  By elaborating on the recent progress made in the area of Feynman integrals,
we apply the intersection theory for twisted de Rham cohomologies to simple
integrals involving orthogonal polynomials, matrix elements of operators in
Quantum Mechanics and Green's functions in Field Theory, showing that the
algebraic identities they obey are related to the decomposition of twisted
cocycles within cohomology groups, and which, therefore, can be derived by
means of intersection numbers. Our investigation suggests an algebraic approach
generically applicable to the study of higher-order moments of probability
distributions, where the dimension of the cohomology groups corresponds to the
number of independent moments; the intersection numbers for twisted cocycles
can be used to derive linear and quadratic relations among them. Our study
offers additional evidence of the intertwinement between physics, geometry, and
statistics.
