Arithmetic invariant theory and applications
The origins of "arithmetic invariant theory" come from the work of Gauss, who used integer binary quadratic forms to study ideal class groups of quadratic fields. The underlying philosophy---parametrizing arithmetic and geometric objects by orbits of group representations---has now been used to study higher degree number fields, curves, and higher-dimensional varieties. We will discuss some of these constructions and highlight the applications to topics such as bounding ranks of elliptic curves and dynamics on K3 surfaces.
This talk is intended for a general mathematical audience.