Lots of real world problems lead to group valued data...500 people ranking 5 types of chocolate chip cookies gives data on S(5). Daily birth rate data in New York City to data on C(365). There is even monster valued data- does the product replacement algorithm algorithm generate random looking elements. Fourier analysis on the groups calls for detailed knowledge of both characters and representations. It needs non-commutative analogs of the FFT. I will try to explain the statistics to people who know about groups (but not statistics). Again, there are many open problems