A few years ago, Granville and Soundararajan introduced the
notion of what they called "pretentious behavior" - they proved that
the magnitude of a character sum (i.e. a sum of chi(n) over 0 < n < x,
where chi is primitive Dirichlet character) is large for some x if and
only if the character chi 'pretends' to be a primitive character of
opposite parity and small conductor. This led to several breakthroughs
in the study of character sums, including the first unconditional
improvement of the Polya-Vinogradov inequality in almost 90 years.
Since then they have continued to explore pretentious behavior in
other contexts, perhaps the most notable application being to a key
step ('weak subconvexity') in Holowinsky and Soundararajan's recent
proof of the holomorphic analogue of the Quantum Unique Ergodicity
conjecture. I will discuss pretentious behavior in number theory and
describe some applications of the theory.