Dynamic Spatio-Topological Metrology: Electromagnetic Inversion and Axion Electrodynamics

I develop a comprehensive theoretical framework unifying classical electromagnetic inversion, the method of image charges, and axion electrodynamics in topological materials. I first review classical image methods in planar, cylindrical, and spherical geometries of dielectric media, deriving image-charge solutions and boundary conditions. I then extend to topological magneto-electric materials: three-dimensional topological insulators exhibit an axion term (θ) in the Lagrangian, yielding novel “image” effects such as induced magnetic monopoles for nearby electric charges. I discuss quantum sensing applications – e.g. nitrogen-vacancy (NV) center diamond magnetometry – that can detect the weak fields (~10–100 mG) generated by these topological effects. Next, I examine inverse electromagnetic problems: advanced solvers (regularized Newton, gradient-based, Bayesian/PINN approaches) use multi-frequency data to recover subsurface material properties. I highlight the emerging use of topological data analysis (TDA) to preprocess measurement curves via their persistent homology. Finally, I introduce Dynamic Spatio-Topological Metrology (DSTM): a novel concept combining spatial scanning, temporal modulation, and topological invariants to enhance inversion robustness. I propose concrete DSTM schemes (e.g. homology-based feature extraction from frequency-dependent coil data) and derive analytic inversion formulas for coupled coils and layered media. Numerical examples illustrate my approach. This template sets a rigorous blueprint for a detailed scientific article, with proofs in appendices and extensive citations of recent quantum sensing and topological materials literature.