowed with its natural non-formal star-product. It is done by a direct resolution of the defining equation of the star-exponential and produces an expression with Bessel functions. This yields a continuous group homomorphism from $SL(2,\mathbb{R})$ into the von Neumann algebra of multipliers of the Hilbert algebra associated to this natural star-product. As an application, we prove a new identity on Bessel functions. This is the joint work with Axel de Goursac and Pierre Bieliavsky, which is still in progress.

We review some tentative steps towards a (horizontal and vertical) categorification of non-commutative geometry, starting for now only at the “topological level”, with the description of suitable notions of “strict higher $C^*$-categories”.

$C^*$-categories were introduced long ago by John E.Roberts and, despite the quite remarkable developments in higher category theory (that has recently evolved along lines closer to classical higher homotopy)… still no satisfactory notion of $n-C^*$-category (for $n>2$) has been produced and no marriage between “higher” categories and operator algebras has been in sight.

We propose a possible (and questionable) solution to the problem and also suggest some physical areas (relational quantum physics, quantum cybernetics), where higher $C^*$-categories might find natural application.