This is an introductory talk related to algebraic analysis. We will explain how Cauchy problem of certain differential equations such as Schrodinger equation leads to interesting symplectic topology of Lagrangian or Legendrian submanifolds. The evolution of those Lagrangian or Legendrian submanifolds can be regarded as a classical geometry counterpart of the evolution of the solution of the equations. We will also emphasize the use of homological algebra in the sense that it provides an alternative way to "complete the metric space".