A new characterization of characters and a converse to a theorem of Gauss

Dirichlet originally introduced characters in his famous proof of the infinitude of primes in arithmetic progressions. Since then they have become foundational throughout number theory (and also launched representation theory). In this talk I'll give an overview of the theory of Dirichlet characters---why they're important and what properties are known or conjectured. In particular, we'll discuss a recent result (joint with Jonathan Bober) that characters are essentially the only functions whose Fourier transform has magnitude 1 somewhere; a special case of this yields a converse to a famous theorem of Gauss on the magnitude of Gauss sums. The talk will be broadly accessible.