A theorem of Ramanan says that there is no universal bundle on any Zariski open subset of the moduli space of stable bundles over a curve. In this talk I will outline a proof of this fact due to Nori. The idea is to restrict a family to an analytic neighbourhood of the trivial bundle. Luna's etale slice theorem comes to the rescue and gives a very concrete description of this analytic neighbourhood which one uses to prove the theorem.

The most famous representative of public-key cryptography is the RSA cryptosystem. However, it is now losing its predominance: elliptic curve cryptography is more efficient, in terms of speed and key size.

Using curves of genus 2 may provide systems that are faster than their elliptic analogues. I will describe current efforts to find a "secure" curve of genus 2, and give an overview of the extensive computations involved.

Joint work with Pierrick Gaudry