Alain Connes explores the algebraic relations between the algebra of functions on a manifold and its infinitesimal length element \( ds \) in both commutative and non-commutative cases. In the commutative case, \( ds \) is the Dirac propagator, and its unitary representations correspond to Riemannian metrics and Spin structures. The spectral action, which is the trace of a function of the length element in Planck units, is shown to yield the Standard Model Lagrangian coupled to gravity when applied to the non-commutative geometry of the Standard Model. The internal fluctuations of the non-commutative geometry are trivial in the commutative case but yield the full bosonic sector of the SM with all correct quantum numbers in the slightly non-commutative case. The group of local gauge transformations appears spontaneously as a normal subgroup of the diffeomorphism group. Connes defines the axioms for commutative geometry and extends them to the non-commutative case, leading to a non-commutative geometry characterized by a triple \((\mathcal{A}, \mathcal{H}, D)\) with a real structure \( J \). He provides examples of non-commutative geometries, including finite-dimensional commutative algebras and the irrational rotation algebra, and discusses the internal diffeomorphisms and Morita equivalence in non-commutative geometry. The non-commutative geometry of the irrational rotation algebra is shown to depend on a complex number \(\tau\) and exhibits unique properties such as Cantor spectrum elements and Morita equivalence.Alain Connes explores the algebraic relations between the algebra of functions on a manifold and its infinitesimal length element \( ds \) in both commutative and non-commutative cases. In the commutative case, \( ds \) is the Dirac propagator, and its unitary representations correspond to Riemannian metrics and Spin structures. The spectral action, which is the trace of a function of the length element in Planck units, is shown to yield the Standard Model Lagrangian coupled to gravity when applied to the non-commutative geometry of the Standard Model. The internal fluctuations of the non-commutative geometry are trivial in the commutative case but yield the full bosonic sector of the SM with all correct quantum numbers in the slightly non-commutative case. The group of local gauge transformations appears spontaneously as a normal subgroup of the diffeomorphism group. Connes defines the axioms for commutative geometry and extends them to the non-commutative case, leading to a non-commutative geometry characterized by a triple \((\mathcal{A}, \mathcal{H}, D)\) with a real structure \( J \). He provides examples of non-commutative geometries, including finite-dimensional commutative algebras and the irrational rotation algebra, and discusses the internal diffeomorphisms and Morita equivalence in non-commutative geometry. The non-commutative geometry of the irrational rotation algebra is shown to depend on a complex number \(\tau\) and exhibits unique properties such as Cantor spectrum elements and Morita equivalence.