RESEARCH PROGRAM
Research Program
The Concept and Formalization of Constellatory Unfolding as a Cross-Cutting Principle
of an Autogenetic Universe
Albrecht von Müller and Elias Zafiris
The present project is part of our research program on "The Concept and Formalization of Constellatory Unfolding as a Cross-Cutting Principle of an Autogenetic Universe", and a further vindication of its far-reaching interpretational power.
It is planned to constitute the second book volume of our research program targeting the foundations of mathematical thinking. The first book volume targets the foundations of physics and has been published by Springer in 2018. The third book volume targets the foundations of ontophainetics and the role of primes in the present.
By consulting the original sources, based on the writings of the mathematical figures in focus that - as universally recognized - have enriched and developed the body of mathematical thinking in novel ways, we explicate in detail the semantic constellatory unfolding processes permeating these genuine advances in the following characteristic instances:
Riemann's geometric function theory on the complex domain;
Galois' theory of field extensions via permutation groups;
Grothendieck's unification of Riemann's and Galois theories in the topological domain of covering spaces and the fundamental group;
Poincare's uniformization strategy in complex surface geometry;
Cayley's geometrization of group theory with emphasis on the non-commutative free group on two generators;
Klein's uniformization strategy of Riemann surfaces based on potential theory and the free group;
Goedel's first incompleteness theorem in mathematical logic;
Cohen's method of forcing and the emergence of topoi as non-standard models of set theory;
Grassmann's extension theory in the context of lineal geometric analysis;
Leray's topological theory of sheaves and the compatiple passage from the local to the global;
Grothendieck's theory of sheaves as coefficient systems for cohomology;
de Rham's theory of differential forms and cohomology;
Weyl's program of localizing gauge symmetry to articulate matter;
Chern's theory of bundles with connection and the emergence of integrality;
Steenrod's theory of integrable connections and local systems;
Hilbert's program relating the monodromy action of the fundamental group with the theory of differential equations.
On the Foundations of Mathematical Thinking from the Perspective of Constellatory Unfolding
Program History and Forthcoming Work
(Published by Springer 2018)
On the Foundations of Physics
The first book volume of our project, published by Springer in 2018, targets the Foundations of Physics proposing and enunciating a fundamentally different categorial framework of conceptualizing time and reality, whose two most predominant characteristics are “constellatory unfolding” and “strong self-referentiality”. This is particularly relevant to the recent debate around the “ER=EPR” conjecture targeting the relation between Quantum Physics and General Relativity Theory. The novelty of this radically different framework is that it allows quantum reduction and singularities to be addressed as inverse transitions into and out of the factual layer of reality. In a nutshell, this book elaborates that there exists a universal mechanism of a topological nature operating beyond the factual layer, although linked with it, which is able to manifest these two inverse types of transition, based on the logical and topological characteristics of the Borromean Rings. The mathematical approach draws upon concepts and methods of algebraic topology, including the theory of covering spaces, sheaves, links, and free groups.
On the Foundations of Ontophainetics and the Role of Primes in the Present
The unveiling of the invariants, literally making them appear in light, called "epiphaneia'' in ancient Greek, in its philosophical and technical meaning as a spectrum of distinguishable appearances following the analysis of distinctions incident to the architectonics of adjoining and modular substitution, requires for its articulation a well-defined framework of ontophainetics, called "chroma'', pre-conditioning the process of spectral resolution and qualifying the "ana-phenomenon" in the time-space of the present. In particular, the process of constellatory unfolding of the chromatic invariants requires tempering within uniformly-partitioned chromatic intervals, underlying in turn the notion of an objective uniform probability distribution in relation to a directly inaccessible domain, like the quantum one. We will show that the progressive unveiling of the chromatic invariants in the time-space of the present - characterizing genuine novelty in this manner - is subordinate to the appropriate qualification of the "primitive roots of unity" in the complex arithmetic domain, and thus congruent with the novel features unveiled through the relative primes.
