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In the past decade, J. Huh solved several long-standing open problems onlog-concave sequences in combinatorics. The ground-breaking techniquesdeveloped in those work are from algebraic geometry: "We believe that behindany log-concave sequence that appears in nature there is such a Hodge structureresponsible for the log-concavity".  A function is called completely monotone if its derivatives alternate insigns; e.g., $e^{-t}$. A fundamental conjecture in mathematical physics andShannon information theory is on the complete monotonicity of Gaussiandistribution (GCMC), which states that $I(X+Z_t)$\footnote{The probabilitydensity function of $X+Z_t$ is called "heat flow" in mathematical physics.} iscompletely 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 isreformulated in the form of a log-convex sequence. In general, a completelymonotone function can admit a log-convex sequence and a log-convex sequence canfurther induce a log-concave sequence. The new formulation may guide GCMC tothe marvelous temple of algebraic geometry. Moreover, to make GCMC moreaccessible to researchers from both information theory andmathematics\footnote{The author was not familiar with algebraic geometry. Thepaper is also aimed at providing people outside information theory of necessarybackground on the history of GCMC in theory and application.}, together withsome new findings, a thorough summary of the origin, the implication andfurther study on GCMC is presented.

A Reformulation of Gaussian Completely Monotone Conjecture: A Hodge Structure on the Fisher Information along Heat Flow