**Lattices play a central role in number theory and its applications. The aim of this school is to introduce participants to the ubiquity of lattices in number theory, algebra, arithmetic algebraic geometry, cryptography and coding theory. The theory of lattices will be developed from its very beginning and the basic notions required for the applications in number theory, algebra, arithmetic algebraic geometry will be provided. Appearances of lattices that we intend to cover include: The natural lattice structure of Mordell-Weil groups and unit groups. Lie algebra root lattices. The lattice basis reduction algorithm "LLL", which has many applications to many areas of mathematics and finally the construction of the famous Leech lattice. On the applied side we plan to cover constructions of good error-correcting codes and of good sphere packings via dense lattices.**

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