Title: Pretentious behavior in number theory

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 a
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.