The Langlands program predicts a deep connection between representation
theory and number theory. At its heart is the idea that spaces of
functions that are highly symmetric can be broken into "atomic" pieces,
and that number theory governs how this decomposition works. This idea is
a vast generalization of the decomposition of periodic functions into
frequencies using Fourier analysis.

In this talk, I will start by discussing how to organize representations
using linear algebra and harmonic analysis. I will then explain how
introducing number theory into the picture leads to the proof of Fermat's
Last Theorem, sheds light on the Ramanujan Conjecture, and culminates in
the Arthur multiplicity formula. This last conjecture completely describes
a spectral decomposition for representations of Lie groups defined over
the rational numbers. Along the way, I will illustrate how modern number
theory provides a unifying framework linking these seemingly different
problems and conclude by discussing my contributions towards the most
general refinement of the multiplicity formula.

No prior background in the Langlands program will be assumed.