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The workshop \emph{Explicit Methods in Number Theory\/} was organised by Henri Cohen (Talence), Hendrik W. Lenstra (Leiden), and Don B. Zagier (Bonn), with the assistance of Karim Belabas (Talence), and it took place July 15--21, 2007. Four previous workshops on the topic had been held in 1999, 2001, 2003, and 2005. The goal of the meeting was to present new methods and results on concrete aspects of number theory. In several cases, this included algorithmic and experimental work, but the emphasis was on the implications for number theory. There were three `mini-series' highlighting important recent developments: one of three hours, by Jordan Ellenberg and Akshay Venkatesh, on asymptotics for field extensions and class numbers; one of two hours by Mark Watkins, on random matrices and LLL-functions; and one of three hours by Noam Elkies, on the construction of elliptic curves of high Mordell--Weil rank. Some of the other themes were: \begin{itemize} \item Modular forms \item Rational and integral points on curves and higher-dimensional varieties \item Counting points on varieties over finite fields \item Fast multiplication \item Aspects of lattice basis reduction. \end{itemize} As always in Oberwolfach, the atmosphere was lively and active, providing an ideal environment for the exchange of ideas and productive discussions. The meeting was well-attended, with 51 participants from a variety of backgrounds, including a large number of younger researchers. There were 32 talks of various lengths, and ample time was allotted to informal collaboration.
