Abstract differential geometry
Encyclopedia
The adjective abstract has often been applied to differential geometry before, but the abstract differential geometry (ADG) of this article is a form of differential geometry without the calculus
notion of smoothness, developed by Anastasios Mallios and others from 1998 onwards.
Instead of calculus, an axiomatic treatment of differential geometry is built via sheaf theory and sheaf cohomology
using vector sheaves in place of bundles
based on arbitrary topological space
s. Mallios says noncommutative geometry
can be considered a special case of ADG, and that ADG is similar to synthetic differential geometry
.
and propose this as a route to quantum gravity
.
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
notion of smoothness, developed by Anastasios Mallios and others from 1998 onwards.
Instead of calculus, an axiomatic treatment of differential geometry is built via sheaf theory and sheaf cohomology
Sheaf cohomology
In mathematics, sheaf cohomology is the aspect of sheaf theory, concerned with sheaves of abelian groups, that applies homological algebra to make possible effective calculation of the global sections of a sheaf F...
using vector sheaves in place of bundles
Fiber bundle
In mathematics, and particularly topology, a fiber bundle is intuitively a space which locally "looks" like a certain product space, but globally may have a different topological structure...
based on arbitrary topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
s. Mallios says noncommutative geometry
Noncommutative geometry
Noncommutative geometry is a branch of mathematics concerned with geometric approach to noncommutative algebras, and with construction of spaces which are locally presented by noncommutative algebras of functions...
can be considered a special case of ADG, and that ADG is similar to synthetic differential geometry
Synthetic differential geometry
In mathematics, synthetic differential geometry is a reformulation of differential geometry in the language of topos theory, in the context of an intuitionistic logic characterized by a rejection of the law of excluded middle. There are several insights that allow for such a reformulation...
.
ADG Gravity
Mallios and Raptis use ADG to avoid the singularities in general relativityGeneral relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
and propose this as a route to quantum gravity
Quantum gravity
Quantum gravity is the field of theoretical physics which attempts to develop scientific models that unify quantum mechanics with general relativity...
.
Further reading
- Space-time foam dense singularities and de Rham cohomology, A Mallios, EE Rosinger, Acta Applicandae Mathematicae, 2001