I follow with a lot of interest the stream of posts by Terence Tao on the Hilbert’s fifth problem and I am waiting impatiently to see how it connects with the field of approximate groups.
In his latest post Tao writes that
… Hilbert’s fifth problem is a manifestation of the “rigidity” of algebraic structure (in this case, group structure), which turns weak regularity (continuity) into strong regularity (smoothness).
This is something amazing and worthy of exploration!
I propose the following “explanation” of this phenomenon, taking the form of the:
Here are more explanations (adapted from the first paper on emergent algebras):
A differentiable algebra, is an algebra (set of operations A) over a manifold X with the property that all the operations of the algebra are differentiable with respect to the manifold structure of X. Let us denote by D the differential structure of the manifold X.
From a more computational viewpoint, we may think about the calculus which can be
done in a differentiable algebra as being generated by the elements of a toolbox with two compartments “A” and “D”:
– “A” contains the algebraic information, that is the operations of the algebra, as
well as algebraic relations (like for example ”the operation ∗ is associative”, or ”the operation ∗ is commutative”, and so on),
– “D” contains the differential structure informations, that is the information needed in order to formulate the statement ”the function f is differentiable”.
The compartments “A” and “D” are compatible, in the sense that any operation from “A” is differentiable according to “D”.
I 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 uniform irq is a family of irqs indexed by elements of a commutative group (with an absolute), such that the third Reidemeister move is related to a statement in terms of uniform limits of composites of operations of the family of irqs.
An emergent algebra is an 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 uniform irq, possibly by taking limits which are uniform with respect to a set of parameters. In this approach, the usual compatibility condition between algebraic information and differential information, expressed as the differentiability of algebraic operations with respect to the differential structure, is replaced by the “emergence” of algebraic operations and relations from the minimal structure of a uniform irq.
Thus, for example, algebraic operations and the differentiation operation (taking the triple (x,y,f) to Df(x)y , where “x, y” are points and “f” is a function) are expressed as uniform limits of composites of more elementary operations. The algebraic operations appear to be differentiable because of algebraic abstract nonsense (obtained by exploitation of the Reidemeister moves) and because of the uniformity assumptions which allow us to freely permute limits with respect to parameters in the commutative group (as they tend to the absolute), due to the uniformity assumptions.