Hyperelliptic curves are a fundamental class of polynomial equations, that are rapidly becoming a major topic in number theory. Demands for the theory are coming both from within pure mathematics (such as the pioneering work or Bhargava and his collaborators, and the Langlands programme, which needs a supply of high-dimensional Galois representations), as well as from areas bordering to theoretical physics (via hypergeometric motives) and from cryptography, where one of the main methods for modern data encryption is based on hyperelliptic curves. 

The advanced school will focus on the modern techniques in number theory that have previously had major impacts on the theory of elliptic curves and that are likely to have great impact in the theory of hyperelliptic curves in the near future. Specifically, the topics to be covered are the following, with a particular emphasis on the setting of hyperelliptic curves: 

1. L-functions and the Birch-Swinnerton-Dyer Conjecture; 
2. Selmer groups; 
3. Modularity; 
4. Galois representations.