Abstract: I will explain a formula relating the minimal discrepency of Fano cone singularities and Reeb dynamics on the Sasaki link as well as Floer theoric invariants. Such a result can be used to obtain the uniqueness of Kähler compactification of Cn provided the added divisor is smooth. Time permitting, I will also explain a sharp upper bound of minimal discrepency of Fano cone singularities motivated from this formula as well as compactifications of affine varieties beyond Cn. This is based on joint works with Chi Li.