The human brain connectome is an ambitious project to provide a complete map of neural connectivity and a recent source of excitement in the neuroscience community. Just as the human genome is a triumph of marrying technology (high throughput sequencers) with theory (dynamic programming for sequence alignment), the human connectome is a result of a similar union. The technology in question is that of diffusion magnetic resonance imaging (dMRI) while the requisite theory, we shall argue, comes from three areas: PDE, harmonic analysis, and convex algebraic geometry. The underlying mathematical model in dMRI is the Bloch-Torrey PDE but we will approach the 3-dimensional imaging problem directly. The main problems are (i) to reconstruct a homogeneous polynomial representing a real-valued function on a sphere from dMRI data; and (ii) to analyze the homogeneous polynomial via a decomosition into a sum of powers of linear forms. We will focus on the nonlinear approximation associated with (ii) and discuss a technique that combines (i) and (ii) for mapping neural fibers. This is joint work with T. Schultz of MPI Tubingen.