Title: Points of small height in infinite extensions

Abstract: Height functions play a crucial role in Diophantine geometry, as they provide a partial order on arithmetic objects. A classical theme in Diophantine geometry is to study points of small height. A subset S of algebraic numbers is said to have the Bogomolov property if it does not contain points of arbitrarily small Weil height. In this talk, we formulate a criterion for the Bogomolov property, particularly for infinite non-Galois extensions, and also provide a non-archimedean perspective on the equidistribution of roots. If time permits, we will also discuss the analogue of Bogomolov property for elliptic curves over these extensions.