theory of distributions Laurent Schwartz quantum mechanics decidability of the first-order theory of the real numbers Alfred Tarski implement George Collins cylindrical algebraic decomposition real algebraic geometry polynomial equations polynomial equations or was proved algebra pure mathematics numbers analysis geometry real numbers computational complexity equal algebraic geometry prove
An example of the first case is the theory of distributions, introduced by Laurent Schwartz for validating computations done in quantum mechanics, which became immediately an important tool of (pure) mathematical analysis. An example of the second case is the decidability of the first-order theory of the real numbers, a problem of pure mathematics that was proved true by Alfred Tarski, with an algorithm that is impossible to implement because of a computational complexity that is much too high. For getting an algorithm that can be implemented and can solve systems of polynomial equations and inequalities, George Collins introduced the cylindrical algebraic decomposition that became a fundamental tool in real algebraic geometry.
