  The purpose of this article is to expose and further develop a simple yet
surprisingly far-reaching framework for generating monotone quantities for
positive solutions to linear heat equations in euclidean space. This framework
is intimately connected to the existence of a rich variety of algebraic closure
properties of families of sub/super-solutions, and more generally solutions of
systems of differential inequalities capturing log-convexity properties such as
the Li--Yau gradient estimate. Various applications are discussed, including
connections with the general Brascamp--Lieb inequality and the
Ornstein--Uhlenbeck semigroup.
