Since the time of the ancient Greeks, mathematicians have been interested
in understanding solutions of equations with rational coefficients. In
modern times, it is customary to study the field of all solutions to all
equations at once; thanks to Galois, we now know that the absolute Galois
group of the rationals acts on this space, and encodes a wealth of
information about its structure. This is a very rich and complicated
group, so to
glimpse it's inner workings we are led to study its representations. The
1-dimensional case was the
crown jewel of nineteenth century algebraic number theory, and
there is a largely conjectural framework for classifying arbitrary
representations, initiated by Langlands in the 1970's,
which asserts a profound link between algebra, geometry, and analysis.

I will survey in a down-to-earth manner a specific instance of this
framework, namely Serre's conjecture on 2-dimensional
mod p Galois representations, which provides a precise dictionary for
translating between such representations and
holomorphic differential forms on certain algebraic curves. I will then
explain how this dictionary translates refined representation-theoretic
phenomena into congruence conditions between differential forms.

&ldquo The Sumner B. Myers
prize is awarded for the best Ph. D. thesis in mathematics each year. The award honors the memory of Professor Myers, a well known differential geometer who served in the department from 1936 until his untimely death in 1955. No award is made unless a thesis is judged to be truly distinguished.&rdquo^{
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