** **

**Abstract:**

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.