I will present on-going work applying topological data analysis (TDA) and machine learning to identify transitions in cell organization and cell state within the context of developmental biology. First, using cell positions obtained from agent-based simulations of cell sorting and skin pigmentation, the complex relationship between cell-cell interactions and emergent patterns is automatically discovered via unsupervised classification of persistence images. This approach is used to analyze phase transitions in proliferating, heterogeneous populations and found to be empirically robust to random perturbations and finite-size effects. Next, I will discuss challenges associated with TDA of high-dimensional single cell sequencing datasets. In particular, lack of suitable techniques for intrinsic dimension and curvature estimation is limiting the use of multi-parameter filtration as a tool for understanding these data. I will briefly outline a novel approach for tackling this problem, using graph diffusion probabilities to predict curvature on toy data consisting of points sampled from quadric surfaces.