Title: The Effective Shafarevich Conjecture
 
Abstract:  Let $K$ be a number field, $d$ a positive integer, and $S$ a finite set of primes of $K$. One of the crowning achievements of 20th century arithmetic geometry was Faltings's proof that there are only finitely many isomorphism classes of dimension $d$ abelian varieties $A/K$ with good reduction away from $S$. Whilst several effective algorithms have been developed to explicitly classify elliptic curves with good reduction outside a finite set of primes, effectively solving this problem in higher dimensions remains a challenge.  In this talk, I will give a brief survey on some known methods for classifying abelian varieties, and will present some work in progress on classifying isogeny classes of abelian surfaces over $\mathbb{Q}$ with good reduction away from 2.