First, we will consider the problem of persistence of stationary real valued Gaussian processes and estimate the probability that it does not cross zero in a long interval. We will see that the behaviour of this probability depends on the nature of the spectral measure near origin. Next, we will define a generalized notion of the Restricted Isometry Property (RIP) in sparse recovery and estimate the number of rows needed for a random matrix with symmetric alpha stable entries to satisfy the same with high probability. The result highlights certain limitations of the RIP framework. Time permitting, we will also touch upon limit theorems describing the phase transitions in the Betti numbers of Gaussian process excursions.