January - June 2010

**Organisers:** Michel Balazard *(CNRS, Laboratoire Poncelet)*, Xavier Caruso *(CNRS, Laboratoire Poncelet)*, Michael Tsfasman *(CNRS, Laboratoire Poncelet, Institute for Information Transmission Problems)*, Alexey Zykin *(Laboratoire Poncelet, State University Higher School of Economics)*

Number theory has long been a central area of research in modern mathematics deeply connected to almost all of its branches. There are many different approaches to the field, utilizing a wide variety of techniques: complex and real analysis, algebraic and arithmetic geometry, group theory, representation theory, combinatorics, the theory of differential equations and dynamical systems, topology and differential geometry all play important roles.

Quite often number theoretic approaches establish unexpected connections between different mathematical domains. It is not unusual that such connections allow solving difficult problems in these domains. This multiplicity of techniques and interconnections creates a difficulty in grasping the overall picture of number theory. The fact that new methods and approaches tend to appear with extreme rapidity also contributes to this problem. One of the goals of the semester is to convey the essential unity of the subject, especially to young mathematicians who use number theory in their research.

Mathematical activities during the semester include a weekly number theory seminar, a course by Xavier Caruso and two conferences. The first conference will be devoted to algorithmic problems in arithmetic and the second one to zeta functions. These areas are currently undergoing very active development and constitute a large part of research in number theory.

The number theory semester benefited from the support of the EADS Foundation Chair, the Russian Foundation for Basic Research, Centre National de la Recherche Scientifique. It is organized within the framework of the year of French-Russian cooperation.