  In the past decade, J. Huh solved several long-standing open problems on
log-concave sequences in combinatorics. The ground-breaking techniques
developed in those work are from algebraic geometry: "We believe that behind
any log-concave sequence that appears in nature there is such a Hodge structure
responsible for the log-concavity".
  A function is called completely monotone if its derivatives alternate in
signs; e.g., $e^{-t}$. A fundamental conjecture in mathematical physics and
Shannon information theory is on the complete monotonicity of Gaussian
distribution (GCMC), which states that $I(X+Z_t)$\footnote{The probability
density function of $X+Z_t$ is called "heat flow" in mathematical physics.} is
completely monotone in $t$, where $I$ is Fisher information, random variables
$X$ and $Z_t$ are independent and $Z_t\sim\mathcal{N}(0,t)$ is Gaussian.
  Inspired by the algebraic geometry method introduced by J. Huh, GCMC is
reformulated in the form of a log-convex sequence. In general, a completely
monotone function can admit a log-convex sequence and a log-convex sequence can
further induce a log-concave sequence. The new formulation may guide GCMC to
the marvelous temple of algebraic geometry. Moreover, to make GCMC more
accessible to researchers from both information theory and
mathematics\footnote{The author was not familiar with algebraic geometry. The
paper is also aimed at providing people outside information theory of necessary
background on the history of GCMC in theory and application.}, together with
some new findings, a thorough summary of the origin, the implication and
further study on GCMC is presented.
