Arithmetic of plane quartic curves.
Bhargava, Gross and Wang have studied the group J(k)/2J(k), J the Jacobian of a hyperelliptic curve over a field k, using representation theory and invariant theory. In this talk, I will outline a similar program for smooth plane quartic curves (which are non-hyperelliptic of genus 3) with a marked rational point.
For further info please see the London Number Theory website.