Holonic Theory and Holistic Education

This paper presents the holonic theory, which is an attempt to develop in a single model the explanation of the evolution in the physic, biologic and cultural dimensions. The purpose of this development is to understand the traits of nowadays common holism, which is considered a necessary thinking practice in different domains, among them education. A phenomenological study has been developed connecting diverse noospheric holons. The results allow for a characterization of the holonic structure of a holistic consciousness act. This characterization is used to define holistic education: An education with the dimensions of preservation, profundity, projective action and span. The article can be read as a contribution to provide ontological and epistemological bases for going beyond the modern worldview and the widespread postmodern syncretism. These findings help in providing a framework for today’s holistic pedagogical debates and developments.

Two-cocycle twists and Atiyah–Patodi–Singer index theory

We construct η- and ρ-invariants for Dirac operators, on the universal covering of a closed manifold, that are invariant under the projective action associated to a 2-cocycle of the fundamental group. We prove an Atiyah–Patodi–Singer index theorem in this setting, as well as its higher generalisation. Applications concern the classification of positive scalar curvature metrics on closed spin manifolds. We also investigate the properties of these twisted invariants for the signature operator and the relation to the higher invariants.

TWISTED INDEX THEORY ON GOOD ORBIFOLDS, I: NONCOMMUTATIVE BLOCH THEORY

We study the twisted index theory of elliptic operators on orbifold covering spaces of compact good orbifolds, which are invariant under a projective action of the orbifold fundamental group. We apply these results to obtain qualitative results on real and complex hyperbolic spaces in two and four dimensions, related to generalizations of the Bethe–Sommerfeld conjecture and the Ten Martini Problem, on the spectrum of self adjoint elliptic operators which are invariant under a projective action of a discrete cocompact group.

Classical actions for forty-two asymptotic symmetry groups of four-dimensional space-times

In a recent paper, forty-two generalizations of the Bondi–Metzner–Sachs group B were abstractly defined, and examined in a quantum (Hilbert space) setting. However, B itself is often interpreted via its classical action B x I + -I + on Penrose’s future null infinity , J + and the question arises: do the generalizations have analagous classical actions? Here, it is shown that each generalization of B does indeed have such a classical action. In fact, two actions are given in each case, the first being a ‘lifting’ of the second ‘projective’ action. The actions are given in a uniform manner, and the spaces on which the groups act are interpreted not as ‘boundaries at infinity’, but rather as ‘blow-ups’ at the origin of parts of punctured flat spaces.