Characterization of algebraic varieties by their groups of symmetries

An old question steaming from Klein's Erlangen Program can be phrased in modern terms as: Is a given geometric object uniquely determined by its group of symmetries? The first part of this talk consists of an introduction to the problem with some selected examples from outside algebraic geometry.   In the second part of the talk, we come to the setting of algebraic geometry, where we show that, in general, the answer to the above question is negative. After restricting the class of varieties, we will show an instance where the answer is affirmative. Indeed, we show that complex affine toric surfaces are determined by the abstract group structure of their regular automorphism groups in the category of complex normal affine surfaces using properties of the Cremona group. We will also show that the normality assumption in the above result cannot be removed.