The Hopf index theorem counts solutions of equations via a winding number. I’ll review this and give 8 equivalent formulations. Then I’ll describe 8 analogues for equations which form “isotropic sections” of “bundles with quadratic form”. There are applications to “cosection localised virtual cycles” and to “DT4 virtual cycles”, but mainly I’ll focus on more elementary topics. This is joint work with Martijn Kool, Jeongseok Oh and Jørgen Rennemo.
