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Equidistribution in Number Theory, An Introduction

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From July 11th to July 22nd, 2005, a NATO advanced study institute, as part of the series “S´eminaire de math´ematiques sup´erieures”, was held at the Universit ´e de Montr´eal, on the subject Equidistribution in the theory of numbers. There were about one hundred participants from sixteen countries around the world. This volume presents details of the lecture series that were given at the school.

Across the broad panorama of topics that constitute modern number theory one finds shifts of attention and focus as more is understood and better questions are formulated. Over the last decade or so we have noticed increasing interest being paid to distribution problems, whether of rational points, of zeros of zeta functions, of eigenvalues, etc. Although these problems have been motivated from very different perspectives, one finds that there is much in common, and presumably it is healthy to try to view such questions as part of a bigger subject. It is for this reason we decided to hold a school on “Equidistribution in number theory” to introduce junior researchers to these beautiful questions, and to determine whether different approaches can influence one another.

There are far more good problems than we had time for in our schedule. We thus decided to focus on topics that are clearly inter-related or do not require a lot of background to understand. Since there were two major number theory research programs taking place during the academic year 2005–2006, on Analysis in number theory at the Centre de recherches math´ematiques in Montr´eal, and during the spring semester 2006 on Rational and Integral Points on Higher-Dimensional Varieties at the Mathematical Sciences Research Institute in Berkeley, California, we decided to help prepare junior participants by inviting some lecturers who would go on to be senior participants at those programs.
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