1 On some possible mathematical connections concerning Noncommutative Minisuperspace Cosmology, Noncommutative Quantum Cosmology in low-energy String Action, Noncommutative Kantowsky-Sachs Quantum Model, Spectral Action Principle associated with a Noncommutative Space and some aspects concerning the Loop Quantum Gravity Michele Nardelli 2 , 1 1 Dipartimento di Scienze della Terra Università degli Studi di Napoli Federico II, Largo S. Marcellino, 10 80138 Napoli, Italy 2 Dipartimento di Matematica ed Applicazioni “R. Caccioppoli” Università degli Studi di Napoli “Federico II” – Polo delle Scienze e delle Tecnologie Monte S. Angelo, Via Cintia (Fuorigrotta), 80126 Napoli, Italy Abstract This paper is a review of some interesting results that has been obtained in various sectors of noncommutative cosmology, string theory and loop quantum gravity. In the Section 1, we have described some results concerning the noncommutative model of the closed Universe with the scalar field. In the Section 2, we have described some results concerning the low-energy string effective quantum cosmology. In the Section 3, we have showed some results regarding the noncommutative Kantowsky-Sachs quantum model. In Section 4, we have showed some results regarding the spectral action principle associated with a noncommutative space and applied to the Einstein-Yang-Mills system. Section 5 is a review of some results regarding some aspects of loop quantum gravity. In Section 6, we’ve described some results concerning the dynamics of vector mode perturbations including quantum corrections based on loop quantum gravity. In Section 7, we’ve described some equations concerning matrix models as a non-local hidden variables theories. In Section 8, we have showed some results concerning the quantum supergravity and the role of a “free” vacuum in loop quantum gravity. In Section 9, we’ve described various results concerning the unifying role of equivariant cohomology in the Topological Field Theories. In conclusion, in Section 10 we have showed the possible mathematical connections between the arguments above mentioned and some relationship with some equations concerning some sectors of Number Theory. 1. On some equations concerning the Noncommutative model of the closed Universe with the scalar field. [1] We remember that for any nD minisuperspace model the ordering parameter ξ in the conformally invariant Wheeler-De Witt (WDW) equation (Planck’s constant 1 =  ): ( ) () ( ) 0 2 1 = Ψ       + + ∆ − = Ψ A A q q U R q H ξ , (1.1)