  We consider the convex geometry of the cone of nonnegative quadratics over
Stanley-Reisner varieties. Stanley-Reisner varieties (which are unions of
coordinate planes) are amongst the simplest real projective varieties, so this
is potentially a starting point that can generalize to more complicated real
projective varieties. This subject has some suprising connections to algebraic
topology and category theory, which we exploit heavily in our work.
  These questions are also valuable in applied math, because they directly
translate to questions about positive semidefinite (PSD) matrices. In
particular, this relates to a long line of work concerning the extent to which
it is possible to approximately check that a matrix is PSD by checking that
some principle submatrices are PSD, or to check if a partial matrix can be
approximately completed to full PSD matrix.
  We systematize both these practical and theoretical questions using a
framework based on algebraic topology, category theory, and convex geometry. As
applications of this framework we are able to classify the extreme nonnegative
quadratics over many Stanley-Reisner varieties. We plan to follow these
structural results with a paper that is more focused on quantitative questions
about PSD matrix completion, which have applications in sparse semidefinite
programming.
