In [1], Connes presented axioms governing noncommutative geometry. He went on to claim that when specialised to the commutative case, these axioms recover spin or spinc geometry depending on whether the geometry is “real” or not. We attempt to flesh out the details of Connes’ ideas. As an illustration we present a proof of his claim, partly extending the validity of the result to pseudo-Riemann...

We formulate two new criteria of amenability of discrete equivalence relations: in terms of asymptotically invariant families of leafwise probability measures and in terms of isoperimetric properties of leafwise graph structures. These criteria lead to a geometric proof of the Connes{Feldman{Weiss theorem on coincidence of amenability and hyperrniteness for equivalence relations. Nous consid er...

Almost forty years ago, Connes, Feldman and Weiss proved that for measurable equivalence relations the notions of amenability hyperfiniteness coincide. In this paper we define uniform version graphed bounded vertex degrees prove these two coincide as well. Roughly speaking, a measured graph $\mathcal {G}$ is uniformly hyperfinite if any ${\varepsilon }>0$ there exists $K\geq 1$ such not only {G...

An ensemble of 2 × 2 pseudo-Hermitian random matrices is constructed that possesses real eigenvalues with level-spacing distribution exactly as for the Gaussian unitary ensemble found by Wigner. By a re-interpretation of Connes’ spectral interpretation of the zeros of Riemann zeta function, we propose to enlarge the scope of search of the Hamiltonian connected with the celebrated Riemann Hypoth...

Let X be a manifold on which a discrete (pseudo)group of diffeomorphisms Γ acts, and let E be a Γ-equivariant vector bundle on X. We give a construction of cyclic cocycles on the cross product algebra C∞ 0 (X)o Γ representing the equivariant characteristic classes of E. Our formulas can be viewed as a higher-dimensional analogue of Connes’ Godbillon-Vey cyclic cocycle. An essential tool for our...