I will discuss a strong unique continuation theorem for zero order perturbations of the fractional heat equation based on a monotonicity property of a generalized Almgren frequency in weighted Gauss space. This is a joint work with Nicola Garofalo. If time permits, I will also talk on how such a truncated variant of the Almgren type
monotonicity formula combined with new Weiss and Monneau type monotonicity formulas allows to study the structure of the singular set on the free boundary in the obstacle problem for the fractional heat operator. This part of the talk ( if at all I get time !) is joint with Donatella Danielli, Nicola Garofalo and Arshak Petrosyan.