I will talk about the following result:

For large enough k, every symmetric Boolean function {0,1}^k -> {0,1}
(with 4 exceptions) has a non-zero Fourier Coefficient of order o(k)
(and order at least 1). The four exceptions are the two PARITY
functions and the two constant functions. The o(k) term is ck/\lg{k},
for some constant c>0.

The result was motivated by an application in PAC learning: learning
of symmetric juntas, which are Boolean functions on n variables but
which depend only on an unknown set of k < n variables. This gives the
first n^{o(k)} algorithm for learning symmetric juntas, running in
time n^{k/\log{k}}.

(This result is joint work with Mihalis Kolountzakis, Richard Lipton,
Vangelis Markakis and Nisheeth Vishnoi)