We present in this talk a theoretical framework for studying the persistent homology of point clouds from time-delay (or sliding window) embeddings. We will show that maximum 1-d persistence yields a suitable measure of periodicity at the signal level, and present theorems which relate the resulting diagrams to the choices of window size, embedding dimension and field of coefficients. If time permits, we will demonstrate how this methodology can be applied to the study of periodicity on time series from gene expression data.