### School of Mathematics and Statistics

### Brian Conrad

Harvard University

### Deformation theory of Galois Representations for the non-expert

*Friday 24th July, 12-1pm, Carslaw 273.*
Our aim is to give some idea (to a non-number theorist) of what
Mazur's deformation theory of Galois representations is about and
how its applicability to the study of elliptic curves
has been extended by Wiles in his work on Fermat's Last Theorem,
and modularity of elliptic curves.

The Taniyama-Shimura Conjecture relates, in a very non-trivial way,
the algebraic theory of elliptic curves and the analytic theory of modular
forms. We will use Wiles' reformulation of this conjecture in terms of
deformation theory as a means of indicating what the deformation theory can
do.

Extending these methods to more elliptic curves has required some
generalizations of the deformation problems considered. We will
conclude with some remarks about this work.