The concept of a group is central to essentially all of modern mathematics.
In Number theory and geometry, where groups take central stage
in various shapes such as symmetry groups, Galois groups, fundamental groups,
reflection groups and permutation groups, the conceptual unification
that it provides is most strikingly illustrated.
The School will help the students acquiring a good background on the Langlands program, which, after all, is about
relations between symmetries in geometry, analysis and number theory.
In this school, we present groups and the natural objects they act on
in a variety of arithmetic and geometric contexts.
Special emphasis will be given to concrete examples, and
practical and computational aspects of groups and their actions
will be stressed.
The topics to be treated include finite fields, coding theory, covering spaces,
representation theory, modular forms and Galois theory.