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Consider the evolution of quantitative finance through the lens of evolution equations of the form $(D_t + A)u(t,x) = F(u)$ for time component $D_t$ and the space component $A$.  From Bachelier 1900 to Samuelson et. al. 1970s into 2000s we find that the probability density of the financial asset returns and volatilities being modeled with $F=0$, and the time derivative goes from the familiar $D_t = \frac{\partial}{\partial t}$ to various forms of Riemann-Liouville or Caputo or other fractional derivatives; the space derivative goes from $-\Delta$ the Laplacian to various fractional powers.  The evolution addresses non-Markovian and fat-tailed (compared to Gaussian of Brownian motion) and jumps as well for the Levy stable distributions all of which are improvements, but none of these models with $F=0$ can possibly explain turbulence in the global financial volatility, so a next step for quantitative finance as a science has to be seeking the behavior of $F$.