OUTLINE ABOUT A SHEAF APPROACH OF THE “ARROW OF TIME” AND CREATIVITY: FRACTIONAL OPERATORS, TOPOS AND GROTHENDIECK SCHEMES APPROACH.

The purpose of this note is to show why and how the fractional operators required for formalizing dynamic issues in complex environment, mobilize among the most advanced mathematical concepts : topos, site, spectrum, sheaf, ringed spaces, p-adic numbers, etc. In the context of physical exchanges in spaces crumpled by long range interaction (hyperbolic geometries), geodesics no longer respond to Noether invariance principles but to memory effects and to the categorical limits imposed by some requirements for completion. This completion is in fine based upon algebra-topology coupling. This coupling may be expressed through the zeta function which, as a bridge, ensures the formal closure of the representation and implicitly the space-time characteristics. The analysis points out the existence of a particular case corresponding to the Riemann hypothesis (and Martin’s axiom). Above approach involves a relationship between quantum mechanics and 2D self-similarity. In non integer cases, time is discretized and an arrow of time naturally emerges from the analysis. The note relates this arrow to anti entropic properties of dissipative complex systems. [ABSTRACT FROM AUTHOR]