A new way to solve linear programming problems that promises to be faster than the simplex method for large problems. An ordinal number is just a well-ordered set. Finite ordinals are the familiar whole numbers. But all kinds of infinities are easy to define and do all the standard arithmetic with. A generalization of complex numbers.
Multiplication is noncommutative.
Quaternions can be used to represent rotations in 3D. Lebesgue integral. This is the definition of integral mathemeticians really use. Recently claimed to be solved in 3 dimensions, but still unknown in higher dimensions. Mathematics of Elections. See the book Chaotic Elections by Donald Saari.