## Final Activity Report Summary - GTEM (Galois Theory and Explicit Methods)

The traditional focus of pure mathematics is on deciding whether certain mathematical objects exist or not. For instance, one wants to know if given equations have solutions, or whether structures with given properties exist. In the past few decades however an increasing need has become apparent for methods that actually construct these mathematical objects if they exist. This trend has not meant a shift from pure to applied mathematics, but it is rather changing the nature of pure mathematics itself: algorithmic thinking and the language of complexity theory are entering into pure mathematics.

There are two forces driving this trend towards explicit methods in the area of number theory and arithmetic geometry. The first is the desire of mathematicians to understand the mathematical universe in a more thorough way than before, and to develop computer labs to allow computer experiments with mathematical objects of a greater variety than in traditional numerical computation. Secondly, the vastly increased use and complexity of electronic communication and networking has created needs in data security and coding theory that are often met by applications of unexpected branches of number theory and arithmetic geometry. For instance the subjects of lattice basis reduction and algorithms for elliptic curves over finite fields are having a profound impact on cryptology.

For four years the GTEM network provided a European research platform in explicit methods in Galois theory, number theory, and arithmetic geometry, including the training of 12 PhD students, and numerous workshops and summer schools. Research achievements include improved understanding and an improved algorithmic handle on Galois representations, point counting on curves over finite fields, and on the Arakelov class group in number theory.

There are two forces driving this trend towards explicit methods in the area of number theory and arithmetic geometry. The first is the desire of mathematicians to understand the mathematical universe in a more thorough way than before, and to develop computer labs to allow computer experiments with mathematical objects of a greater variety than in traditional numerical computation. Secondly, the vastly increased use and complexity of electronic communication and networking has created needs in data security and coding theory that are often met by applications of unexpected branches of number theory and arithmetic geometry. For instance the subjects of lattice basis reduction and algorithms for elliptic curves over finite fields are having a profound impact on cryptology.

For four years the GTEM network provided a European research platform in explicit methods in Galois theory, number theory, and arithmetic geometry, including the training of 12 PhD students, and numerous workshops and summer schools. Research achievements include improved understanding and an improved algorithmic handle on Galois representations, point counting on curves over finite fields, and on the Arakelov class group in number theory.