Adinkras and pure spinors

The nilpotence variety for extended supersymmetric quantum mechanics is a cone over a quadric in projective space. The pure spinor correspondence, which relates the description of off-shell supermultiplets to the classification of modules over the corresponding hypersurface ring, reduces to a classical problem of linear algebra. Spinor bundles, which correspond to maximal Cohen--Macaulay modules, serve as basic building blocks. Koszul duality appears as a deformed version of the Bernstein--Gel'fand--Gel'fand correspondence that we make fully concrete. We illustrate in numerous examples the close relationship between these connections and the powerful graphical technology of Adinkras, which appears as a decategorification of special complexes on quadrics. We emphasize the role of R-symmetry for recovering higher-dimensional gauge and gravity multiplets.