Hamilton's Ricci flow, used in the proof of the Poincare conjecture, is sometimes known as "[this equation] for metrics.
In geometry, this equation can be used to give a local formula for the index of an elliptic complex; a paper of Atiyah, Bott, and Patodi took advantage of this fact to provide a new proof of the index theorem. Hamilton's Ricci flow, used in the proof of the Poincare conjecture, is sometimes known as "[this equation] for metrics." One numerical method for solving this partial differential equation discretizes space and uses a (*) finite difference method to approximate partial derivatives; that is the Crank-Nicolson algorithm. Fourier series were introduced to solve the one-dimensional version of this equation, which is a prototypical parabolic PDE. Stating that the time partial derivative of the function under consideration is proportional to the Laplacian, for 10 points, name this equation which describes thermal diffusion.
In geometry, this equation can be used to give a local formula for the index of an elliptic complex; a paper of Atiyah, Bott, and Patodi took advantage of this fact to provide a new proof of the index theorem.
heat equation