Hilbert's 10th Problem
Can a procedure be devised that will indicate if there are solutions to a Diophantine equation (an equation where whole-number solutions are sought)? This question on a famous list has now been answered
Nonstandard Analysis
This mathematical theory has restored infinitesimals to good standing. They had been employed since antiquity, but often with doubts, to solve such problems as finding a circle's area
Brownian Motion and Potential Theory
The discovery that these two apparently unrelated branches of physics are in some sense mathematically equivalent has led to a new subject known as probabilistic potential theory
Non-Cantorian Set Theory
In 1963 it was proved that a celebrated mathematical hypothesis put forward by Georg Cantor could not be proved. This profound development is explained by analogy with non-Euclidean geometry