Algebraic geometry, in a naive setting, could be described as the study of spaces that locally are the solutions of systems of polynomial equations.
Similarly, locally any smooth manifolds can be described as the zero set of some smooth functions (If I recall well this is an exercise in chapter 3 of Pollack ). Hence differential topology could be described as the study of spaces that are locally the solutions of a smooth system of equations.
I wonder if we get a similar description for symplectic manifolds. It would be nice, for example, if the existence of a non-degenerate two-form added constraint to what the smooth functions that locally describe the manifold can be.
Does someone know? Thank you.