A riemannian manifold (X,g) is a length metric space (X,d) by Hopf-Rinow thm.

• • (1935, A. Wald) problem 1 for 2-dim manifolds.
• • (1948, A.D. Alexandrov) a metric notion of (sectional) curvature + smoothness solves 2-dim manifolds.
• • (1982, A.D. Alexandrov) conjecture that the same is true for n-dim manifolds.
• • (1998, I.G. Nikolaev) proves (Alexandrov conjecture) for n-dim manifolds.

but (1996, M. Gromov) asks for a solution of

• • (X,D,g) sub-riemannian if D completely non-integrable distribution and g a metric on D.
• • by Hopf-Rinow (X,D,g) is a length metric space (X,d), with d the CC distance.

Sub-riemannian spaces are weird! (except when riemannian)

• • metric (Hausdorff) dimension > topological dimension
• • not Alexandrov spaces
• • tangent spaces are nilpotent (Carnot) groups, not vector spaces
• • Carnot groups have a peculiar differential calculus (Pansu derivative)

Sub-riemannian spaces (techniques) are useful!

• • (1981, M. Gromov) finitely generated groups of polynomial growth same as those groups which have nilpotent subgroups of finite index
• • (1989, P. Pansu) proves Rademacher thm for Carnot groups, which implies a short proof for Margulis-Mostow rigidity
• • (2006, J.R. Lee, A. Naor) counter-example to Goemans-Linial conjecture by using the Heisenberg group as a SR space
• • (2010, E. Hrushovski) (2011, E. Breuillard, B. Green, T. Tao) Approximate groups are essentially Carnot groups

Used in: A characterization of sub-riemannian spaces as length dilatation structures constructed via coherent projections a solution of the problem of intrinsic characterization of sub-riemannian manifolds posed by M. Gromov, 1996.

introduced as algebras in arXiv:0907.1520, as a λ calculus in arXiv:1807.02058.

groups with:


an action by automorphisms


a ⋅ (x * y) = (a ⋅ x) * (a ⋅ y)


which are uniformly contractive:


ε → 0, ε ⋅ x → e, uniformly wrt x in compact set

• • (normed) finite dim vector space X, * = +, Γ = (0, ∞)
• • Heisenberg groups:
  • - take (H, <,>) complex Hilbert space, X = H × R
  • (x,u) * (y,v) = (x + y, u + v + (1/2) Im <x,y>)
  • - distribution D from left translate of H in X, is completely nonintegrable
  • - metric on D from Re <,>
  • - Γ = C ∖ {0}
  • a ⋅ (x,u) = (a x , ∣a∣² u )
• • or just any Carnot group

The operation * is commutative iff any of the following:

• - we can do the shuffle trick:
• 
• - rays are semigroups, uniformly
• ∀ a, b ∈ Γ, ∃ a+b ∈ Γ, (a ⋅ x) * (b ⋅ x) = (a+b) ⋅ x
• - barycentric condition: ∀ a ∈ Γ, ∃ 1-a ∈ Γ

Let's denote the 3 ports of a dilation node as:

port 1: "from", port 2: "see", port 3: "as"



The operation * is commutative iff all the 6 permutations of ports are also dilations, with coefficients from the anharmonic group: Pure See!

(1936, A. Church) Untyped λ calculus is a term rewrite system

Terms:

• variable: x, y, z, ...
• term:
  • - variable  
  • - A B where A, B terms     (application)
  • - λx.A where x var, A term     (abstraction)

Term rewrite rule:

• β-reduction: (λx.D)B → D[x=B]

(1936, A. Church) Pure λ calculus is a term rewrite system

Term rewrite rule:

• β-reduction: (λx.D)B → D[x=B]

(1971, C.P. Wadsworth, 1990, J. Lamping) graph rewrite system

Aplication can be seen as:



(all the 6 permutations of ports are also dilations, with coefficients from the anharmonic group: Pure See!)

Abstraction can be seen as:



(all the 6 permutations of ports are also dilations, with coefficients from the anharmonic group: Pure See!)

THANK YOU!

More:

• Pure See
• Graph rewrites, from emergent algebras to chemlambda
• Chemlambda