TITLE: Recent Applications of Mathematics
QUESTION [41 upvotes]: What are the recent and new applications of Mathematics in other Sciences ?
Let me try to be more precise about the question: 

By "recent" I mean the last 15 years. 
By "new" I want to exclude the standard answers like cryptography or finance
By "applications" I mean a mathematical concept (or even a trick) successfully used in another field (preferably not Theoretical Physics) to solve a problem or to shed a new light on a phenomenon. 
I would prefer to see recent applications of modern Mathematics, but new applications of classical results should be considered as a valid answer too. 
the answer should not simply be "XXX was successfully applied to YYY". it should contain a short explanation of the mathematical concept involve, and a description of the problem/phenomenon it solved/enlightened. 

The typical example I have in mind does not strictly answer the question (first it is an application of Theoretical Physics to condensed matter, and then I am not giving the required details): it seems that some 2d quantum field theories which had a priori no physical meaning were successfully used to understand properties of graphene (I think this was really unexpected: 2d conformal field theory was considered as a toy model to approach the understanding of more relevant field theories in higher dimension). 

v2: I hope this version is better than the previous one.

REPLY [2 votes]: Pseudodifferential (PDO) and Fourier integral operators (FIO) and also Wick and antiWick operators have applications in signal processing, especially in analysis of non stationnary signals like speech signals. Examples of such time frequency representation are spectrograms, scalograms which use wavelet theory and similar tools like Gaborets, chirplets, ridglets,...etc. This tools are also used to solve PDEquations as they serve to almost diagonalization of a large class of PDO and FIO.