During the last decade it became clear that methods of p-adic Analytic Geometry
play more and more important role in Representation Theory, including Langlands’ program.
These methods produce very powerful new tools to construct and study the representations,
and I have a filling that we have to learn these tools to continue to work in Representation Theory.
The p-adic analytic theory by now is very rich and highly developed theory.
In this course I will try to describe basic notions and results of this theory. In my exposition I will
try to emphasize how this theory is related to the Representation Theory, but I am not sure
that I will have time to describe this relation in some details.
There are two main directions in the development of this theory.
1. Rigid Analytic Geometry over p-adic numbers.
This is a p-adic analogue of the theory of complex manifolds.
2. p-adic Hodge Theory.
This is an analogue of complex Hodge theory.
Of course these two directions are highly intertwined.
In my course I will mostly discuss the Hodge theory. It is closely related to the theory of representations of Galois groups.
I think that proper understanding of this relation gives a new insight into the standard theory of representations of p-adic groups.
In my lectures I will try to formulate most of concepts and results that I need in the lectures.
However, since the material of the course is rather advanced, some preliminary knowledge of many of these topics will be very helpful.