Non Commutative Geometry is what we need

Deep networks learn on spaces whose quotients are too singular for classical geometry. We are building the theory that treats them as noncommutative spaces — algebras first, points second.

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Fig. 1 — Kronecker foliation of 𝕋², slope θ = {{ thetaLabel }}

Fig. 2 — The leaf space: an orbit of x ↦ x + θ, dense in S¹

I — The idea

When the quotient breaks, keep the algebra.

Wind a line around a torus at an irrational slope and every leaf is dense. The space of leaves has no meaningful topology — every function on it is constant. Connes’ answer was to study the convolution algebra of the foliation instead: the noncommutative torus 𝒜_θ.

Learned representations exhibit the same pathology: symmetries, equivalences and invariances whose quotients are wild. We think the right language for them is spectral triples, cyclic cohomology and operator K-theory.

II — An atlas

The machinery, in motion

Five objects we lean on. Each figure is computed live — the arithmetic on screen is the arithmetic of the theory.

Fig. 3An étale groupoid becomes its convolution C*-algebra (f ∗ g)(γ) = Σ_{γ₁γ₂ = γ} f(γ₁) g(γ₂)

Fig. 4Modular flow Matrix units turn at the gaps of the modular Hamiltonian; the centraliser — the diagonal — stands still. σ_t^φ(x) = Δ^it x Δ^−it

Fig. 5Traces The trace of a projection in 𝒜_θ lands in ℤ + θℤ. Row by row, its points cut the circle into gaps of at most three lengths. τ_*(K₀(𝒜_θ)) = ℤ + θℤ

Fig. 6A 2-cocycle twist Walk a loop U, V, U*, V* on ℤ². The phase carried home is the enclosed area times θ. UV = e^2πiθ VU, ω ∈ Z²(ℤ², 𝕋)

Fig. 7A spectral triple The Dirac spectrum of a flat torus, counted as Λ grows. Its growth recovers dimension; its commutators, distance. d(p,q) = sup { |f(p) − f(q)| : ‖[D, f]‖ ≤ 1 }

III — Directions

Three lines of inquiry

Spectral geometry of learned representations

Recovering metric and dimension of a model’s latent space from a Dirac-type operator — geometry read off the spectrum, not the points.

ii.

Foliated training dynamics

Parameter symmetries foliate the loss landscape. We model the space of leaves as a groupoid C*-algebra and study optimisation on it.

iii.

K-theoretic invariants of networks

Index-theoretic quantities that persist under reparametrisation — candidate invariants for comparing what different models have learned.

IV — Notes

From the notebook

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V — Correspond

Write to us. We read everything.

Operator algebraists, geometers, and machine-learning researchers — we are looking for collaborators, critics and early readers.

research@noncommutativegeometry.com

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