Theory of computing and more specifically computational complexity and the NP=!P problem represet an important new paradigm in mathematics. This offers new look on classical issues like the study of algorithms, optimization, and randomness, and provide an interesting lens for many areas/examples/results of math.
Among other things, computational complexity describes different novel notions of "proofs" and among them the possibility to prove a mathematical statement only "beyond a reasonable doubt."
As already mentioned in another answer, quantum information/computation and mathematics related to it represent a major new programme/paradigm.