Recovering the 3D geometry of points, curves, surfaces from uncalibrated perspective views is a core problem in computer vision, lying at the intersection of projective differential geometry and algebraic geometry. Leveraging the highly optimized framework MINUS initiated at ICERM since 2019, modern methods from numerical algebraic geometry (NAG) comprising dynamical systems on complex varieties — namely fast homotopy continuation (HC) and numerical monodromy — have recently made globally convergent multivariate polynomial solving feasible for real-time applications. Because symbolic-oriented solvers have been largely shown to break down under the weight of nonlinear complexity, fast HC has become the mandatory engine for safety-critical vision tasks like autonomous driving, with many problems currently on the verge of real-time tractability. We argue that further unlocking extreme sequential computational speeds is possible, but requires a radical departure from generic local path-tracking heuristics. This talk describes a mathematical foundation suitable for fast homotopy continuation and 3D vision. We start with the modeling of continuous structure from motion involving jets of families of curves observed from video using tools from fiber bundles, connections, gauge theory, Riemannian metrics, and singularity-aware variants. We then build these differential-geometric tools up to the tracking of complex solutions to families of polynomials, using Hermitian bundles and metrics as guides to fast algorithms. Numerical path tracking in HC is framed as generalized ODE flows of sections defined by various connections in a bundle. While certain connections naturally recover the usual Davidenko flow, others minimize an action and define geodesics or general minimal paths. Optimizing the connection itself to minimize an action elevates optimal homotopy generation to a PDE problem. We show how the approach integrates the classical condition length of Shub into a geometrically coherent framework, and how Hermitian, Kähler and Hamiltonian structure relate to reducing sequential complexity bounds.