Inspired by the work of Iwasawa on growth of class groups in $\mathbb{Z}_p$-extensions, Mazur developed an analogous theory to study the growth of Selmer groups of Abelian varieties in such extensions. In my pre-talk, I will introduce Iwasawa theory and briefly explain the work of Iwasawa and Mazur. I will finally introduce the fine Selmer group whose systematic study was initiated by Coates-Sujatha in 2005. This is a subgroup of the Selmer group obtained by imposing stronger conditions at primes above $p$.

In the main talk, I will explain the growth of the fine Selmer group in towers of number fields and relate it to the growth of class groups in such towers. This will show the close relationship between the conjectures in classical Iwasawa theory and the Iwasawa theory of Abelian varieties. I will report on some modest progress made towards some of these conjectures. If time permits, I will also talk about Control Theorems of Fine Selmer Groups

For an introductory lecture on this topic, please see: https://youtu.be/CiwR-YcEetI

For Introductory slides on this topic, please see: http://www.fields.utoronto.ca/sites/default/files/uploads/pre-talk_0.pdf