Title: Endomorphism algebras of abelian varieties

Abstract: The main part of this talk is a survey of some (well-known) questions and methods of studying endomorphism algebras of abelian varieties. Then we discuss  a few explicit results over finite fields  joint in progress with Jiangwei Xue. We determine possible endomorphism algebras of abelian surfaces, and the number of the endomorphism rings of superspecial abelian surfaces.