Applications of TCS to classical mathematics? We in TCS often use powerful results and ideas from classical mathematics (algebra, topology, analysis, geometry, etc.).  


  What are some examples of when it has gone the other way around?  


Here are some I know of (and also to give a flavor of the type of results I'm asking about):


Cubical foams (Guy Kindler, Ryan O'Donnell, Anup Rao, and Avi Wigderson: Spherical Cubes and Rounding in High Dimensions, FOCS 2008)
The Geometric Complexity Theory Program.  (Although this is technically an application of algebraic geometry and representation theory to TCS, they were led to introduce new quantum groups and new purely algebro-geometric and representation-theoretic ideas in their pursuit of P vs NP.)
Work on metric embeddings inspired by approximation algorithms and inapproximability results


I am in particular not looking for applications of TCS to logic (finite model theory, proof theory, etc.) unless they are particularly surprising -- the relationship between TCS and logic is too close and standard and historical for the purposes of this question.
Expanders were developed to a large extent in TCS and they have profound connections and applications to mathematics.