TOPOLOGICAL HARMONIC ANALYSIS: FOURIER TRANSFORM ON THE SPACE OF GEOMETRIES AND ITS PHYSICAL APPLICATIONS

We develop a rigorous framework for topological harmonic analysis—a generalisation of classical Fourier analysis to the space of Riemannian geometries. The central object is the topological Fourier transform, which maps a compact Riemannianmanifold (M, g) to its spectral density {an, bm} on the topological basis of noncommutative tori T 2θnand round spheres S2Rm.We prove the Plancherel formula for the topological Fourier series: the L2-norm of the scalar curvature equals the ℓ2-norm of the topological spectrum. We establish the topological convolution theorem: the convolution of two geometries correspondsto the pointwise product of their spectra, endowing the space of compact Riemannian manifolds with the structure of a commutative algebra.We prove that the topological basis simultaneously diagonalises all canonical geometric operators—the Hodge Laplacian, the Dirac operator, the Lichnerowicz Laplacian, the linearised Ricci flow, and the Einstein operator—and that their eigenvalues are givenexplicitly in terms of the topological parameters. We show that the Riccati operator, the fundamental dynamical operator of the unified theory, does not diagonalise: its quadratic term couples different topological modes, and this non-diagonalisability isthe geometric origin of physical interactions (generation mixing, graviton–photon con-version, vacuum tunnelling).We establish the exact relation between the topological Fourier decomposition and the spectral action of Connes: the spectral action is the generating functional for the topological amplitudes, and the Seeley–de Witt coefficients are the moments of thetopological spectrum.As physical applications, we develop spectral diagnostics of black holes: the quasi-normal mode spectrum determines the topological Fourier coefficients, allowing the reconstruction of the horizon geometry from gravitational wave observations. Weshow that the cosmic microwave background power spectrum is the topological spectral density of the primordial Universe. We propose a quantisation of geometry based on the discrete spectrum of topological amplitudes, providing a bridge between the spectral geometry of Connes and the canonical quantisation of general relativity.The framework is constructive, uses no free parameters, and reduces geometric questions to algebraic operations on the topological spectrum.