Corpus ID: 14695358

Emergent algebras as generalizations of differentiable algebras, with applications

@inproceedings{Buliga2009EmergentAA,
  title={Emergent algebras as generalizations of differentiable algebras, with applications},
  author={Marius Buliga},
  year={2009}
}
We propose a generalization of differentiable algebras, where the underlying differential structure is replaced by a uniform idempotent right quasigroup (irq). Algebraically, irqs are related with racks and quandles, which appear in knot theory (the axioms of a irq correspond to the first two Reidemeister moves). An emergent algebra is a algebra A over the uniform irq X such that all operations and algebraic relations from A can be constructed or deduced from combinations of operations in the… Expand
Uniform refinements, topological derivative and a differentiation theorem in metric spaces
For the importance of differentiation theorems in metric spaces (starting with Pansu Rademacher type theorem in Carnot groups) and relations with rigidity of embeddings see the section 1.2 in CheegerExpand
Deformations of normed groupoids and differential calculus. First part
Differential calculus on metric spaces is contained in the algebraic study of normed groupoids with $\delta$-structures. Algebraic study of normed groups endowed with dilatation structures isExpand
Braided spaces with dilations and sub-riemannian symmetric spaces
Braided sets which are also spaces with dilations are presented and explored in this paper, in the general frame of emergent algebras arxiv:0907.1520. Examples of such spaces are the sub-riemannianExpand
Maps of metric spaces
This is a pedagogical introduction covering maps of metric spaces, Gromov-Hausdorff distance and its "physical" meaning, and dilation structures as a convenient simplification of an exhaustiveExpand
How to add space to computation? A tangle formalism for chora and difference
2 From maps to dilation structures 5 2.1 Accuracy, precision, resolution, Gromov-Hausdorff distance . . . 6 2.2 Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 ScaleExpand
Computing with space: a tangle formalism for chora and difference
TLDR
Here, inspired by Bateson, the point of view that there is no difference between the map and the territory, but instead the transformation of one into another can be understood by using a formalism of tangle diagrams is explored. Expand
What is a space? Computations in emergent algebras and the front end visual system
With the help of link diagrams with decorated crossings, I explain computations in emergent algebras, introduced in arXiv:0907.1520, as the kind of computations done in the front end visual system.
More than discrete or continuous: a bird's view
TLDR
I try to give mathematical evidence to the following equivalence, which is based on ideas from Plato (Timaeus): reality emerges from a more primitive, non-geometrical, reality in the same way as the brain construct the image of reality, starting from intensive properties. Expand
Normed groupoids with dilations
We study normed groupoids with dilations and their induced deformations. 1 Normed groupoids 2 1.1 Groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2Expand
Introduction to metric spaces with dilations
This paper gives a short introduction into the metric theory of spaces with dilations.

References

SHOWING 1-10 OF 32 REFERENCES
THE DIFFERENTIAL OF A QUASI-CONFORMAL MAPPING OF A CARNOT-CARATHEODORY SPACE
The theory of quasi-conformal mappings has been used to prove rigidity theorems on hyperbolic n space over the division algebras ℝ, ℂ, ℍ, and \({\Bbb O}\), by studying quasi-conformal mappings onExpand
Hardy spaces on homogeneous groups
The object of this monograph is to give an exposition of the real-variable theory of Hardy spaces (HP spaces). This theory has attracted considerable attention in recent years because it led to aExpand
RACKS AND LINKS IN CODIMENSION TWO
A rack, which is the algebraic distillation of two of the Reidemeister moves, is a set with a binary operation such that right multiplication is an automorphism. Any codimension two link has aExpand
Infinitesimal affine geometry of metric spaces endowed with a dilatation structure
We study algebraic and geometric properties of metric spaces endowed with dilatation structures, which are emergent during the passage through smaller and smaller scales. In the limit we obtain aExpand
Contractible groups and linear dilatation structures
A dilatation structure on a metric space, arXiv:math/0608536v4, is a notion in between a group and a differential structure, accounting for the approximate self-similarity of the metric space. TheExpand
Hypoelliptic second order differential equations
that is, if u must be a C ~ function in every open set where Pu is a C ~ function. Necessary and sufficient conditions for P to be hypoelliptic have been known for quite some time when theExpand
A characterization of sub-riemannian spaces as length dilatation structures constructed via coherent projections
We introduce length dilatation structures on metric spaces, tempered dilatation structures and coherent projections and explore the relations between these objects and the Radon-Nikodym property andExpand
Métriques de Carnot-Carthéodory et quasiisométries des espaces symétriques de rang un
We exhibit a rigidity property of the simple groups Sp(n, 1) and F7-20 which implies Mostow rigidity. This property does not extend to O(n, 1) and U(n, 1). The proof relies on quasiconformal theoryExpand
The tangent space in sub-riemannian geometry
Tangent spaces of a sub-Riemannian manifold are themselves sub-Riemannian manifolds. They can be defined as metric spaces, using Gromov’s definition of tangent spaces to a metric space, and they turnExpand
A classifying invariant of knots, the knot quandle
Abstract The two operations of conjugation in a group, x▷y=y -1 xy and x▷ -1 y=yxy -1 satisfy certain identities. A set with two operations satisfying these identities is called a quandle. TheExpand
...
1
2
3
4
...