Alain Connes presents a framework for non-commutative geometry that unifies gravity and the Standard Model. He begins by establishing algebraic relations between the algebra of functions on a manifold and its infinitesimal length element, ds. In the commutative case, ds is the Dirac propagator, D⁻¹, where D is the Dirac operator. These relations are extended to the non-commutative case using Tomita's involution J. A spectral action, defined as the trace of a function of ds in Planck units, is introduced. When applied to the non-commutative geometry of the Standard Model, this action yields the Standard Model Lagrangian coupled to gravity. In the commutative case, internal fluctuations of the geometry are trivial, but in the slightly non-commutative case, they yield the full bosonic sector of the Standard Model with correct quantum numbers. The group of local gauge transformations appears spontaneously as a normal subgroup of the diffeomorphism group. Riemann's geometric space is based on a manifold M with coordinates x^μ. The metric is given by ds² = g_{μν}dx^μdx^ν, allowing distance measurement. Connes builds a dual notion of geometry on the pair (A, ds), where A is the algebra of coordinates and ds is the infinitesimal length element. For the commutative case, A is the algebra of smooth functions on M, and ds is represented by the Dirac operator D. The unitary representation of (A, ds) defines a geometry, with the distance between points measured by the supremum of |f(x) - f(y)| over functions f with bounded commutators with D⁻¹. The spectral action is defined as the trace of a function of ds in Planck units. For a suitable algebra A, this action gives Einstein gravity coupled with the Standard Model Lagrangian. The algebra is a non-commutative refinement of the smooth functions on a compact 4-manifold, related to the quantum group refinement of the Spin covering of SO(4). The group of gauge transformations arises as a normal subgroup of the diffeomorphism group. Connes introduces axioms for commutative geometry, including smoothness, orientability, finiteness, and reality. These axioms are extended to non-commutative geometry, incorporating Tomita's involution and Hochschild cycles. The non-commutative geometry is defined by a triple (A, H, D), with real structure J. The axioms ensure that the geometry is well-defined and satisfies Poincaré duality. Examples of non-commutative geometries are provided, including finite-dimensional algebras and matrix-valued functions on manifolds. The non-commutative geometry of the irrational rotation of the circle is discussed, showing that the algebra contains non-trivial idempotents and smooth real elements with CantorAlain Connes presents a framework for non-commutative geometry that unifies gravity and the Standard Model. He begins by establishing algebraic relations between the algebra of functions on a manifold and its infinitesimal length element, ds. In the commutative case, ds is the Dirac propagator, D⁻¹, where D is the Dirac operator. These relations are extended to the non-commutative case using Tomita's involution J. A spectral action, defined as the trace of a function of ds in Planck units, is introduced. When applied to the non-commutative geometry of the Standard Model, this action yields the Standard Model Lagrangian coupled to gravity. In the commutative case, internal fluctuations of the geometry are trivial, but in the slightly non-commutative case, they yield the full bosonic sector of the Standard Model with correct quantum numbers. The group of local gauge transformations appears spontaneously as a normal subgroup of the diffeomorphism group. Riemann's geometric space is based on a manifold M with coordinates x^μ. The metric is given by ds² = g_{μν}dx^μdx^ν, allowing distance measurement. Connes builds a dual notion of geometry on the pair (A, ds), where A is the algebra of coordinates and ds is the infinitesimal length element. For the commutative case, A is the algebra of smooth functions on M, and ds is represented by the Dirac operator D. The unitary representation of (A, ds) defines a geometry, with the distance between points measured by the supremum of |f(x) - f(y)| over functions f with bounded commutators with D⁻¹. The spectral action is defined as the trace of a function of ds in Planck units. For a suitable algebra A, this action gives Einstein gravity coupled with the Standard Model Lagrangian. The algebra is a non-commutative refinement of the smooth functions on a compact 4-manifold, related to the quantum group refinement of the Spin covering of SO(4). The group of gauge transformations arises as a normal subgroup of the diffeomorphism group. Connes introduces axioms for commutative geometry, including smoothness, orientability, finiteness, and reality. These axioms are extended to non-commutative geometry, incorporating Tomita's involution and Hochschild cycles. The non-commutative geometry is defined by a triple (A, H, D), with real structure J. The axioms ensure that the geometry is well-defined and satisfies Poincaré duality. Examples of non-commutative geometries are provided, including finite-dimensional algebras and matrix-valued functions on manifolds. The non-commutative geometry of the irrational rotation of the circle is discussed, showing that the algebra contains non-trivial idempotents and smooth real elements with Cantor