The heat equation was first proposed by FOURIER [VI.25] as a mathematical description of the transfer of heat in solid bodies. Its influence has subsequently been felt in many corners of mathematics: it provides explanations for such disparate phenomena as the formation of ice (the *Stefan problem),* the theory of incompressible viscous fluids (the NAVIERSTOKES EQUATION [III.23]), geometric flows (e.g., curve shortening, and the harmonic-map heat flow problem), BROWNIAN MOTION [IV.24], liquid filtration in porous media (the *Hele-Shaw problem*), index theorems (e.g., the *Gauss-Bonnet-Chern formula),* the price of stock options (the BLACK-SCHOLES FORMULA [VII.9 §2]), and the topology of three-dimensional manifolds ...

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