Computing with higher genus curves, abelian varieties and K3 surfaces

The Simons Foundation recently funded a project in computational number theory with the aim of developing algorithms for studying objects in arithmetic geometry beyond elliptic curves; I will be starting at MIT in February as part of this collaboration.  In this talk, I will give an introduction to higher genus curves, abelian varieties and K3 surfaces, and share some of the computational questions surrounding them that we hope to address.