Hypercontractivity made easy, and sharp threshold theorems : Lecture 2

This lecture will further develop the randomization/symmetrization
technique in the setting of functions on general product probability
spaces. Our motivation will be the desire to prove "sharp threshold
theorems". These are the kinds of statements first proved by Friedgut
for several interesting graph/hypergraph properties: E.g., for a
randomly chosen k-CNF on n variables, there is a constant c such that
the CNF is almost surely satisfiable if the clause density is
(c/n)(1-o(1)) and almost surely unsatisfiable if the clause density is
(c/n)(1+o(1)).
The key analytical tools for proving such sharp threshold theorems
were developed by Friegut, Bourgain, and Hatami. I will show the
proof of Bourgain's version, and explain why it is basically just the
KKL Theorem with a "randomization/symmetrization twist". It took me
10 years to fully understand this theorem, but now I think it's easy!
If you make to the end of this lecture, I hope you will find it easy
too.