Boolean functions are of great interest in extremal combinatorics,
probability theory and complexity theory.
We will define and discuss the WalshFourier coefficients f^(S)
of a Boolean function f and what can we learn from them. Some
results and applications will be presented but I will emphasize open
problems.
Among the fact we mention:
 (Parseval) The sum of squares of the Fourier coefficients is the
probability that f=1.
 The sum of f^(S) S is the sensitivity
(also known as the sum of influences, edgeexpansion) of f.
 If the support of f is small then the WalshFourier
coefficients are supported on "large" coefficients (coefficients where
S is large).
 If f can be described by a bounded depth circuit of small size
then the WalshFourier coefficients are supported on "small" coefficients.
(Open problem: they are also supported on a few coefficients.)
 The connection of WalshFourier coefficients to stability of
f under noise,
applications to questions in probability.
Among the question we ask: Better description of FourierWalsh
coefficients of functions in AC0, "noise stability" for functions
expressable with threshold circuits and more.
The lecture will be selfcontained and elementary.
